Upcoming SlideShare
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Standard text messaging rates apply

# Sol Purcell Ingles

1,953
views

Published on

1 Like
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total Views
1,953
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
51
0
Likes
1
Embeds 0
No embeds

No notes for slide

### Transcript

• 1. Derivatives for Functions of CHAPTER 12 Two or More Variables 12.1 Concepts Review 4. a. 6 1. real-valued function of two real variables b. 12 2. level curve; contour map c. 2 3. concentric circles d. (3cos 6)1/ 2 + 1.44 ≈ 3.1372 4. parallel lines 5. F (t cos t , sec2 t ) = t 2 cos 2 t sec2 t = t 2 , cos t ≠ 0 Problem Set 12.1 6. F ( f (t ), g (t )) = F (ln t 2 , et / 2 ) 1. a. 5 = exp(ln t 2 ) + (et / 2 ) 2 = t 2 + et , t ≠ 0 b. 0 7. z = 6 is a plane. c. 6 d. a6 + a 2 e. 2 x2 , x ≠ 0 f. Undefined The natural domain is the set of all (x, y) such that y is nonnegative. 2. a. 4 8. x + z = 6 is a plane. b. 17 17 c. 16 d. 1 + a2 , a ≠ 0 e. x 3 + x, x ≠ 0 f. Undefined The natural domain is the set of all (x, y) such 9. x + 2y + z = 6 is a plane. that x is nonzero. 3. a. sin(2π ) = 0 ⎛π⎞ b. 4sin ⎜ ⎟ = 2 ⎝6⎠ ⎛π⎞ c. 16sin ⎜ ⎟ = 16 ⎝2⎠ d. π2 sin(π2 ) ≈ –4.2469 744 Section 12.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 2. 10. z = 6 – x 2 is a parabolic cylinder. 15. z = exp[–( x 2 + y 2 )] 11. x 2 + y 2 + z 2 = 16 , z ≥ 0 is a hemisphere. x2 16. z = ,y>0 y x2 y 2 z 2 12. + + = 1 , z ≥ 0 is a hemi-ellipsoid. 4 16 16 17. x 2 + y 2 = 2 z; x 2 + y 2 = 2k 13. z = 3 – x 2 – y 2 is a paraboloid. 18. x = zy, y ≠ 0 ; x = ky, y ≠ 0 14. z = 2 – x – y 2 19. x 2 = zy, y ≠ 0; x 2 = ky, y ≠ 0 Instructor’s Resource Manual Section 12.1 745 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 3. 20. x 2 = –( y – z ); x 2 = –( y – k ) 24. ( x – 2)2 + ( y + 3)2 = 16 V2 25. a. San Francisco and St. Louis had a x2 + 1 temperature between 70 and 80 degrees 21. z = , k = 1, 2, 4 x2 + y 2 Fahrenheit. k = 1: y 2 = 1 or y = ±1 ; b. Drive northwest to get to cooler two parallel lines temperatures, and drive southeast to get k = 2: 2 x 2 + 2 y 2 = x 2 + 1 warmer temperatures. x2 y 2 c. Since the level curve for 70 runs southwest + = 1 ; ellipse 1 1 to northeast, you could drive southwest or 2 northeast and stay at about the same k = 4: 4 x 2 + 4 y 2 = x 2 + 1 temperature. x2 y2 The lowest barometric pressure, 1000 + = 1; ellipse 26. a. 1 1 3 4 millibars and under, occurred in the region of the Great Lakes, specifically near Wisconsin. The highest barometric 22. y = sin x + z; y = sin x + k pressure, 1025 millibars and over, occurred on the east coast, from Massachusetts to South Carolina. b. Driving northwest would take you to lower barometric pressure, and driving southeast would take you to higher barometric pressure. c. Since near St. Louis the level curves run southwest to northeast, you could drive 23. x = 0, if T = 0: southwest or northeast and stay at about the ⎛1 ⎞ y 2 = ⎜ – 1⎟ x 2 , if y ≠ 0 . same barometric pressure. ⎝T ⎠ 27. x 2 + y 2 + z 2 ≥ 16 ; the set of all points on and outside the sphere of radius 4 that is centered at the origin 28. The set of all points inside (the part containing the z-axis) and on the hyperboloid of one sheet; x2 y 2 z 2 + – = 1. 9 9 9 x2 y 2 z 2 29. + + ≤ 1 ; points inside and on the 9 16 1 ellipsoid 746 Section 12.1 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 4. 30. Points inside (the part containing the z-axis) or on x2 y2 37. 4 x 2 – 9 y 2 = k , k in R; – = 1, if k ≠ 0; x2 y 2 z 2 k k the hyperboloid of one sheet, + – = 1, 4 9 9 9 16 2x excluding points on the coordinate planes planes y = ± (for k = 0) and all hyperbolic 3 31. Since the argument to the natural logarithm cylinders parallel to the z-axis such that the ratio function must be positive, we must have ⎛1⎞ ⎛1⎞ a:b is ⎜ ⎟ : ⎜ ⎟ or 3:2 (where a is associated x 2 + y 2 + z 2 > 0 . This is true for all ( x, y , z ) ⎝ 2⎠ ⎝3⎠ except ( x, y, z ) = ( 0, 0, 0 ) . The domain consists with the x-term) 3 all points in except the origin. 2 + y2 + z2 38. e x = k, k > 0 32. Since the argument to the natural logarithm x 2 + y 2 + z 2 = ln k function must be positive, we must have xy > 0 . concentric circles centered at the origin. This occurs when the ordered pair ( x, y ) is in the first quadrant or the third quadrant of the 39. a. All ( w, x, y, z ) except ( 0, 0, 0, 0 ) , which xy-plane. There is no restriction on z. Thus, the would cause division by 0. domain consists of all points ( x, y, z ) such that x and y are both positive or both negative. b. All ( x1 , x2 ,… , xn ) in n-space. 33. x 2 + y 2 + z 2 = k , k > 0; set of all spheres c. All ( x1 , x2 ,… , xn ) that satisfy centered at the origin 2 2 x1 + x2 + 2 + xn ≤ 1 ; other values of 34. 100 x 2 + 16 y 2 + 25 z 2 = k , k > 0; ( x1 , x2 ,… , xn ) would lead to the square root of a negative number. x2 y2 z2 + + = 1; set of all ellipsoids centered k 100 k 16 k 25 40. If z = 0, then x = 0 or x = ± 3 y . at origin such that their axes have ratio ⎛ 1 ⎞ ⎛1⎞ ⎛1⎞ ⎜ ⎟ : ⎜ ⎟ : ⎜ ⎟ or 2:5:4. ⎝ 10 ⎠ ⎝ 4 ⎠ ⎝ 5 ⎠ x2 y2 z2 35. + – = k ; the elliptic cone 1 1 1 16 4 2 2 x y z2 + = and all hyperboloids (one and two 9 9 16 sheets) with z-axis for axis such that a:b:c is 41. a. AC is the least steep path and BC is the most steep path between A and C since the level ⎛1⎞ ⎛1⎞ ⎛1⎞ curves are farthest apart along AC and ⎜ ⎟ : ⎜ ⎟ : ⎜ ⎟ or 3:3:4. ⎝ 4⎠ ⎝ 4⎠ ⎝ 3⎠ closest together along BC. x2 y2 z2 AC ≈ (5750) 2 + (3000)2 ≈ 6490 ft 36. – – = k ; the elliptical cone b. 1 1 1 9 4 BC ≈ (580) 2 + (3000) 2 ≈ 3060 ft 2 2 y z x2 + = and all hyperboloids (one and two 9 36 4 sheets) with x-axis for axis such that a:b:c is ⎛1⎞ ⎛1⎞ ⎜ ⎟ : ⎜ ⎟ :1 or 2:3:6 ⎝3⎠ ⎝ 2⎠ 42. Completing the squares on x and y yields the equivalent equation f ( x, y ) + 25.25 = ( x – 0.5) 2 + 3( y + 2)2 , an elliptic paraboloid. Instructor’s Resource Manual Section 12.1 747 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 5. 43. (2 x – y 2 ) exp(– x 2 – y 2 ) sin 2x 2 + y 2 46. 44. sin x sin y (1 + x 2 + y 2 ) sin( x 2 + y 2 ) x2 + y2 12.2 Concepts Review [( f ( x0 + h, y0 ) – f ( x0 , y0 )] 1. lim ; partial h →0 h derivative of f with respect to x 2. 5; 1 ∂2f 3. 45. ∂ y∂ x 4. 0 Problem Set 12.2 1. f x ( x, y ) = 8(2 x – y )3 ; f y ( x, y ) = –4(2 x – y )3 2. f x ( x, y ) = 6(4 x – y 2 )1/ 2 ; f y ( x, y ) = –3 y (4 x – y 2 )1/ 2 748 Section 12.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 6. ( xy )(2 x) – ( x 2 – y 2 )( y ) x2 + y2 f s ( s, t ) = –2set 2 – s2 ; f s ( s, t ) = 2tet 2 – s2 3. f x ( x, y ) = = 14. ( xy )2 x2 y ( xy )(−2 y ) − ( x 2 − y 2 )( x) 15. Fx ( x, y ) = 2 cos x cos y; Fy ( x, y ) = –2sin x sin y f y ( x, y ) = ( xy )2 16. f r (r , θ ) = 9r 2 cos 2θ ; fθ (r , θ ) = –6r 3 sin 2θ ( x2 + y2 ) =− xy 2 17. f x ( x, y ) = 4 xy 3 – 3x 2 y5 ; f xy ( x, y ) = 12 xy 2 – 15 x 2 y 4 4. f x ( x, y ) = e x cos y; f y ( x, y ) = – e x sin y f y ( x, y ) = 6 x 2 y 2 – 5 x 3 y 4 ; 5. f x ( x, y ) = e y cos x; f y ( x, y ) = e y sin x f yx ( x, y ) = 12 xy 2 – 15 x 2 y 4 ⎛ 1⎞ 6. f x ( x, y ) = ⎜ – ⎟ (3 x 2 + y 2 ) –4 / 3 (6 x) 18. f x ( x, y ) = 5( x3 + y 2 ) 4 (3 x 2 ); ⎝ 3⎠ f xy ( x, y ) = 60 x 2 ( x3 + y 2 )3 (2 y ) = –2 x(3 x 2 + y 2 ) –4 / 3 ; ⎛ 1⎞ = 120 x 2 y ( x3 + y 2 )3 f y ( x, y ) = ⎜ – ⎟ (3 x 2 + y 2 ) –4 / 3 (2 y ) ⎝ 3⎠ f y ( x, y ) = 5( x3 + y 2 ) 4 (2 y ); ⎛ 2y ⎞ 2 2 –4 / 3 = ⎜– ⎟ (3 x + y ) f yx ( x, y ) = 40 y ( x3 + y 2 )3 (3x 2 ) ⎝ 3 ⎠ = 120 x 2 y ( x3 + y 2 )3 2 2 −1/ 2 7. f x ( x, y ) = x ( x − y ) ; f y ( x, y ) = – y ( x 2 – y 2 ) –1/ 2 19. f x ( x, y ) = 6e2 x cos y; f xy ( x, y ) = –6e2 x sin y f y ( x, y ) = –3e2 x sin y; f yx ( x, y ) = –6e2 x sin y 8. fu (u , v) = ve ; f v (u , v) = ue uv uv 20. f x ( x, y ) = y (1 + x 2 y 2 ) –1; – xy – xy 9. g x ( x, y ) = – ye ; g y ( x, y ) = – xe f xy ( x, y ) = (1 − x 2 y 2 )(1 + x 2 y 2 ) −2 10. f s ( s, t ) = 2 s ( s 2 – t 2 ) –1 ; f x ( x, y ) = x(1 + x 2 y 2 ) –1; ft ( s, t ) = −2t ( s 2 − t 2 )−1 f xy ( x, y ) = (1 − x 2 y 2 )(1 + x 2 y 2 ) −2 11. f x ( x, y ) = 4[1 + (4 x – 7 y ) 2 ]–1; ( xy )(2) – (2 x – y )( y ) y2 1 21. Fx ( x, y ) = = = ; 2 2 2 f y ( x, y ) = –7[1 + (4 x – 7 y ) ] 2 –1 ( xy ) x y x2 1 Fx (3, − 2) = 1 ⎛1⎞ –1 ⎛ w ⎞ 9 12. Fw ( w, z ) = w ⎜ ⎟ + sin ⎜ ⎟ ( xy )(–1) – (2 x – y )( x) –2 x 2 2 1– ( ) w 2 z ⎝ z⎠ ⎝z⎠ Fy ( x, y ) = ( xy ) 2 = 2 2 x y =– x2 ; w ⎛ w⎞ 1 = z + sin –1 ⎜ ⎟ ; Fy (3, − 2) = − 2 ( w) ⎝z⎠ 2 1– z 22. Fx ( x, y ) = (2 x + y )( x 2 + xy + y 2 ) –1; ( w) 2 1 ⎛ w⎞ – Fz = ( w, z ) = w z 2 ⎜– 2 ⎟ = Fx (–1, 4) = ≈ 0.1538 ( ) w 2 ⎝ z ⎠ 1– ( w) 2 1– 13 z z Fy ( x, y ) = ( x + 2 y )( x 2 + xy + y 2 ) –1 ; 13. f x ( x, y ) = –2 xy sin( x 2 + y 2 ); Fy (–1, 4) = 7 ≈ 0.5385 2 2 2 2 2 13 f y ( x, y ) = –2 y sin( x + y ) + cos( x + y ) Instructor’s Resource Manual Section 12.2 749 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 7. 23. f x ( x, y ) = – y 2 ( x 2 + y 4 ) –1; 31. P (V , T ) = kT V fx ( ) 5, – 2 = – 4 21 ≈ –0.1905 k P (V , T ) = ; T V f y ( x, y ) = 2 xy ( x 2 + y 4 ) –1; k P (100, 300) = lb/in.2 per degree ( ) 4 5 T fy 5, – 2 = – ≈ –0.4259 100 21 32. V [ P (V , T )] + T [ P (V , T )] V T 24. f x ( x, y ) = e y sinh x; = V (– kTV –2 ) + T (kV –1 ) = 0 f x (–1, 1) = e sinh(–1) ≈ –3.1945 ⎛ kT ⎞ ⎛ k ⎞ ⎛ V ⎞ kT PV f y ( x, y ) = e y cosh x; P VT TP = ⎜ – V ⎟⎜ ⎟⎜ ⎟=– =– = –1 ⎝ V 2 ⎠⎝ P ⎠⎝ k ⎠ PV PV f y (–1, 1) = e cosh(–1) ≈ 4.1945 x2 y 2 33. f x ( x, y ) = 3 x 2 y – y 3 ; f xx ( x, y ) = 6 xy; 25. Let z = f ( x, y ) = + . 9 4 f y ( x, y ) = x3 – 3xy 2 ; f yy ( x, y ) = –6 xy f y ( x, y ) = y Therefore, f xx ( x, y ) + f yy ( x, y ) = 0. 2 The slope is f y (3, 2) = 1. 34. f x ( x, y ) = 2 x( x 2 + y 2 ) –1 ; f xx ( x, y ) = –2( x 2 – y 2 )( x 2 + y 2 ) –1 26. Let z = f ( x, y ) = (1/ 3)(36 – 9 x 2 – 4 y 2 )1/ 2 . ⎛ 4⎞ f y ( x, y ) = 2 y ( x 2 + y 2 ) –1 ; f y ( x, y ) = ⎜ − ⎟ y (36 − 9 x 2 − y 2 ) −1/ 2 ⎝ 3⎠ f yy ( x, y ) = 2( x 2 − y 2 )( x 2 + y 2 ) −1 8 The slope is f y (1, – 2) = ≈ 0.8040. 3 11 35. Fy ( x, y ) = 15 x 4 y 4 – 6 x 2 y 2 ; ⎛1⎞ Fyy ( x, y ) = 60 x 4 y 3 − 12 x 2 y; 27. z = f ( x, y ) = ⎜ ⎟ (9 x 2 + 9 y 2 − 36)1/ 2 ⎝2⎠ Fyyy ( x, y ) = 180 x 4 y 2 – 12 x 2 9x f x ( x, y ) = 2(9 x + 9 y 2 – 36)1/ 2 2 36. f x ( x, y ) = [– sin(2 x 2 – y 2 )](4 x) f x (2, 1) = 3 = –4 x sin(2 x 2 – y 2 ) ⎛5⎞ 28. z = f ( x, y ) = ⎜ ⎟ (16 – x 2 )1/ 2 . f xx ( x, y ) = (–4 x)[cos(2 x 2 – y 2 )](4 x) ⎝4⎠ ⎛ 5⎞ + [sin(2 x 2 – y 2 )](–4) f x ( x, y ) = ⎜ – ⎟ x(16 – x 2 ) –1/ 2 ⎝ 4⎠ f xxy ( x, y ) = –16 x 2 [– sin(2 x 2 – y 2 )](–2 y ) 5 f x (2, 3) = – ≈ –0.7217 – 4[cos(2 x 2 – y 2 )](–2 y ) 4 3 = –32 x 2 y sin(2 x 2 – y 2 ) + 8 y cos(2 x 2 – y 2 ) 29. Vr (r , h) = 2πrh; Vr (6, 10) = 120π ≈ 376.99 in.2 ∂3f 37. a. ∂ y3 2 30. Ty ( x, y ) = 3 y ; Ty (3, 2) = 12 degrees per ft ∂ 3y b. ∂ y∂ x 2 ∂ 4y c. ∂ y 3∂ x 750 Section 12.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 8. 38. a. f yxx 45. Domain: (Case x < y) The lengths of the sides are then x, y – x, and b. f yyxx 1 – y. The sum of the lengths of any two sides must be greater than the length of the remaining side, leading to three inequalities: c. f yyxxx 1 x + (y – x) > 1 – y ⇒ y > 2 39. a. f x ( x, y, z ) = 6 xy – yz 1 (y – x) + (1 – y) > x ⇒ x < 2 b. f y ( x, y, z ) = 3 x 2 – xz + 2 yz 2 ; 1 x + (1 – y) > y – x ⇒ y < x + f y (0, 1, 2) = 8 2 c. Using the result in a, f xy ( x, y, z ) = 6 x – z. 40. a. 12 x 2 ( x3 + y 2 + z )3 b. f y ( x , y , z ) = 8 y ( x 3 + y 2 + z )3 ; f y (0, 1, 1) = 64 c. f z ( x, y, z ) = 4( x3 + y 2 + z )3 ; The case for y < x yields similar inequalities 2 f zz ( x, y, z ) = 12( x + y + z )2 2 (x and y interchanged). The graph of DA , the domain of A is given above. In set notation it is f x ( x, y, x) = – yze – xyz – y ( xy – z 2 ) –1 ⎧ 1 1 1⎫ 41. D A = ⎨ ( x, y ) : x < , y > , y < x + ⎬ ⎩ 2 2 2⎭ ⎛ 1 ⎞⎛ xy ⎞ –1/ 2 ⎛ y⎞ ⎧ 1 1 1⎫ f x ( x, y, z ) = ⎜ ⎟⎜ ⎟ ∪ ⎨ ( x, y ) : x > , y < , x < y + ⎬ . 42. ⎜ ⎟; ⎩ 2 2 2⎭ ⎝ 2 ⎠⎝ z ⎠ ⎝z⎠ –1/ 2 Range: The area is greater than zero but can be ⎛ 1 ⎞⎛ 1 ⎞ ⎛ 1⎞ 1 arbitrarily close to zero since one side can be f x (–2, – 1, 8) = ⎜ ⎟ ⎜ ⎟ ⎜– ⎟ = – ⎝ 2 ⎠⎝ 4 ⎠ ⎝ 8⎠ 8 arbitrarily small and the other two sides are bounded above. It seems that the area would be 43. If f ( x, y ) = x 4 + xy3 + 12, f y ( x, y ) = 3 xy 2 ; largest when the triangle is equilateral. An 1 f y (1, – 2) = 12. Therefore, along the tangent line equilateral triangle with sides equal to has 3 Δy = 1 ⇒ Δz = 12, so 0, 1, 12 is a tangent 3 ⎛ 3⎤ area . Hence the range of A is ⎜ 0, ⎜ ⎥ . (In vector (since Δx = 0). Then parametric equations 36 ⎝ 36 ⎦ ⎧ x =1 ⎫ Sections 8 and 9 of this chapter methods will be ⎪ ⎪ of the tangent line are ⎨ y = –2 + t ⎬ . Then the presented which will make it easy to prove that ⎪ z = 5 + 12t ⎪ the largest value of A will occur when the triangle ⎩ ⎭ is equilateral.) point of xy-plane at which the bee hits is (1, 0, 29) [since y = 0 ⇒ t = 2 ⇒ x = 1, z = 29]. 46. a. u = cos (x) cos (ct): u x = – sin( x ) cos(ct ) ; ut = – c cos( x) sin(ct ) 44. The largest rectangle that can be contained in the circle is a square of diameter length 20. The edge u xx = – cos( x) cos(ct ) of such a square has length 10 2, so its area is utt = – c 2 cos( x) cos(ct ) 200. Therefore, the domain of A is Therefore, c 2 u xx = utt . 2 2 {( x, y ) : 0 ≤ x + y < 400}, and the range is u = e x cosh(ct ) : (0, 200]. u x = e x cosh(ct ), ut = ce x sinh(ct ) u xx = e x cosh(ct ), utt = c 2 e x cosh(ct ) Therefore, c 2u xx = utt . Instructor’s Resource Manual Section 12.2 751 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 9. b. u = e – ct sin( x) : 48. a. sin( x + y 2 ) u x = e – ct cos x u xx = – e – ct sin x ut = – cect sin x Therefore, cu xx = ut . 2 u = t –1/ 2 e – x / 4ct : 2 / 4ct ⎛ x ⎞ u x = t –1/ 2 e – x ⎜– ⎟ b. Dx (sin( x + y 2 )) ⎝ 2ct ⎠ ( x 2 – 2ct ) u xx = 2 / 4ct (4c 2 t 5 / 2 e x ) 2 ( x – 2ct ) ut = 2 (4ct 5 / 2 e x / 4ct ) Therefore, cu xx = ut . 47. a. Moving parallel to the y-axis from the point c. D y (sin( x + y 2 )) (1, 1) to the nearest level curve and Δz approximating , we obtain Δy 4–5 f y (1, 1) = = –4. 1.25 – 1 b. Moving parallel to the x-axis from the point (–4, 2) to the nearest level curve and Δz d. Dx ( D y (sin( x + y ) 2 )) approximating , we obtain Δx 1– 0 2 f x (–4, 2) ≈ = . –2.5 – (–4) 3 c. Moving parallel to the x-axis from the point (–5, –2) to the nearest level curve and Δz approximately , we obtain Δx 1– 0 2 49. a. f y ( x, y , z ) f x (–4, – 5) ≈ = . –2.5 – (–5) 5 f ( x , y + Δy , z ) − f ( x , y , z ) = lim Δy →0 Δy d. Moving parallel to the y-axis from the point (0, –2) to the nearest level curve and b. f z ( x, y , z ) Δz approximating , we obtain f ( x , y , z + Δz ) − f ( x , y , z ) Δy = lim Δz →0 Δz 0 –1 8 f y (0, 2) ≈ = . –19 – (–2) 3 8 c. Gx ( w, x, y, z ) G ( w, x + Δx, y , z ) − G ( w, x, y, z ) = lim Δx →0 Δx 752 Section 12.2 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 10. ∂ 4. The limit does not exist because of Theorem A. d. λ ( x, y , z , t ) The function is a rational function, but the limit ∂z of the denominator is 0, while the limit of the λ ( x , y , z + Δz , t ) − λ ( x , y , z , t ) = lim numerator is -1. Δz →0 Δz 5 ∂ 5. – e. S (b0 , b1 , b2 ,… , bn ) = 2 ∂b2 ⎛ S (b0 , b1 , b2 + Δb2 ,… , bn ) ⎞ 6. −1 ⎜ ⎟ − S (b0 , b1 , b2 ,… , bn ) ⎟ 7. 1 = lim ⎜ Δb2 →0 ⎜ Δb2 ⎟ ⎜ ⎟ tan( x 2 + y 2 ) ⎝ ⎠ 8. lim ( x, y )→(0, 0) ( x2 + y2 ) ∂ sin( x 2 + y 2 ) 50. a. ( sin w sin x cos y cos z ) = lim 1 ∂w ( x, y )→(0, 0) ( x + y ) cos( x + y 2 ) 2 2 2 = cos w sin x cos y cos z = (1)(1) = 1 ∂ wyz 9. The limit does not exist since the function is not b. ⎡ x ln ( wxyz ) ⎤ = x ⋅ ⎣ ⎦ + 1 ⋅ ln ( wxyz ) ∂x wxyz defined anywhere along the line y = x. That is, there is no neighborhood of the origin in which = 1 + ln ( wxyz ) the function is defined everywhere except possibly at the origin. c. λt ( x, y, z , t ) (1 + xyzt ) cos x − t ( cos x ) xyz ( x 2 + y 2 )( x 2 – y 2 ) = 10. lim ( x, y )→(0, 0) x2 + y 2 (1 + xyzt )2 cos x = lim ( x2 – y 2 ) = 0 = ( x, y )→(0, 0) (1 + xyzt )2 11. Changing to polar coordinates, xy r cos θ ⋅ r sin θ lim = lim 12.3 Concepts Review ( x, y )→(0,0) x 2 + y 2 r →0 r 1. 3; (x, y) approaches (1, 2). = lim r cos θ ⋅ sin θ = 0 r →0 2. lim f ( x, y ) = f (1, 2) ( x, y )→(1, 2) 12. If ( x, y ) approaches (0, 0) along the line y = x , 3. contained in S x2 1 lim = lim = +∞ 2 2 2 ( x, x )→(0,0) 4 x 2 +x ) ( x, x )→(0,0) ( x 4. an interior point of S; boundary points Thus, the limit does not exist. 13. Use polar coordinates. Problem Set 12.3 r 7 / 3 ( cos θ ) 7/3 x7 / 3 = r1/ 3 ( cos θ ) 7/3 = 1. –18 x +y2 2 r 2 2. 3 r 1/ 3 ( cos θ ) 7/3 → 0 as r → 0 , so the limit is 0. ⎡ 2 ⎛ xy ⎞ ⎤ 14. Changing to polar coordinates, 3. lim ⎢ x cos xy – sin ⎜ 3 ⎟ ⎥ ( x, y )→(2, π) ⎣ ⎝ ⎠⎦ r 2 cos 2 θ − r 2 sin 2 θ lim r 2 cos θ sin θ ⋅ ⎛ 2π ⎞ 3 r →0 r2 = 2 cos 2 2π – sin ⎜ ⎟ = 2 – ≈ 1.1340 ⎝ 3 ⎠ 2 = lim r 2 cos θ sin θ cos 2θ = 0 r →0 Instructor’s Resource Manual Section 12.3 753 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 11. r 4 cos 2 θ sin 2 θ 26. Require 4 − x 2 − y 2 − z 2 > 0; 15. f ( x, y ) = r 2 cos 2 θ + r 4 sin 4 θ x 2 + y 2 + z 2 < 4. S is the space in the interior ⎛ cos 2 θ sin 2 θ ⎞ of the sphere centered at the origin with radius 2. = r2 ⎜ ⎟ ⎜ cos 2 θ + r 2 sin 4 θ ⎟ ⎝ ⎠ 27. The boundary consists of the points that form the If cos θ = 0 , then f ( x, y ) = 0 . If cos θ ≠ 0 , outer edge of the rectangle. The set is closed. hen this converges to 0 as r → 0 . Thus the limit is 0. 16. As ( x, y ) approaches (0,0) along x = y2, y4 1 lim = . Along the x-axis, 4 24 y +y ( x, x )→(0,0) 0 however, lim = 0. Thus, the limit does ( x,0)→(0,0) x 2 not exist. 17. f ( x, y ) is continuous for all ( x, y ) since 28. The boundary consists of the points of the circle for all ( x, y ), x 2 + y 2 + 1 ≠ 0. shown. The set is open. 18. f ( x, y ) is continuous for all ( x, y ) since for all ( x, y ), x 2 + y 2 + 1 > 0. 19. Require 1 – x 2 – y 2 > 0; x 2 + y 2 < 1. S is the interior of the unit circle centered at the origin. 20. Require 1 + x + y > 0; y > − x − 1. S is the set of all ( x, y ) above the line y = − x − 1. 29. The boundary consists of the circle and the 2 21. Require y – x ≠ 0. S is the entire plane except origin. The set is neither open (since, for example, (1, 0) is not an interior point), nor the parabola y = x 2 . closed (since (0, 0) is not in the set). 22. The only points at which f might be discontinuous occur when xy = 0. sin( xy ) lim = 1 = f (a, 0) for all nonzero ( x, y )→( a , 0) xy a in , and then sin( xy ) lim = 1 = f (0, b) for all b in . ( x, y )→(0, b ) xy Therefore, f is continuous on the entire plane. 23. Require x – y + 1 ≥ 0; y ≤ x + 1. S is the region below and on the line y = x + 1. 24. Require 4 – x 2 – y 2 > 0; x 2 + y 2 < 4. S is the interior of the circle of radius 2 centered at the origin. 25. f ( x, y, z ) is continuous for all ( x, y, z ) ≠ ( 0, 0, 0 ) since for all ( x, y, z ) ≠ ( 0, 0, 0 ) , x 2 + y 2 + z 2 > 0. 754 Section 12.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 12. 30. The boundary consists of the points on the line 0 x = 1 along with the points on the line x = 4. The 35. Along the x-axis (y = 0): lim =0. ( x, y )→(0, 0) x 2 +0 set is neither closed nor open. Along y = x: x2 1 1 lim = = . lim ( x, y )→(0, 0) 2 x 2 ( x, y )→(0, 0) 2 2 Hence, the limit does not exist because for some points near the origin f(x, y) is getting closer to 0, 1 but for others it is getting closer to . 2 0 36. Along y = 0: lim = 0. Along y = x: x→ 0 x 2 +0 31. The boundary consists of the graph of ⎛1⎞ x 2 + x3 1+ x 1 y = sin ⎜ ⎟ along with the part of the y-axis for lim = lim = . x→ 0 x 2 +x 2 x→ 0 2 2 ⎝ x⎠ which y ≤ 1. The set is open. x 2 (mx) mx3 37. a. lim = lim x→ 0 x 4 + (mx)2 x→ 0 x 4 + m2 x 2 mx = lim =0 x→ 0 x 2 + m2 x2 ( x2 ) x4 1 1 b. lim = lim = lim = x→ 0 x 4 + (x )2 2 x→ 0 2 x 4 x→ 0 2 2 32. The boundary is the set itself along with the x2 y origin. The set is neither open (since none of its c. lim does not exist. ( x, y )→( 0, 0) x 4 +y points are interior points) nor closed (since the origin is not in the set). 38. f is discontinuous at each overhang. More interesting, f is discontinuous along the Continental Divide. 39. a. {( x, y, z ) : x 2 + y 2 = 1, z in [1, 2]} [For x 2 + y 2 < 1, the particle hits the hemisphere and then slides to the origin (or bounds toward the origin); for x 2 + y 2 = 1, it 2 2 bounces up; for x 2 + y 2 > 1, it falls straight x – 4y ( x + 2 y )( x – 2 y ) 33. = = x + 2 y (if x ≠ 2y) down.] x – 2y x – 2y If x = 2 y , x + 2 y = 2 x . Take g ( x ) = 2 x . b. {( x, y, z ) : x 2 + y 2 = 1, z = 1} (As one moves at a level of z = 1 from the rim of the bowl 34. Let L and M be the latter two limits. toward any position away from the bowl [ f ( x, y ) + g ( x, y )] – [ L + M ] there is a change from seeing all of the interior of the bowl to seeing none of it.) ε ε ≤ f ( x, y ) – L + f ( x , y ) – M ≤ + 2 2 c. {(x, y, z): z = 1} [f(x, y, z) is undefined for (x, y) in some δ-neighborhood of (a, b). (infinite) at (x, y, 1).] Therefore, lim [ f ( x, y ) + g ( x, y )] = L + M . d. φ (Small changes in points of the domain ( x , y ) →( a , b ) result in small changes in the shortest path from the points to the origin.) Instructor’s Resource Manual Section 12.3 755 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 13. 40. f is continuous on an open set D and P0 is in D implies that there is neighborhood of P0 with radius r on which f is continuous. f is continuous at P0 ⇒ lim f ( P) = f ( P0 ). Now let ε = f ( P0 ) which is positive. Then there is a δ P → P0 such that 0 < δ < r and f ( p ) – f ( P0 ) < f ( P0 ) if P is in the δ-neighborhood of P0 . Therefore, – f ( P0 ) < f ( p ) – f ( P0 ) < f ( P0 ), so 0 < f(p) (using the left-hand inequality) in that δ-neighborhood of P0 . ⎧( x 2 + y 2 )1/ 2 + 1 if y ≠ 0 ⎫ ⎪ ⎪ 41. a. f ( x, y ) = ⎨ ⎬ . Check discontinuities where y = 0. ⎪ ⎩ x –1 if y = 0 ⎪ ⎭ As y = 0, ( x 2 + y 2 )1/ 2 + 1 = x + 1, so f is continuous if x + 1 = x – 1 . Squaring each side and simplifying yields x = – x, so f is continuous for x ≤ 0. That is, f is discontinuous along the positive x-axis. b. Let P = (u, v) and Q = (x, y). ⎧ ⎪ OP + OQ if P and Q are not on same ray from the origin and neither is the origin ⎫ ⎪ f (u, v, x, y ) = ⎨ ⎬. ⎪ PQ ⎩ otherwise ⎪ ⎭ This means that in the first case one travels from P to the origin and then to Q; in the second case one travels directly from P to Q without passing through the origin, so f is discontinuous on the set {(u, v, x, y ) : u, v = k x, y for some k > 0, u , v ≠ 0, x, y ≠ 0}. ⎛ hy ( h2 – y 2 ) ⎞ ⎜ 2 2 –0⎟ 2 2 42. a. f x (0, y ) = lim ⎜ h +y ⎟ = lim y (h − y ) = − y h →0 ⎜ h ⎟ h →0 h 2 + y 2 ⎜ ⎟ ⎝ ⎠ ⎛ xh ( x 2 – h 2 ) ⎞ ⎜ –0⎟ x 2 + h2 y( x2 – h2 ) b. f y ( x, 0) = lim ⎜ ⎟ = lim =x h →0 ⎜ h ⎟ h →0 x 2 + y 2 ⎜ ⎟ ⎝ ⎠ f y (0 + h, y ) – f y (0, y ) h–0 c. f yx (0, 0) = lim = lim =1 h →0 h h →0 h f x ( x, 0 + h) – f x ( x, 0) –h – 0 d. f xy (0, 0) = lim = lim = –1 h →0 h h →0 h Therefore, f xy (0, 0) ≠ f yx (0, 0). 43. b. 44. a. 45. 756 Section 12.3 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 14. 46. A function f of three variables is continuous at a point (a, b, c ) if f (a, b, c) is defined and equal to the limit of f ( x, y, z ) as ( x, y, z ) approaches (a, b, c) . In other words, lim f ( x, y, z ) = f (a, b, c) . ( x , y , z ) → ( a ,b , c ) A function of three variables is continuous on an open set S if it is continuous at every point in the interior of the set. The function is continuous at a boundary point P of S if f (Q) approaches f (P) as Q approaches P along any path through points in S in the neighborhood of P. 47. If we approach the point ( 0, 0, 0 ) along a straight path from the point ( x, x, x ) , we have x ( x )( x) x3 1 lim = lim = 3 3 3 3 3 x +x +x ( x, x, x )→(0,0,0) ( x , x, x )→(0,0,0) 3 x Since the limit does not equal to f (0, 0, 0) , the function is not continuous at the point (0, 0, 0) . 48. If we approach the point (0, 0, 0) along the x-axis, we get ( x 2 − 02 ) x2 lim (0 + 1) = lim =1 ( x,0,0)→(0,0,0) ( x 2 + 02 ) ( x,0,0)→(0,0,0) x 2 Since the limit does not equal f (0, 0, 0) , the function is not continuous at the point (0, 0, 0). 12.4 Concepts Review 1. gradient 2. locally linear ∂f ∂f 3. (p)i + (p) j; y 2 i + 2 xyj ∂x ∂y 4. tangent plane Problem Set 12.4 1. 2 xy + 3 y, x 2 + 3 x 2. 3 x 2 y , x3 – 3 y 2 3. ∇f ( x, y ) = ( x)(e xy y ) + (e xy )(1), xe xy x = e xy xy + 1, x 2 4. 2 xy cos y, x 2 (cos y – y sin y ) 5. x( x + y ) –2 y ( x + 2), x 2 6. ∇f ( x, y ) = 3[sin 2 ( x 2 y )][cos( x 2 y )](2 xy ), 3[sin 2 ( x 2 y )][cos( x 2 y )]( x 2 ) = 3x sin 2 ( x 2 y ) cos( x 2 y ) 2 y, x Instructor’s Resource Manual Section 12.4 757 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 15. 7. ( x 2 + y 2 + z 2 ) –1/ 2 x, y, z 8. 2 xy + z 2 , x 2 + 2 yz , y 2 + 2 xz 9. ∇f ( x, y ) = ( x 2 y )(e x – z ) + (e x – z )(2 xy ), x 2 e x – z , x 2 ye x – z (–1) = xe x – z y ( x + 2), x, – xy 10. xz ( x + y + z ) –1 + z ln( x + y + z ), xz ( x + y + z ) –1 , xz ( x + y + z ) –1 + x ln( x + y + z ) 11. ∇f ( x, y ) = 2 xy – y 2 , x 2 – 2 xy ; ∇f (–2, 3) = –21, 16 z = f (–2, 3) + –21, 16 ⋅ x + 2, y – 3 = 30 + (–21x – 42 + 16 y – 48) z = –21x + 16y – 60 12. ∇f ( x, y ) = 3x 2 y + 3 y 2 , x3 + 6 xy , so ∇f (2, – 2) = (–12, – 16). Tangent plane: z = f (2, – 2) + ∇ (2, – 2) ⋅ x – 2, y + 2 = 8 + –12, – 16 ⋅ x – 2, y + 2 = 8 + (–12x + 24 – 16y – 32) z = –12x – 16y 13. ∇f ( x, y ) = – π sin(πx) sin(πy ), π cos(πx) cos(πy ) + 2π cos(2πy ) ⎛ 1⎞ ∇f ⎜ –1, ⎟ = 0, – 2π ⎝ 2⎠ ⎛ 1⎞ 1 z = f ⎜ –1, ⎟ + 0, – 2π ⋅ x + 1, y – = –1 + (0 – 2πy + π); ⎝ 2⎠ 2 z = –2 π y + ( π – 1) 2x x2 14. ∇f ( x, y ) = ,− ; ∇f (2, − 1) = −4, − 4 y y2 z = f (2, – 1) + −4, – 4 ⋅ x – 2, y + 1 = –4 + (–4x + 8 –4y – 4) z = –4x – 4y 15. ∇f ( x, y, z ) = 6 x + z 2 , – 4 y, 2 xz , so ∇f (1, 2, – 1) = 7, – 8, – 2 Tangent hyperplane: w = f (1, 2, – 1) + ∇f (1, 2, – 1) ⋅ x – 1, y – 2, z + 1 = –4 + 7, – 8, – 2 ⋅ x – 1, y – 2, z + 1 = –4 + (7x – 7 – 8y + 16 – 2z – 2) w = 7x – 8y – 2z + 3 16. ∇f ( x, y, z ) = yz + 2 x, xz , xy ; ∇f (2, 0, – 3) = 4, – 6, 0 w = f (2, 0, – 3) + 4, – 6, 0 ⋅ x – 2, y, z + 3 = 4 + (4x – 8 – 6y + 0) w = 4x – 6y – 4 ⎛ f ⎞ gf x − fg x , gf y − fg y , gf z − fg z g fx , f y , fz – f gx , g y , gz g ∇f – f ∇g 17. ∇ ⎜ ⎟ = = = ⎝g⎠ g2 g2 g2 18. ∇( f r ) = rf r –1 f x , rf r –1 f y , rf r –1 f z = rf r –1 f x , f y , f z = rf r –1∇f 758 Section 12.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 16. 19. Let F ( x, y, z ) = x 2 − 6 x + 2 y 2 − 10 y + 2 xy − z = 0 respectively. Again we find the cross product of the normal vectors: ∇F ( x, y, z ) = 2 x − 6 + 2 y, 4 y − 10 + 2 x, −1 12,10, −1 × 1, 0, 0 = 0,10,10 The tangent plane will be horizontal if Thus, parametric equations for the desired ∇F ( x, y, z ) = 0, 0, k , where k ≠ 0 . Therefore, tangent line are x = 2 we have the following system of equations: y = 1 + 10t 2x + 2 y − 6 = 0 z = 9 + 10t 2 x + 4 y − 10 = 0 c. Using the equation for the tangent plane from Solving this system yields x = 1 and y = 2 . the first part, we now want the vertical plane Thus, there is a horizontal tangent plane at to be parallel to the line y = x , but still pass ( x, y ) = (1, 2 ) . through the projected point (2,1, 0) . The vertical plane now has equation y − x + 1 = 0 . 20. Let F ( x, y, z ) = x3 − z = 0 The normal equations are given by ∇F ( x, y , z ) = 3 x 2 , 0, −1 12,10, −1 and 1, −1, 0 for the tangent and The tangent plane will be horizontal if vertical planes respectively. Again we find the cross product of the normal vectors: ∇F ( x, y, z ) = 0, 0, k , where k ≠ 0 . Therefore, 12,10, −1 × 1, −1, 0 = −1, −1, −22 we need only solve the equation 3x 2 = 0 . There Thus, parametric equations for the desired is a horizontal tangent plane at ( x, y ) = (0, y ). tangent line are x = 2 − t (Note: there are infinitely many points since y y = 1− t can take on any value). z = 9 − 22t 21. a. The point (2,1,9) projects to (2,1, 0) on the xy plane. The equation of a plane containing this 22. a. The point (3, 2, 72) on the surface is the point point and parallel to the x-axis is given by (3, 2, 0) when projected into the xy plane. The y = 1 . The tangent plane to the surface at the equation of a plane containing this point and point (2,1,9) is given by parallel to the x-axis is given by y = 2 . The z = f (2,1) + ∇f (2,1) ⋅ x − 2, y − 1 tangent plane to the surface at the point (3, 2, 72) is given by = 9 + 12,10 x − 2, y − 1 z = f (3, 2) + ∇f (3, 2) ⋅ x − 3, y − 2 = 12 x + 10 y − 25 = 72 + 48,108 x − 3, y − 2 The line of intersection of the two planes is the tangent line to the surface, passing = 48 x + 108 y − 288 through the point (2,1,9) , whose projection in The line of intersection of the two planes is the xy plane is parallel to the x-axis. This line the tangent line to the surface, passing of intersection is parallel to the cross product through the point (3, 2, 72) , whose projection of the normal vectors for the planes. The in the xy plane is parallel to the x-axis. This normal vectors are 12,10, −1 and 0,1, 0 for line of intersection is parallel to the cross the tangent plane and vertical plane product of the normal vectors for the planes. respectively. The cross product is given by The normal vectors are 12,10, −1 × 0,1, 0 = 1, 0,12 48,108, −1 and 0, 2, 0 for the tangent plane Thus, parametric equations for the desired and vertical plane respectively. The cross tangent line are x = 2 + t product is given by 48,108, −1 × 0, 2, 0 = 2, 0,96 y =1 Thus, parametric equations for the desired z = 9 + 12t tangent line are b. Using the equation for the tangent plane from x = 3 + 2t the previous part, we now want the vertical y=2 plane to be parallel to the y-axis, but still pass z = 72 + 96t through the projected point (2,1, 0) . The vertical plane now has equation x = 2 . The normal equations are given by 12,10, −1 and 1, 0, 0 for the tangent and vertical planes Instructor's Resource Manual Section 12.4 759 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 17. b. Using the equation for the tangent plane from c. Using the equation for the tangent plane from the previous part, we now want the vertical the first part, we now want the vertical plane plane to be parallel to the y-axis, but still pass to be parallel to the line x = − y , but still pass through the projected point (3, 2, 72) . The through the projected point (3, 2, 72) . The vertical plane now has equation x = 3 . The vertical plane now has equation y + x − 5 = 0 . normal equations are given by The normal equations are given by 48,108, −1 and 3, 0, 0 for the tangent and 48,108, −1 and 1,1, 0 for the tangent and vertical planes respectively. Again we find vertical planes respectively. Again we find the cross product of the normal vectors: the cross product of the normal vectors: 48,108, −1 × 3, 0, 0 = 0, −3, −324 48,108, −1 × 1,1, 0 = 1, −1, −60 Thus, parametric equations for the desired Thus, parametric equations for the desired tangent line are tangent line are x=3 x = 3+t y = 2 − 3t y = 2−t z = 72 − 324t z = 72 − 60t ⎛ 1 xy ⎞ ⎛ 1 xy ⎞ –5 xy ⎡ a a ⎤ 23. ∇f ( x, y ) = –10 ⎜ y ⎟ , – 10 ⎜ x⎟ = y, x ⎢ Note that = .⎥ ⎜ 2 xy xy ⎟ ⎜ 2 xy xy ⎟ xy 3/ 2 ⎢ ⎣ a a ⎥ ⎦ ⎝ ⎠ ⎝ ⎠ ∇f (1, – 1) = –5, 5 Tangent plane: z = f (1, – 1) + ∇f (1, – 1) ⋅ x – 1, y + 1 = –10 + –5, 5 ⋅ x – 1, y + 1 = –10 + (–5 x + 5 + 5 y + 5) z = –5x + 5y 24. Let a be any point of S and let b be any other 26. f(b) – f(a) = f 2, 6 − f 0, 0 = 0 − 2 = −2 point of S. Then for some c on the line segment between a and b: −x ∇f ( x , y ) = , 0 ; b – a = 2, 6 f (b) − f (a) = ∇f (c) ⋅ (b − a) = 0 ⋅ (b − a) = 0, so 4 − x2 f(b) = f(a) (for all b in S). The value c = cx , c y will be the solution to 25. f(a) – f(b) = f 2,1 − f 0, 0 = 4 − 9 = −5 −c x ∇f ( x, y ) = −2 x, −2 y ; b – a = 2,1 −2 = ,0 2, 6 4 − cx 2 The value c = cx , c y will be a solution to −2cx −5 = −2cx , −2c y 2,1 −2 = ⇒ cx = 2 4 − cx 2 { c ∈ cx , c y : 4cx + 2c y = 5 } Since c must be between a and b, c must lie on In order for c to be between a and b, c must lie the line y = 3x. Since cx = 2, c y = 3 2. on the line y = 1 x . Consequently, c will be the 2 Thus, c = 2,3 2 . solution to the following system of equations: 4cx + 2c y = 5 and c y = 1 c x . The solution is 2 27. ∇f (p) = ∇g (p) ⇒ ∇[ f (p) – g (p)] = 0 ⇒ f (p) – g (p) is a constant. c = 1, 1 . 2 760 Section 12.4 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 18. 28. ∇f (p) = p ⇒ ∇f ( x, y ) = x, y 31. a. (i) ⇒ f x ( x , y ) = x, f y ( x, y ) = y ∂( f + g ) ∂( f + g ) ∂( f + g ) ∇[ f + g ] = i+ j+ k ∂x ∂y ∂z 1 2 ⇒ f ( x, y ) = x + α ( y ) for any function of y, ∂f ∂g ∂f ∂g ∂f ∂g 2 = i+ i+ j+ j+ k + k ∂x ∂x ∂y ∂y ∂z ∂z 1 and f ( x, y ) = y 2 + β ( x) for any function of x. ∂f ∂f ∂f ∂g ∂g ∂g 2 = i+ j+ k + i+ j+ k 1 ∂x ∂y ∂z ∂x ∂y ∂z ⇒ f ( x, y ) = ( x 2 + y 2 ) + C for any C in . = ∇f + ∇g 2 29. ∂[α f ] ∂[α f ] ∂[α f ] (ii) ∇[α f ] = i+ j+ k ∂x ∂y ∂z ∂[ f ] ∂[ f ] ∂[ f ] =α i +α j+α k ∂x ∂y ∂z = α∇f (iii) − xy ∂ ( fg ) ∂ ( fg ) ∂ ( fg ) ∇[ fg ] = i+ j+ k ∂x ∂y ∂z ⎛ ∂g ∂f ⎞ ⎛ ∂g ∂f ⎞ =⎜ f + g ⎟i + ⎜ f + g ⎟j ⎝ ∂x ∂x ⎠ ⎝ ∂y ∂y ⎠ ⎛ ∂g ∂f ⎞ +⎜ f + g ⎟k ⎝ ∂z ∂z ⎠ ⎛ ∂g ∂g ∂g ⎞ = f ⎜ i+ j+ k ⎟ ⎝ ∂x ∂y ∂z ⎠ ⎛ ∂f ∂f ∂f ⎞ +g ⎜ i + j+ k ⎟ a. The gradient points in the direction of ⎝ ∂x ∂y ∂z ⎠ greatest increase of the function. = f ∇g + g ∇f b. No. If it were, 0 + h – 0 = 0 + h δ (h) where b. (i) δ (h) → 0 as h → 0, which is possible. ∂( f + g ) ∂( f + g ) ∇[ f + g ] = i1 + i2 ∂x1 ∂x2 30. ∂( f + g ) + + in ∂xn ∂f ∂g ∂f ∂g = i1 + i1 + i2 + i2 ∂x1 ∂x1 ∂x2 ∂x2 ∂f ∂g + + in + in ∂xn ∂xn sin(x) + sin(y) – sin(x + y) ∂f ∂f ∂f = i1 + i2 + + in ∂x1 ∂x2 ∂xn ∂g ∂g ∂g + i1 + i2 + + in ∂x1 ∂x2 ∂xn = ∇f + ∇ g Instructor's Resource Manual Section 12.4 761 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 19. (ii) 4. Du f ( x, y ) ∂[α f ] ∂[α f ] ∂[α f ] ⎡⎛ 1 ⎞ ⎤ ∇[α f ] = i1 + i2 + + in ∂x1 ∂x2 ∂xn = 2 x – 3 y , – 3 x + 4 y ⋅ ⎢⎜ ⎟ 2, – 1 ⎥ ; ⎣⎝ 5 ⎠ ⎦ ∂[ f ] ∂[ f ] ∂[ f ] =α i1 + α i2 + +α in 27 ∂x1 ∂x2 ∂xn Du f (–1, 2) = – ≈ –12.0748 5 = α∇f ⎡⎛ 1 ⎞ ⎤ (iii) 5. Du f ( x, y ) = e x sin y, cos y ⋅ ⎢⎜ ⎟ 1, 3 ⎥ ; ⎣ ⎝2⎠ ⎦ ∂ ( fg ) ∂ ( fg ) ∂ ( fg ) ∇[ fg ] = ∂x1 i1 + ∂x2 i2 + + ∂xn in ⎛ π⎞ Du f ⎜ 0, ⎟ = ( 2+ 6 ) ≈ 0.9659 ⎛ ∂g ∂f ⎞ ⎛ ∂g ∂f ⎞ ⎝ 4⎠ 4 =⎜ f +g ⎟ i1 + ⎜ f +g ⎟ i2 ⎝ ∂x1 ∂x1 ⎠ ⎝ ∂x2 ∂x2 ⎠ –1, 3 ⎛ ∂g ∂f ⎞ 6. Du f ( x, y ) = – ye – xy – xe – xy ⋅ + +⎜ f +g ⎟ in 2 ⎝ ∂xn ∂xn ⎠ –1, 3 ⎛ ∂g ∂g ∂g ⎞ Du f (1, –1) = e, – e ⋅ = –e – e 3 = f⎜ i1 + i2 + + in ⎟ ⎝ ∂x1 ∂x2 ∂xn ⎠ 2 2 ≈ –3.7132 ⎛ ∂f ∂f ∂f ⎞ +g ⎜ i1 + i2 + + in ⎟ ⎝ ∂x1 ∂x2 ∂xn ⎠ 7. Du f ( x, y, z ) = = f ∇ g + g ∇f ⎡⎛ 1 ⎞ ⎤ = 3 x 2 y, x3 – 2 yz 2 , – 2 y 2 z ⋅ ⎢⎜ ⎟ 1, – 2, 2 ⎥ ; ⎣⎝ 3 ⎠ ⎦ 52 Du f (–2, 1, 3) = 12.5 Concepts Review 3 [ f (p + hu) – f (p)] ⎡⎛ 1 ⎞ ⎤ 1. 8. Du f ( x, y, z ) = 2 x, 2 y, 2 z ⋅ ⎢⎜ ⎟ 2, –1, –1 ⎥ ; h ⎣⎝ 2 ⎠ ⎦ Du f (1, – 1, 2) = 2 – 1 ≈ 0.4142 2. u1 f x ( x, y ) + u2 f y ( x, y ) 9. f increases most rapidly in the direction of the 3. greatest increase gradient. ∇f ( x, y ) = 3 x 2 , – 5 y 4 ; 4. level curve ∇f (2, – 1) = 12, – 5 12, – 5 Problem Set 12.5 is the unit vector in that direction. The 13 rate of change of f(x, y) in that direction at that 3 4 8 1. Du f ( x, y ) = 2 xy, x 2 ⋅ , − ; Du f (1, 2) = point is the magnitude of the gradient. 5 5 5 12, – 5 = 13 ⎡⎛ 1 ⎞ ⎤ 2. Du f ( x, y ) = x –1 y 2 , 2 y ln x ⋅ ⎢⎜ ⎟ 1, – 1 ⎥ ; 10. ∇f ( x, y ) = e y cos x, e y sin x ; ⎣⎝ 2 ⎠ ⎦ Du f (1, 4) = 8 2 ≈ 11.3137 ⎛ 5π ⎞ 3 1 ∇f ⎜ , 0 ⎟ = – , , which is a unit vector. ⎝ 6 ⎠ 2 2 ⎛ a⎞ The rate of change in that direction is 1. 3. Du f ( x, y ) = f ( x, y ) ⋅ u ⎜ where u = ⎟ ⎜ a⎟ ⎝ ⎠ 1, – 1 = 4 x + y, x – 2 y ⋅ ; 2 1, – 1 3 Du f (3, – 2) = 10, 7 ⋅ = ≈ 2.1213 2 2 762 Section 12.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 20. 11. ∇f ( x, y, z ) = 2 xyz , x 2 z , x 2 y ; 16. At (2, 1), x 2 + 4 y 2 = 8 is the level curve. ∇f ( x , y ) = 2 x , 8 y f (1, – 1, 2) = –4, 2, – 1 A unit vector in that direction is ∇f (2, 1) = 4 1, 2 , which is perpendicular to the ⎛ 1 ⎞ level curve at (2, 1). ⎜ ⎟ −4, 2, − 1 . The rate of change in that ⎝ 21 ⎠ direction is 21 ≈ 4.5826. 12. f increases most rapidly in the direction of the gradient. ∇f ( x, y, z ) = e yz , xze yz , xye yz ; ∇f (2, 0, – 4) = 1, – 8, 0 1, – 8, 0 is a unit vector in that direction. 65 2 2 1 1, – 8, 0 = 65 ≈ 8.0623 is the rate of change 17. u = ,– , 3 3 3 of f(x, y, z) in that direction at that point. 2 2 1 Du f ( x, y, z ) = y, x, 2 z ⋅ ,– , 13. –∇f ( x, y ) = 2 x, y ; –∇f (–1, 2) = 2 –1, 2 is 3 3 3 2 the direction of most rapid decrease. A unit Du f (1, 1, 1) = ⎛ 1 ⎞ 3 vector in that direction is u = ⎜ ⎟ –1, 2 . ⎝ 5⎠ ⎛ π⎞ 18. ⎜ 0, ⎟ is on the y-axis, so the unit vector toward ⎝ 3⎠ 14. – ∇f ( x, y ) = –3cos(3 x – y ), cos(3 x – y ) ; the origin is –j. ⎛π π⎞ ⎛ 1 ⎞ Du ( x, y ) = – e – x cos y , – e – x sin y ⋅ 0, – 1 – ∇f ⎜ , ⎟ = ⎜ ⎟ –3, 1 is the direction of ⎝6 4⎠ ⎝ 2⎠ most rapid decrease. A unit vector in that = e – x sin y; ⎛ 1 ⎞ ⎛ π⎞ 3 direction is ⎜ ⎟ –3, 1 . Du ⎜ 0, ⎟ = ⎝ 10 ⎠ ⎝ 3⎠ 2 y 19. a. Hottest if denominator is smallest; i.e., at the 15. The level curves are = k . For p = (1, 2), origin. 2 x y k = 2, so the level curve through (1, 2) is =2 –200 2 x, 2 y, 2 z x2 b. ∇T ( x , y , z ) = ; (5 + x 2 + y 2 + z 2 )2 or y = 2 x (x ≠ 0). 2 ⎛ 25 ⎞ ∇T (1, – 1, 1) = ⎜ – ⎟ 1, – 1, 1 ∇f ( x, y ) = –2 yx –3 , x –2 ⎝ 4 ⎠ ∇f (1, 2) = –4, 1 , which is perpendicular to the −1,1, −1 is one vector in the direction of parabola at (1, 2). greatest increase. c. Yes 20. – ∇V ( x, y, z ) 2 + y2 + z 2 ) = –100e –( x –2 x, – 2 y, – 2 z –( x 2 + y 2 + z 2 ) = 200e x, y, z is the direction of greatest decrease at (x, y, z), and it points away from the origin. Instructor’s Resource Manual Section 12.5 763 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 21. 21. ∇ f ( x, y , z ) 24. The unit vector from (2, 4) toward (5, 0) is 1 3 4 ( ) − , – . Then = x x2 + y 2 + z 2 2 cos x 2 + y 2 + z 2 , 5 5 3 4 Du f (2, 4) = –3, 8 ⋅ , – = –8.2. 1 5 5 ( ) − y x2 + y 2 + z 2 2 cos x 2 + y 2 + z 2 , 25. The climber is moving in the direction of 1 ⎛ 1 ⎞ ( ) − z x2 + y2 + z2 2 cos x 2 + y 2 + z 2 u=⎜ ⎟ −1, 1 . Let ⎝ 2⎠ 2 + 2 y 2 ) /100 ⎛ 1 ⎞ f ( x, y ) = 3000e−( x . ( ) − = ⎜ x 2 + y 2 + z 2 2 cos x 2 + y 2 + z 2 ⎟ x, y, z 2 + 2 y 2 ) /100 x y ⎜ ⎟ ∇f ( x, y ) = 3000e –( x – ,– ; ⎝ ⎠ 50 25 which either points towards or away from the origin. f (10, 10) = –600e –3 1, 2 She will move at a slope of 22. Let D = x + y + z be the distance. Then we 2 2 2 ⎛ 1 ⎞ Du (10, 10) = –600e –3 1, 2 ⋅ ⎜ ⎟ –1, 1 have ⎝ 2⎠ ∇T = ∂T ∂T ∂T , , = dT ∂D dT ∂D dT ∂D , , ( ) = –300 2 e –3 ≈ –21.1229 . ∂x ∂y ∂z dD ∂x dD ∂y dD ∂z She will descend. Slope is about –21. 1 1 = dT dD x x2 + y 2 + ( z 2 −2 ) , dT dD ( y x2 + y2 + z ) 2 −2 , 26. dx dt = dy dt ; dx dy = ; ln x = – ln y + C 2x –2 y x –y 1 dT dD z x +y +z 2 2 ( 2 −2 ) At t = 0: ln –2 = – ln 1 + C ⇒ C = ln 2. 2 2 ln x = – ln y + ln 2 = ln ; x = ; xy = 2 ⎛ dT ⎞ 1 y y =⎜ ⎜ dD x2 + y2 + ( ⎟ x, y , z ⎟ z 2 −2 ) Since the particle starts at (–2, 1) and neither x nor y can equal 0, the equation simplifies ⎝ ⎠ which either points towards or away from the to xy = −2. ∇T (–2, 1) = –4, – 2 , so the particle origin. moves downward along the curve. 23. He should move in the direction of 1 1 –∇f (p) = – f x (p), f y (p) = – – , – 2 4 ⎛1⎞ = ⎜ ⎟ 2, 1 . Or use 2, 1 . The angle α formed ⎝4⎠ ⎛1⎞ with the East is tan –1 ⎜ ⎟ ≈ 26.57° (N63.43°E). ⎝2⎠ 27. ∇T ( x, y ) = –4 x, – 2 y dx dy = –4 x, = –2 y dt dt dx dy dt = dt has solution x = 2 y 2 . Since the –4 x –2 y particle starts at (–2, 1), this simplifies to x = –2 y 2 . 764 Section 12.5 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 22. 28. f(1, –1) = 5 -1, 1 D u , u f (1, –1) = u1 , u2 ⋅ –5, 5 = –5u1 + 5u2 1 2 ⎛ 1 ⎞ a. -1, 1 (in the direction of the gradient); u = ⎜ ⎟ –1, 1 . ⎝ 2⎠ ⎛ 1 ⎞ b. ± 1, 1 (direction perpendicular to gradient); u = ⎜ ± ⎟ 1, 1 ⎝ 2⎠ 3 4 c. Want Du f (1, – 1) = 1 where u = 1. That is, want –5u1 + 5u2 = 1 and u1 + u2 = 1. Solutions are u = 2 2 , 5 5 4 3 and − , − . 5 5 10(2 x) 10(2 y ) 10(2 z ) 29. a. ∇T ( x , y , z ) = − ,− ,− (x + y + z ) 2 2 2 2 (x + y + z ) 2 2 2 2 ( x + y 2 + z 2 )2 2 20 =– x, y , z ( x + y 2 + z 2 )2 2 r (t ) = t cos π t , t sin π t, t , so r(1) = -1, 0, 1 . Therefore, when t = 1, the bee is at (–1, 0, 1), and ∇T (–1, 0, 1) = –5 –1, 0, 1 . r ′(t ) = cos πt – πt sin πt , sin πt + πt cos πt , 1 , so r ′(1) = –1, – π, 1 . r ′(1) –1, – π, 1 U= = is the unit tangent vector at (–1, 0, 1). r ′(1) 2 + π2 DuT (–1, 0, 1) = u ⋅∇T (–1, 0, 1) –1, – π, 1 ⋅ 5, 0, – 5 10 = =– ≈ –2.9026 2+π 2 2 + π2 Therefore, the temperature is decreasing at about 2.9°C per meter traveled when the bee is at (–1, 0, 1); i.e., when t = 1 s. b. Method 1: (First express T in terms of t.) 10 10 10 5 T= = = = x +y +z 2 2 2 (t cos πt ) + (t sin πt ) + (t ) 2 2 2 2t 2 t2 T (t ) = 5t −2 ; T ′(t ) = −10t −3 ; t ′(1) = −10 Method 2: (Use Chain Rule.) 10 ⎛ 2⎞ = ( DuT ) ( r '(t ) ) , so Dt T (t ) = [ DuT (−1, 0, 1)] ( r '(1) ) = − dT ds Dt T (t ) = ⎜ 2 + π ⎟ = −10 ds dt 2 ⎝ ⎠ 2+π Therefore, the temperature is decreasing at about 10°C per second when the bee is at (–1, 0, 1); i.e., when t = 1 s. 3 4 30. a. Du f = , – ⋅ f x ⋅ f y = –6, so 5 5 3 f x – 4 f y = –30. 4 3 Dv f = , ⋅ f x , f y = 17, so 5 5 4 f x + 3 f y = 85. The simultaneous solution is f x = 10, f y = 15, so ∇f = 10, 15 . Instructor's Resource Manual Section 12.5 765 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 24. 12.6 Concepts Review ∂ z dx ∂ z dy 1. + ∂ x dt ∂ y dt 2. y 2 cos t + 2 xy (– sin t ) = cos3 t – 2sin 2 t cos t ∂z∂x ∂z ∂y 3. + ∂ x ∂t ∂ y ∂t 4. 12 Problem Set 12.6 dw 1. = (2 xy 3 )(3t 2 ) + (3x 2 y 2 )(2t ) dt = (2t 9 )(3t 2 ) + (3t10 )(2t ) = 12t11 dw 2. = (2 xy – y 2 )(– sin t ) + ( x 2 – 2 xy )(cos t ) dt = (sin t + cos t )(1 − 3sin t cos t ) dw 3. = (e x sin y + e y cos x)(3) + (e x cos y + e y sin x)(2) = 3e3t sin 2t + 3e2t cos 3t + 2e3t cos 2t + 2e2t sin 3t dt dw ⎛ 1 ⎞ 2 ⎛ 1 ⎞ sec2 t sec2 t – 2 tan 2 t 1 – tan 2 t 4. = ⎜ ⎟ sec t + ⎜ – ⎟ (2sec2 t tan t ) = – 2 tan t = = dt ⎝ x ⎠ ⎝ y⎠ tan t tan t tan t dw 5. = [ yz 2 (cos( xyz 2 )](3t 2 ) + [ xz 2 cos( xyz 2 )](2t ) + [2 xyz cos( xyz 2 )](1) dt = (3 yz 2 t 2 + 2 xz 2 t + 2 xyz ) cos( xyz 2 ) = (3t 6 + 2t 6 + 2t 6 ) cos(t 7 ) = 7t 6 cos(t 7 ) dw 6. = ( y + z )(2t ) + ( x + z )(−2t ) + ( y + x)(−1) = 2t (2 – t – t 2 ) – 2t (1 – t + t 2 ) – (1) = –4t 3 + 2t –1 dt ∂w 7. = (2 xy )( s) + ( x 2 )(–1) = 2st ( s – t ) s – s 2t 2 = s 2 t (2s – 3t ) ∂t ∂w ⎡ ⎛ s ⎞⎤ 8. = (2 x – x –1 y )(– st –2 ) + (– ln x)( s 2 ) = s 2 ⎢1 – 2t –3 – ln ⎜ ⎟ ⎥ ∂t ⎣ ⎝ t ⎠⎦ ∂w 2 2 2 2 2 2 9. = e x + y (2 x)( s cos t ) + e x + y (2 y )(sin s) = 2e x + y ( xs cos t + y sin s ) ∂t = 2( s 2 sin t cos t + t sin 2 s) exp( s 2 sin 2 t + t 2 sin 2 s ) ∂w 2e s (t +1) ( st –1) 10. = [( x + y ) −1 − ( x − y )−1 ](e s ) + [( x + y )−1 + ( x − y )−1 ]( se st ) = ∂t t 2 e2 s – e2 st Instructor's Resource Manual Section 12.6 767 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 25. ∂w x(– s sin st ) y ( s cos st ) z(s2 ) 11. = + + = s 4 t (1 + s 4 t 2 ) –1/ 2 ∂ t ( x 2 + y 2 + z 2 )1/ 2 ( x 2 + y 2 + z 2 )1/ 2 ( x 2 + y 2 + z 2 )1/ 2 ∂w 2 12. = (e xy + z y )(1) + (e xy + z x)(−1) + (e xy + z )(2t ) = e xy + z ( y − x + 2t ) = e s (0) = 0 ∂t ∂z ⎛∂ z⎞ 13. = (2 xy )(2) + ( x 2 )(–2 st ) = 4(2t + s )(1 − st 2 ) − 2 st (2t + s )2 ; ⎜ ⎟ = 72 ∂t ⎝ ∂ t ⎠ (1, − 2) ∂z ⎛∂ z ⎞ 14. = ( y + 1)(1) + ( x + 1)(rt ) = 1 + rt (1 + 2s + r + t ); ⎜ ⎟ =5 ∂s ⎝ ∂ s ⎠ (1, –1, 2) dw 15. = (2u – tan v)(1) + (– u sec2 v)(π) = 2 x – tan πx – πx sec2 πx dx dw ⎛1⎞ ⎛π⎞ 1+ π = ⎜ ⎟ –1 – ⎜ ⎟ = – ≈ –2.0708 dx x = 1 ⎝ 2 ⎠ ⎝ 2⎠ 2 4 ∂w 16. = (2 xy )(– ρ sin θ sin φ ) + ( x 2 )( ρ cosθ sin φ ) + (2 z )(0) = ρ 3 cos θ sin 3 φ (–2sin 2 θ + cos 2 θ ); ∂θ ⎛∂w⎞ ⎜ ⎟ = –8 ⎝ ∂θ ⎠ (2, π, π ) 2 dr 19. The stream carries the boat along at 2 ft/s with 17. V (r , h) = πr 2 h, = 0.5 in./yr, respect to the boy. dt dh = 8 in./yr dt dV ⎛ dr ⎞ ⎛ dh ⎞ = (2πrh) ⎜ ⎟ + (πr 2 ) ⎜ ⎟ ; dt ⎝ dt ⎠ ⎝ dt ⎠ ⎛ dV ⎞ dx dy ⎜ ⎟ = 11200π in.3/yr = 2, = 4, s 2 = x 2 + y 2 ⎝ dt ⎠ (20, 400) dt dt 11200π in.3 1 board ft ⎛ ds ⎞ ⎛ dx ⎞ ⎛ dy ⎞ 2s ⎜ ⎟ = 2 x ⎜ ⎟ + 2 y ⎜ ⎟ = × ≈ 244.35 board ft/yr ⎝ dt ⎠ ⎝ dt ⎠ ⎝ dt ⎠ 1 yr 144 in.3 ds (2 x + 4 y ) = 18. Let T = e – x –3 y . dt s dT dx dy When t = 3, x = 6, y = 12, s = 6 5. Thus, = e – x –3 y (–1) + e – x –3 y (–3) dt dt dt ⎛ ds ⎞ ⎜ ⎟ = 20 ≈ 4.47 ft/s = e – x –3 y (–1)(2) + e – x –3 y (–3)(2) = –8e – x –3 y ⎝ dt ⎠ t =3 dT = –8, so the temperature is decreasing ⎛1⎞ dh dt (0, 0) 20. V (r , h) = ⎜ ⎟ πr 2 h, = 3 in./min, ⎝3⎠ dt at 8°/min. dr = 2 in./min dt dV ⎛ 2 ⎞ ⎛ dr ⎞ ⎛ 1 ⎞ ⎛ dh ⎞ = ⎜ ⎟ πrh ⎜ ⎟ + ⎜ ⎟ πr 2 ⎜ ⎟ ; dt ⎝ 3 ⎠ ⎝ dt ⎠ ⎝ 3 ⎠ ⎝ dt ⎠ ⎛ dV ⎞ 20,800π ⎜ ⎟ = ≈ 21, 782 in.3/min ⎝ dt ⎠ (40, 100) 3 768 Section 12.6 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 26. 21. Let F ( x, y ) = x3 + 2 x 2 y – y3 = 0 . ⎛1⎞ 29. y = ⎜ ⎟ [ f (u ) + f (v)], ∂F ⎝2⎠ dy − ∂x (3 x 2 + 4 xy ) 3x 2 + 4 xy Then = =− = . where u = x – ct, v = x + ct. ∂F dx ∂y 2 x2 − 3 y 2 3 y2 − 2 x2 ⎛1⎞ ⎛1⎞ y x = ⎜ ⎟ [ f ′(u )(1) + f ′(v)(1)] = ⎜ ⎟ [ f ′(u ) + f ′(v)] ⎝ 2⎠ ⎝2⎠ 22. Let F ( x, y ) = ye – x + 5 x – 17 = 0. ⎛1⎞ y xx = ⎜ ⎟ [ f ′′(u )(1) + f ′′(v)(1)] dy (– ye – x + 5) ⎝2⎠ =– = y – 5e x dx e– x ⎛ 1 ⎞ ′′ = ⎜ ⎟ [ f (u ) + f ′′(v)] ⎝2⎠ 23. Let F(x, y) = x sin y + y cos x = 0. ⎛1⎞ dy (sin y – y sin x) y sin x – sin y yt = ⎜ ⎟ [ f ′(u )(– c) + f ′(v)(c)] =– = ⎝2⎠ dx x cos y + cos x x cos y + cos x ⎛ c⎞ = ⎜ – ⎟ [ f ′(u ) – f ′(u )] ⎝ 2⎠ 24. Let F ( x, y ) = x 2 cos y – y 2 sin x = 0. ⎛ c⎞ ∂F ytt = ⎜ − ⎟ [ f ′′(u )(−c) − f ′′(v)(c)] dy –(2 x cos y – y 2 cos x) ⎝ 2⎠ Then = – ∂x = ∂F dx ∂y – x 2 sin y – 2 y sin x ⎛ c2 ⎞ = ⎜ ⎟ [ f ′′(u ) + f ′′(v )] = c 2 y xx 2 x cos y – y 2 cos x ⎜ 2 ⎟ = . ⎝ ⎠ x 2 sin y + 2 y sin x 30. Let w = f(x, y, z) where x = r – s, y = s – t, z = t – r. Then 25. Let F ( x, y, z ) = 3 x 2 z + y 3 − xyz 3 = 0. wr + ws + wt = ( wx xr + wx xs ) + ( wy ys + wy yt ) ∂z (6 xz – yz 3 ) yz 3 – 6 xz + ( wz zt + wz zr ) =– = ∂x 3x 2 – 3xyz 2 3 x 2 – 3 xyz 2 = [ wx (1) + wx (−1)] + [ wy (1) + wy (−1)] + [ wz (1) + wz (−1)] 26. Let f ( x, y, z ) = ye – x + z sin x = 0. =0 ∂x – sin x sin x = = ∂ z – ye + z cos x ye – z cos x –x –x y x 31. Let w = ∫ f (u )du = – ∫ f (u )du, where x = g(t), x y ∂T ∂T ∂ x ∂T ∂ y ∂T ∂ z ∂T ∂ w y = h(t). 27. = + + + ∂s ∂x ∂s ∂ y ∂s ∂z ∂s ∂w ∂s Then dw ∂w dx ∂w dy = + = − f ( x) g ′(t ) + f ( y )h′(t ) ∂z dt ∂x dt ∂y dt 28. We use the zr notation for . ∂r = f (h(t ))h′(t ) – f ( g (t )) g ′(t ) . zr = z x xr + z y yr = z x cos θ + z y sin θ Thus, for the particular function given zθ = z x xθ + z y yθ = z x (– r sin θ ) + z y (r cos θ ), so F ′(t ) = 9 + (t 2 )4 (2t ) r zθ = – z x sin θ + z y cos θ . Thus, –1 ( ) ( 2π cos ) 4 ( zr ) + ( r 2 –2 )( zθ ) = ( z x cos θ + z y sin θ ) 2 2 – 9 + sin 2πt 2πt ; +(– z x sin θ + z y cos θ )2 F ′ ( 2 ) = (5) ( 2 2 ) – (3) ( 2π ) = ( z x ) 2 + ( z y )2 (expanding and using = 10 2 – 3 2π ≈ 0.8135. sin 2 θ + cos 2 θ = 1). 32. If f (tx, ty ) = tf ( x, y ), then d d [ f (tx, ty )] = [tf ( x, y )]. dt dt That is, [ ftx (tx, ty )][ x] + [ fty (tx, ty )][ y ] = f ( x, y ). Letting t = 1 yields the desired result. Instructor's Resource Manual Section 12.6 769 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 27. 33. c 2 = a 2 + b 2 – 2ab cos 40° (Law of Cosines) where a, b, and c are functions of t. aa ′ + bb′ – (a ′b + ab′) cos 40° 2cc′ = 2aa ′ + 2bb′ – 2(a ′b + ab′) cos 40° so c′ = . c When a = 200 and b = 150, c 2 = (200) 2 + (150)2 – 2(200)(150) cos 40° = 62,500 – 60,000 cos 40°. It is given that a ′ = 450 and b′ = 400, so at that instant, (200)(450) + (150)(400) − [(450)(150) + (200)(400)]cos 40° c′ = ≈ 288 . 62,500 – 60, 000 cos 40° Thus, the distance between the airplanes is increasing at about 288 mph. 2 34. r = x, y, z , so r 2 = r = x 2 + y 2 + z 2 . GMm F= , so x + y2 + z2 2 F ′(t ) = Fm m′(t ) + Fx x ′(t ) + Fy y ′(t ) + Fz z ′(t ) GMm′(t ) 2GMmxx′(t ) = – x +y +z 2 2 ( x2 + y 2 + z 2 )2 2 2GMmyy ′(t ) 2GMmzz ′(t ) – + (x + y + z ) 2 2 2 2 ( x 2 + y 2 + z 2 )2 GM [( x 2 + y 2 + z 2 )m′(t ) – 2m( xx′(t ) + yy ′(t ) + zz ′(t )] = . ( x 2 + y 2 + z 2 )2 12.7 Concepts Review 2. ∇F ( x, y, z ) = 2 8 x, y, 8 z ; 1. perpendicular ⎛ 2⎞ ∇F ⎜ 1, 2, ⎜ ⎟ = 4 4, 1, 2 2 ⎝ 2 ⎟⎠ 2. 3,1, −1 Tangent Plane: ⎛ 2⎞ 3. x − +4( y − 1) + 6( z − 1) = 0 4( x – 1) + 1( y – 2) + 2 2 ⎜ z – ⎜ ⎟ , or ⎝ 2 ⎟ ⎠ ∂f ∂f 4 x + y + 2 2 z = 8. 4. dx + dy ∂x ∂y 3. Let F ( x, y , z ) = x 2 – y 2 + z 2 + 1 = 0. Problem Set 12.7 ∇F ( x , y , z ) = 2 x , – 2 y , 2 z = 2 x , – y , z 1. ∇F ( x, y, z ) = 2 x, y, z ; ( ) ∇F 1, 3, 7 = 2 1, – 3, 7 , so 1, – 3, 7 is ( ) ∇F 2, 3, 3 = 2 2, 3, 3 normal to the surface at the point. Then the tangent plane is ( ) Tangent Plane: 1( x – 1) – 3( y – 3) + 7 z – 7 = 0, or more ( 2( x – 2) + 3( y – 3) + 3 z – 3 = 0, or ) simply, x − 3 y + 7 z = –1. 2 x + 3 y + 3 z = 16 770 Section 12.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 28. 4. ∇f ( x, y, z ) = 2 x, y, – z ; ⎛1⎞ 11. dz = 2 x −1dx + y −1dy = (−1)(0.02) + ⎜ ⎟ (−0.04) ∇f (2, 1, 1) = 2 2, 1, − 1 ⎝ 4⎠ = –0.03 Tangent plane: 2( x − 2) + 1( y − 1) − 1( z − 1) = 0, or 2 x + y − z = 4 . Δz = f (–1.98, 3.96) – f (–2.4) = ln[(–1.98)2 (3.96)] – ln16 ≈ –0.030151 ⎛1⎞ 5. ∇f ( x, y ) = ⎜ ⎟ x, y ; ∇f (2, 2) = 1, 1 ⎝ 2⎠ 12. Let z = f ( x, y ) = tan –1 xy; Tangent plane: z − 2 = 1( x − 2) + 1( y − 2) , or y x x+ y−z = 2. dz = dx + dy; 1+ x y2 2 1 + x2 y2 (−0.5)(−0.03) + (−2)(−0.01) 6. Let f ( x, y ) = xe –2 y . = = 0.0175. 1 + (4)(0.25) ∇f ( x, y ) = e−2 y , − 2 xe−2 y Δz = f (−2.03, −0.51) − f (−2, −0.5) ≈ 0.017342 ∇f (1, 0) = 1, − 2 13. Let Then 1, – 2, – 1 is normal to the surface at F ( x, y , z ) = x 2 – 2 xy – y 2 – 8 x + 4 y – z = 0; (1, 0, 1), and the tangent plane is ∇F ( x, y, z ) = 2 x – 2 y – 8, – 2 x – 2 y + 4, – 1 1( x − 1) − 2( y − 0) − 1( z − 1) = 0 , or x − 2 y − z = 0 . Tangent plane is horizontal if ∇F = 0, 0, k for 7. ∇f ( x, y ) = –4e3 y sin 2 x, 6e3 y cos 2 x ; any k ≠ 0 . 2x – 2y – 8 = 0 and –2x – 2y + 4 = 0 if x = 3 and ⎛π ⎞ y = –1. Then z = –14. There is a horizontal ∇f ⎜ , 0 ⎟ = –2 3, – 3 ⎝3 ⎠ tangent plane at (3, –1, –14). ⎛ π⎞ Tangent plane: z + 1 = –2 3 ⎜ x – ⎟ – 3( y – 0) , 14. 8, −3, −1 is normal to 8x – 3y – z = 0. ⎝ 3⎠ Let F ( x, y, z ) = 2 x 2 + 3 y 2 − z . or 2 3 x + 3 y + z = ( 2 3π – 3 ). ∇F ( x, y, z ) = 4 x, 6 y, – 1 is normal to 3 z = 2 x 2 + 3 y 2 at (x, y, z). 4x = 8 and ⎛1⎞ 1 1 1 1 1 8. ∇f ( x, y ) = ⎜ ⎟ , ; ∇f (1, 4) = , 6y = –3, if x = 2 and y = – ; then ⎝2⎠ x y 2 4 2 ⎛1⎞ ⎛1⎞ ⎛ 1 ⎞ Tangent plane: z − 3 = ⎜ ⎟ ( x − 1) + ⎜ ⎟ ( y − 4), z = 8.75 at ⎜ 2, – , 8.75 ⎟ . ⎝2⎠ ⎝4⎠ ⎝ 2 ⎠ 1 1 3 or x + y − z = − . 15. For F ( x, y , z ) = x 2 + 4 y + z 2 = 0, 2 4 2 ∇F ( x, y, z ) = 2 x, 4, 2 z = 2 x, 2, z . 9. Let z = f ( x, y ) = 2 x y ; 2 3 F(0, –1, 2) = 0, and dz = 4 xy3 dx + 6 x 2 y 2 dy. For the points given, ∇F (0, – 1, 2) = 2 0, 2, 2 = 4 0, 1, 1 . dx = −0.01, dy = 0.02 , For G ( x, y, z ) = x 2 + y 2 + z 2 – 6 z + 7 = 0, dz = 4(−0.01) + 6(0.02) = 0.08. ∇G ( x , y , z ) = 2 x , 2 y , 2 z – 6 = 2 x , y , z – 3 . Δz = f (0.99, 1.02) – f (1, 1) G(0, –1, 2) = 0, and = 2(0.99) (1.02) − 2(1) (1) ≈ 0.08017992 2 3 2 3 ∇G (0, – 1, 2) = 2 0, – 1, – 1 = –2 0, 1, 1 . 0, 1, 1 is normal to both surfaces at 10. dz = (2 x − 5 y )dx + (−5 x + 1)dy (0, –1, 2) so the surfaces have the same tangent = (−11)(0.03) + (−9)(−0.02) = −0.15 plane; hence, they are tangent to each other at Δz = f (2.03, 2.98) − f (2,3) = −0.1461 (0, –1, 2). Instructor’s Resource Manual Section 12.7 771 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 29. 16. (1, 1, 1) satisfies each equation, so the surfaces 20. Let f ( x, y, z ) = x – z 2 , and g ( x, y, z ) = y – z 3 . intersect at (1, 1, 1). For ∇f ( x, y, z ) = 1, 0, − 2 z and z = f ( x, y ) = x 2 y : ∇f ( x, y ) = 2 xy, x 2 ; ∇g ( x, y, z ) = 0, 1, – 3z 2 ∇f (1, 1) = 2, 1 , so 2, 1, – 1 is normal at (1, 1, 1). ∇f (1, 1, 1) = 1, 0, – 2 and For f ( x, y, z ) = x 2 − 4 y + 3 = 0; ∇g (1, 1, 1) = 0, 1, – 3 ∇f ( x, y, z ) = 2, – 4, 0 ; 1, 0, −2 × 0,1, −3 = 2,3,1 ∇f (1, 1, 1) = 2, – 4, 0 so 2, – 4, 0 is normal at Line: x = 1 + 2t , y = 1 + 3t , z = 1 + t (1, 1, 1). 21. dS = S A dA + SW dW 2, 1, – 1 ⋅ 2, – 4, 0 = 0, so the normals, hence W A −WdA + AdW tangent planes, and hence the surfaces, are =− dA + dW = perpendicular at (1, 1, 1). ( A −W ) 2 (A −W ) ( A − W )2 2 At W = 20, A = 36: 17. Let F ( x, y, z ) = x 2 + 2 y 2 + 3 z 2 – 12 = 0; −20dA + 36dW −5dA + 9dW dS = = . ∇F ( x, y, z ) = 2 x, 2 y, 3 z is normal to the 256 64 plane. 5 dA + 9 dW 5(0.02) + 9(0.02) Thus, dS ≤ ≤ A vector in the direction of the line, 64 64 2, 8, – 6 = 2 1, 4, – 3 , is normal to the plane. = 0.004375 x, 2 y,3 z = k 1, 4, -3 and (x, y, z) is on the 1 1 surface for points (1, 2, –1) [when k =1] and 22. V = lwh, dl = dw = , dh = , l = 72, w = 48, 2 4 (–1, –2, 1) [when k = –1]. h = 36 dV = whdl + lhdw + lwdh = 3024 in.3 (1.75 ft3) x2 y2 z2 18. Let F ( x, y, z ) = + = 1. + a 2 b2 c2 23. V = πr 2 h, dV = 2πrh dr + πr 2 dh 2x 2 y 2z ∇F ( x , y , z ) = , , dV ≤ 2πrh dr + πr 2 dh ≤ 2πrh(0.02r ) + πr 2 (0.03h) a 2 b2 c2 = 0.04πr 2 h + 0.03πr 2 h = 0.07V x y z ∇F ( x0 , y0 , z0 ) = 2 0 , 0 , 0 Maximum error in V is 7%. a 2 b2 c 2 The tangent plane at ( x0 , y0 , z0 ) is L 24. T = f ( L, g ) = 2π x0 ( x – x0 ) y0 ( y – y0 ) z0 ( z – z 0 ) g + + = 0. a 2 b 2 c2 dT = f L dL + f g dg x0 x y0 y z0 z ⎛ x0 y0 z0 ⎞ 2 2 2 ⎛ ⎞ ⎛ ⎞ + + –⎜ + + ⎟=0 ⎜ ⎟ ⎜ ⎟⎛ a2 b2 c2 ⎜ a 2 b2 c2 ⎟ ⎝ ⎠ 1 ⎟⎛ 1 ⎞ ⎞ = 2π ⎜ dL + 2π ⎜ ⎟ ⎜ − L ⎟ dg 1 ⎜ ⎜ ⎟ ⎜ g2 ⎟ L ⎟⎝ g ⎠ ⎜ ⎟ ⎟⎝ ⎠ x0 x y0 y z0 z L Therefore, + + = 1, since ⎜2 ⎜ ⎟ ⎟ ⎜2 ⎜ ⎟ a2 b2 c2 ⎝ g⎠ ⎝ g ⎠ 2 x0 2 y0 2 z0 π( gdL – Ldg ) + + = 1. = , so a2 b2 c2 g2 L g 19. ∇f ( x, y, z ) = 2 9 x, 4 y , 4 z ; dT π( gdL – Ldg ) gdL – Ldg = = ∇f (1, 2, 2) = 2 9, 8, 8 T ⎛ L ⎞⎛ 2 L ⎞ 2 gL ⎜ 2π ⎜ ⎟⎜ g ⎟⎜ ⎟ ⎟ ∇g ( x , y , z ) = 2 2 x , – y , 3 z ; ⎝ g ⎠⎝ g⎠ ∇f (1, 2, 2) = 4 1, – 1, 3 1 ⎛ dL dg ⎞ = ⎜ – ⎟. 9, 8, 8 × 1, –1, 3 = 32, –19, –17 2⎝ L g ⎠ Line: x = 1 + 32t , y = 2 -19t , z = 2 -17t Therefore, dT 1 ⎛ dL dg ⎞ 1 ≤ ⎜ + ⎟ = (0.5% + 0.3%) = 0.4%. T 2⎝ L g ⎠ 2 772 Section 12.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 30. R1 R2 27. Let F(x, y, z) = xyz = k; let (a, b, c) be any point 25. Solving for R, R = , so on the surface of F. R1 + R2 k k k ∂R R22 ∂R R12 ∇F ( x, y, z ) = yz , xz , xy = , , = and = . x y z ∂ R1 ( R1 + R2 ) 2 ∂ R2 ( R1 + R2 )2 1 1 1 =k , , R2 dR1 + R1 dR2 2 2 x y z Therefore, dR = ; ( R1 + R2 )2 1 1 1 ∇F ( a , b , c ) = k , , 2 R2 dR1 + R1 2 dR2 a b c dR ≤ . Then at R1 = 25, An equation of the tangent plane at the point is ( R1 + R2 ) 2 ⎛1⎞ ⎛1⎞ ⎛1⎞ (25)(100) ⎜ ⎟ ( x – a ) + ⎜ ⎟ ( x – b) + ⎜ ⎟ ( x – c) = 0, or R2 = 100, dR1 = dR2 = 0.5, R = = 20 ⎝a⎠ ⎝b⎠ ⎝c⎠ 25 + 100 x y z (100)2 (0.5) + (25) 2 (0.5) + + = 3. and dR ≤ = 0.34. a b c (125) 2 Points of intersection of the tangent plane on the coordinate axes are (3a, 0, 0), (0, 3b, 0), and (0, 0, 3c). 26. Let F ( x, y , z ) = x 2 + y 2 + 2 z 2 . The volume of the tetrahedron is ∇F ( x , y , z ) = 2 x , 2 y , 4 z ; ⎛1⎞ 1⎛1 ⎞ ∇F ⎜ ⎟ (area of base)(altitude)= ⎜ 3a 3b ⎟ ( 3c ) ∇F (1, 2, 1) = 2 1, 2, 2 ; 1 2 2 = , , ⎝ 3⎠ 3⎝ 2 ⎠ ∇F 3 3 3 9 abc 9 k = = (a constant). 1 2 2 2 2 Thus, u = , , is the unit vector in the 3 3 3 direction of flight, and 1 2 2 x, y, z = 1, 2, 1 + 4t , , is the location 3 3 3 of the bee along its line of flight t seconds after takeoff. Using the parametric form of the line of flight to substitute into the equation of the plane yields t = 3 as the time of intersection with the plane. Then substituting this value of t into the equation of the line yields x = 5, y = 10, z = 9 so the point of intersection is (5, 10, 9). 1 1 1 28. If F ( x, y , z ) = x + y + z , then ∇F ( x, y , z ) = 0.5 , , . The equation of the tangent is x y z 1 1 1 x y z 0.5 , , ⋅ x – x0 , y – y0 , z – z0 = 0, or + + = x0 + y0 + z0 = a. x0 y0 z0 x0 y0 z0 Intercepts are a x0 , a y0 , a z0 ; so the sum is a ( x0 + y0 + z0 = a 2 . ) Instructor’s Resource Manual Section 12.7 773 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 31. 29. f ( x, y ) = ( x 2 + y 2 )1/ 2 ; f (3, 4) = 5 3 4 f x ( x, y ) = x( x 2 + y 2 ) –1/ 2 ; f x (3, 4) = = 0.6 ; f y = ( x, y ) = y ( x 2 + y 2 ) –1/ 2 ; f x (3, 4) = = 0.8 5 5 16 f xx ( x, y ) = y 2 ( x 2 + y 2 ) –3 / 2 ; f x (3, 4) = = 0.128 ; f xy ( x, y ) = – xy ( x 2 + y 2 ) –3 / 2 ; 125 12 f xy (3, 4) = – = –0.096 125 9 f yy = x 2 ( x 2 + y 2 ) –3 / 2 ; f xx (3, 4) = = 0.072 125 Therefore, the second order Taylor approximation is f ( x, y ) = 5 + 0.6( x – 3) + 0.8( y – 4) + 0.5[0.128( x – 3) 2 + 2(–0.096)( x – 3)( y – 4) + 0.072( y – 4)2 ] a. First order Taylor approximation: f ( x, y ) = 5 + 0.6( x - 3) + 0.8( y - 4) . Thus, f (3.1,3.9) ≈ 5 + 0.6(0.1) + 0.8(-0.1) = 4.98 . b. f (3.1,3.9) ≈ 5 + 0.6(–0.1) + 0.8(0.1) + 0.5[0.128(0.1) 2 + 2(–0.096)(0.1)(–0.1) + 0.072(–0.1) 2 ] = 4.98196 c. f (3.1, 3.9) ≈ 4.9819675 ⎛ x2 + y2 ⎞ 30. f ( x, y ) = tan ⎜ ⎟ ; f (0, 0) = 0 ⎜ 64 ⎟ ⎝ ⎠ x ⎛ x + y2 ⎞ 2 f x ( x, y ) = ⋅ sec2 ⎜ ⎟ ; f x (0, 0) = 0 32 ⎜ 64 ⎟ ⎝ ⎠ y ⎛x +y 2 2⎞ f y ( x, y ) = ⋅ sec2 ⎜ ⎟ ; f y (0, 0) = 0 32 ⎜ 64 ⎟ ⎝ ⎠ 2x 2 ⎛ x2 + y2 ⎞ ⎛ x2 + y2 ⎞ 1 ⎛ x2 + y 2 ⎞ 1 f xx ( x, y ) = sec 2 ⎜ ⎟ tan ⎜ ⎟ + sec 2 ⎜ ⎟ ; f xx (0, 0) = 2 ⎜ 64 ⎟ ⎜ 64 ⎟ 32 ⎜ 64 ⎟ 32 32 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 y2 ⎛ x2 + y 2 ⎞ ⎛ x2 + y 2 ⎞ 1 ⎛ x2 + y2 ⎞ 1 f yy ( x, y ) = sec2 ⎜ ⎟ tan ⎜ ⎟ + sec2 ⎜ ⎟ ; f yy (0, 0) = 2 ⎜ 64 ⎟ ⎜ 64 ⎟ 32 ⎜ 64 ⎟ 32 32 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ When computed, each term of f xy ( x, y ) will contain either x or y, resulting in f xy ( 0, 0 ) = 0 . Therefore, the second-order Taylor approximation is 1⎡ 1 1 2⎤ 1 2 1 2 f ( x, y ) = 0 + 0 ⋅ x + 0 ⋅ y + ⎢ x 2 + 2 ⋅ 0 ⋅ x ⋅ y + y = x + y 2 ⎣ 32 32 ⎥ 64 ⎦ 64 a. The first-order Taylor approximation is f ( x, y ) = 0 + 0 ⋅ x + 0 ⋅ y = 0; Thus, f (0.2, −0.3) ≈ 0. 1 1 b. f (0.2, −0.3) ≈ (0.2) 2 + (−0.3)3 = 0.00203125 64 64 c. f (0.2, −0.3) ≈ 0.0020312528 774 Section 12.7 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 32. 12.8 Concepts Review 1. closed bounded 2. boundary; stationary; singular 3. ∇f ( x0 , y0 ) = 0 2 4. f xx ( x0 , y0 ) f yy ( x0 , y0 ) – f xy ( x0 , y0 ) Problem Set 12.8 1. ∇f ( x, y ) = 2 x – 4, 8 y = 0, 0 at (2, 0), a stationary point. D = f xx f yy – f xy = (2)(8) – (0)2 = 16 > 0 and f xx = 2 > 0. Local minimum at (2, 0). 2 2. ∇f ( x, y ) = 2 x − 2, 8 y + 8 = 0, 0 at (1, –1), a stationary point. D = f xx f yy – f xy = (2)(8) – (0)2 = 16 > 0 and 2 f xx = 2 > 0. Local minimum at (1, –1). 3. ∇f ( x, y ) = 8 x3 – 2 x, 6 y = 2 x(4 x 2 – 1), 6 y = 0, 0 , at (0, 0), (0.5, 0), (−0.5, 0) all stationary points. f xx = 24 x 2 – 2; D = f xx f yy – f xy = (24 x 2 – 2)(6) – (0)2 = 12(12 x 2 – 1). 2 At (0, 0) : D = −12 , so (0, 0) is a saddle point. At ( 0.5, 0 ) and ( −0.5, 0 ) : D = 24 and f xx = 6 , so local minima occur at these points. 4. ∇f ( x, y ) = y 2 – 12 x, 2 xy – 6 y = 0, 0 at stationary points (0, 0), (3, −6) and (3, 6). D = f xx f yy – f xy = (–12)(2 x – 6) – (2 y )2 = –4( y 2 + 6 x – 18), f xx = –12 2 At (0, 0): D = 72, and f xx = –12, so local maximum at (0, 0). At (3, ±6) : D = −144, so (3, ± y ) are saddle points. 5. ∇f ( x, y ) = y, x = 0, 0 at (0, 0), a stationary point. D = f xx f yy – f xy = (0)(0) – (1)2 = –1, so (0, 0) is a saddle point. 2 6. Let ∇f ( x, y ) = 3 x 2 – 6 y, 3 y 2 – 6 x = 0, 0 . Then 3x 2 – 6 y = 0 and 3 y 2 − 6 x = 0. 1 4 3x 2 − 6 y = 0 → 3x 2 = 6 y → x 2 = 2 y → x 4 = 4 y 2 → x = y2 4 ⎛1 ⎞ ⎝4 ⎠ 3 4 3 4 3 ( ) ( 3 y 2 − 6 x = 0 → 3 ⎜ x 4 ⎟ − 6 x = 0 → x 4 − 6 x = 0 → x x3 − 8 = 0 → x ( x − 2 ) x 2 + 2 x + 4 = 0 → x = 0, x = 2 4 ) x = 0 : 3 x 2 − 6 y = 0 → 3 ( 0 ) − 6 y = 0 → −6 y = 0 → y = 0 x = 2 : 3x 2 − 6 y = 0 → 3 ( 2 ) − 6 y → 12 − 6 y = 0 → 12 = 6 y → 2 = y 2 Solving simultaneously, we obtain the solutions (0, 0) and (2, 2). f xx = 6 x; D = f xx f yy – f xy = (6 x)(6 y ) – (–6)2 = 36 ( xy – 1) ; At (0, 0): D = −36 < 0 , so (0, 0) is a saddle point. 2 At (2, 2): D = 108 > 0 , f xx > 0, so a local minimum occurs at ( 2, 2 ) . Instructor’s Resource Manual Section 12.8 775 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 33. x 2 y − 2 xy 2 − 4 7. ∇f ( x, y ) = , = 0, 0 at (1, 2). x2 y2 D = f xx f yy – f xy = (4 x – 3)(8 y – 3) – (1)2 = 32 x −3 y −3 − 1, f xx = 4 x −3 2 At (1, 2): D = 3 > 0 , and f xx > 0, so a local minimum at (1, 2). 8. ∇f ( x, y ) = –2 exp(– x 2 – y 2 + 4 y ) x, y – 2 = 0, 0 at (0, 2). D = f xx f yy – f xy = exp 2(− x 2 − y 2 + 4 y )[(4 x 2 − 2)(4 y 2 − 16 y + 14) − (4 xy − 8 x) 2 ], 2 f xx = (4 x 2 – 2) exp(– x 2 – y 2 + 4 y ) At (0, 2) : D > 0 , and f xx < 0, so local maximum at (0, 2). 9. Let ∇f ( x, y ) 13. ∇f ( x, y ) = 2 x, – 2 y = 0, 0 at (0, 0). = – sin x – sin( x + y ), – sin y – sin( x + y ) = 0, 0 D = f xx f yy – f xy = (2)(–2) – (0)2 < 0, so (0, 0) is 2 ⎛ – sin x – sin( x + y ) = 0 ⎞ a saddle point. A parametric representation of the Then ⎜ ⎟ . Therefore, ⎝ sin y + sin( x + y ) = 0 ⎠ boundary of S is x = cos t, y = sin t, t in π [0, 2π ] . sin x = sin y , so x = y = . However, these values 4 f ( x, y ) = f ( x(t ), y (t )) = cos 2 t – sin 2 t + 1 satisfy neither equation. Therefore, the gradient is = cos 2t – 1 defined but never zero in its domain, and the cos 2t – 1 is maximum if cos 2t = 1, which occurs boundary of the domain is outside the domain, so for t = 0, π , 2π . The points of the curve are there are no critical points. (±1, 0). f (±1, 0) = 2 10. ∇f ( x, y ) = 2 x – 2a cos y, 2ax sin y = 0, 0 at f(x, y) = cos 2t – 1 is minimum if cos 2t = –1, π 3π ⎛ π⎞ which occurs for t = , . The points of the ⎜ 0, ± ⎟ , (a, 0) 2 2 ⎝ 2⎠ curve are (0, ±1). f (0, ±1) = 0 . Global minimum D = f xx f yy – f xy = (2)(2ax cos y ) – (2a sin y )2 , 2 of 0 at (0, ± 1); global maximum of 2 at (±1, 0). f xx = 2 14. ∇f ( x, y ) = 2 x − 6, 2 y − 8 = 0, 0 at (3, 4), ⎛ π⎞ ⎛ π⎞ At ⎜ 0, ± ⎟ : D = –4a 2 < 0, so ⎜ 0, ± ⎟ are which is outside S, so there are no stationary ⎝ 2⎠ ⎝ 2⎠ points. There are also no singular points. saddle points. At (a, 0): D = 4a 2 > 0 and x = cos t , y = sin t , t in [0, 2π ] is a parametric f xx > 0, so local minimum at (a, 0). representation of the boundary of S. f ( x, y ) = f ( x(t ), y (t )) 11. We do not need to use calculus for this one. 3x is minimum at 0 and 4y is minimum at –1. (0, –1) is = cos 2 t – 6 cos t + sin 2 t – 8sin t + 7 in S, so 3x + 4y is minimum at (0, –1); the = 8 − 6 cos t − 8sin t = F (t ) minimum value is –4. Similarly, 3x and 4y are 4 each maximum at 1. (1, 1) is in S, so 3x + 4y is F ′(t ) = 6sin t – 8cos t = 0 if tan t = . t can be 3 maximum at (1, 1); the maximum value is 7. (Use in the 1st or 3rd quadrants. The corresponding calculus techniques and compare.) ⎛ 3 4⎞ points of the curve are ⎜ ± , ± ⎟ . 12. We do not need to use calculus for this one. Each ⎝ 5 5⎠ of x 2 and y 2 is minimum at 0 and (0, 0) is in S, ⎛ 3 4⎞ ⎛ 3 4⎞ f ⎜ – , – ⎟ = 18; f ⎜ , ⎟ = –2 so x 2 + y 2 is minimum at (0, 0); the minimum ⎝ 5 5⎠ ⎝5 5⎠ ⎛3 4⎞ value is 0. Similarly, x 2 and y 2 are maximum at Global minimum of –2 at ⎜ , ⎟ ; global ⎝5 5⎠ x = 3 and y = 4, respectively, and (3, 4) is in S, so ⎛ 3 4⎞ x 2 + y 2 is maximum at (3, 4); the maximum maximum of 18 at ⎜ – , – ⎟ . ⎝ 5 5⎠ value is 25. (Use calculus techniques and compare.) 776 Section 12.8 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 34. 15. Let x, y, z denote the numbers, so x + y + z = N. 17. Let S denote the surface area of the box with Maximize dimensions x, y, z. P = xyz = xy ( N – x – y ) = Nxy – x 2 y – xy 2 . S = 2xy + 2xz + 2yz and V0 = xyz , so Let ∇P ( x, y ) = Ny – 2 xy – y 2 , Nx – x 2 – 2 xy S = 2( xy + V0 y –1 + V0 x –1 ). Minimize f ( x, y ) = xy + V0 y –1 + V0 x –1 subject = 0, 0 . to x > 0, y > 0. ⎛ Ny – 2 xy – y 2 = 0 ⎞ Then ⎜ ⎟. ∇f ( x, y ) = y – V0 x –2 , x – V0 y –2 = 0, 0 at ⎜ Nx – x 2 – 2 xy = 0 ⎟ ⎝ ⎠ (V01/ 3 , V01/ 3 ). N ( x, − y ) = x − y = ( x + y )( x − y ). x = y or 2 2 D = f xx f yy – f xy = 4V02 x –3 y –3 – 1, 2 N = x + y. Therefore, x = y (since N = x + y would mean that f xx = 2V0 x –3 . ( ) P = 0, certainly not a maximum value). At V0 3 , V0 3 : D = 3 > 0, f xx = 2 > 0, so 1/ 1/ Then, substituting into Nx – x 2 – 2 xy = 0, we local minimum. obtain Nx – x 2 – 2 x 2 = 0, from which we obtain Conclusion: The box is a cube with edge V0 3 . 1/ N x(N – 3x) = 0, so x = (since x = 0 ⇒ P = 0). 3 18. Let L denote the sum of edge lengths for a box of Pxx = –2 y; dimensions x, y, z. Minimize L = 4x + 4y + 4z, D = Pxx Pyy – Pxy 2 subject to V0 = xyz. 4V0 = (−2 y )(−2 x) − ( N − 2 x − 2 y )2 L ( x, y ) = 4 x + 4 y + , x > 0, y > 0 xy = 4 xy − ( N − 2 x − 2 y )2 Let ∇L( x, y ) = 4 x −1 y −1 x −1 ( x 2 y − V0 ), y −1 ( xy 2 − V0 ) 2 N N 2N At x = y = :D = > 0, Pxx = – < 0 (so 3 3 3 local maximum) = 0,0 . N If x = y = , then z = . N Then x 2 y = V0 and xy 2 = V0 , from which it 3 3 follows that x = y. Therefore x = y = z = V0 3 . 1/ N Conclusion: Each number is . (If the intent is 8V0 3 Lxx = ; to find three distinct numbers, then there is no x3 y maximum value of P that satisfies that 2 ⎛ 8V ⎞ ⎛ 8V ⎞ ⎛ 4V0 ⎞ D= Lxx L yy − L2 = ⎜ 0 ⎟⎜ 0 ⎟ − ⎜ ⎜ x3 y ⎟ ⎜ xy3 ⎟ ⎜ x 2 y 2 ⎟ condition.) xy ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ( ) 16. Let s be the distance from the origin to (x, y, z) on At V0 3 , V0 3 : D > 0, Lxx > 0 (so local 1/ 1/ the plane. s 2 = x 2 + y 2 + z 2 and x + 2 y + 3z = 12. Minimize minimum). There are no other critical points, and as s 2 = f ( y , z ) = (12 – 2 y – 3 z )2 + y 2 + z 2 . (x, y) → boundary, L → ∞ . Hence, the optimal ∇f ( y, z ) = –48 + 12 x + 10 y, – 72 + 12 y + 20 z box is a cube with edge V0 3 . 1/ ⎛ 12 18 ⎞ = 0, 0 at ⎜ , ⎟ . 19. Let S denote the area of the sides and bottom of ⎝7 7⎠ the tank with base l by w and depth h. D = f yy f zz – f yz = 56 > 0 and f yy = 10 > 0; 2 S = lw + 2lh + 2wh and lwh = 256. ⎛ 12 18 ⎞ ⎛ 256 ⎞ ⎛ 256 ⎞ local maximum at ⎜ , ⎟ S (l , w) = lw + 2l ⎜ ⎟ + 2w ⎜ ⎟ , w > 0, l > 0. ⎝7 7⎠ ⎝ lw ⎠ ⎝ lw ⎠ 504 S (l w) = w – 5121–2 , l – 512w –2 = 0, 0 at s2 = , so the shortest distance is 49 (8, 8). h = 4 there. At (8, 8) D > 0 and S11 > 0, s= 6 14 ≈ 3.2071. so local minimum. Dimensions are 8’ × 8’ × 4’. 7 Instructor’s Resource Manual Section 12.8 777 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 35. 20. Let V denote the volume of the box and (x, y, z) denote its 1st octant vertex. V = (2x)(2y)(2z) = 8xyz and 24 x 2 + y 2 + z 2 = 9. ⎡⎛ 1 ⎞ ⎤ V 2 = 64 ⎢⎜ ⎟ (9 – y 2 – z 2 ) ⎥ y 2 z 2 ⎣⎝ 24 ⎠ ⎦ Maximize f ( y , z ) = (9 – y 2 – z 2 ) y 2 z 2 , y > 0, z > 0. 2 ∇f ( y, z ) = 2 yz 2 (9 – 2 y 2 – z 2 ), y 2 z (9 – y 2 – 2 z 2 = 0, 0 at ( 3, 3). x = 4 At ( ) 3, 3 , D = f yy f zz – f yz > 0 and f yy < 0, so local maximum. The greatest possible volume is 2 ⎛ 2⎞ 8⎜ ⎜ 4 ⎟ ⎟ ( 3 )( 3 ) = 6 2. ⎝ ⎠ 21. Let x, y, z denote the vector; let S be the sum of its components. x 2 + y 2 + z 2 = 81, so z = (81 – x 2 – y 2 )1/ 2 . Maximize S ( x, y ) = x + y + (81 – x 2 – y 2 )1/ 2 , 0 ≤ x 2 + y 2 ≤ 9. Let ∇S ( x, y ) = 1 – x(81 – x 2 – y 2 ) –1/ 2 , 1 – y (81 – x 2 – y 2 ) –1/ 2 = 0, 0 . Therefore, x = (81 – x 2 – y 2 )1/ 2 and y = (81 – x 2 – y 2 )1/ 2 . We then obtain x = y = 3 3 as the only stationary point. For these values of x and y, z = 3 3 and S = 9 3 ≈ 15.59. The boundary needs to be checked. It is fairly easy to check each edge of the boundary separately. The largest 18 value of S at a boundary point occurs at three places and turns out to be ≈ 12.73. 2 Conclusion: the vector is 3 3 1, 1, 1 . 22. Let P ( x, x, z ) be any point in the plane 2 x + 4 y + 3z = 12. The square of the distance between the origin and P is d 2 = x 2 + y 2 + z 2 . Consequently, d 2 = f ( x, y ) = x 2 + y 2 + (12 − 2 x − 4 y ) 2 9. To find the critical points, set f x ( x, y ) = 2 x + 9 (12 − 2 x − 4 y )(−2) = 0 and f y ( x, y ) = 2 y + 9 (12 − 2 x − 4 y )(−4) = 0 The resulting system of 2 2 equations is 13x + 8 y = 24 and 8 x + 25 y = 48, which leads to a critical point of( 24 , 29 ) . Since f xx ( x, y) = 26 , 29 48 9 f yy ( x, y ) = 9 ( 29 29 ) 9 ( 29 29 ) 50 , and f ( x, y ) = 16 , D 24 , 48 = 116 Since D 24 , 48 > 0 and f xy 9 xx ( 29 29 ) ( 29 29 ) 24 , 48 > 0, 24 , 48 yields a minimum distance. The point on the plane 2 x + 4 y + 3z = 12 that is closest to the origin is ( 24 , 29 , 36 ) 29 48 29 and this minimum distance is approximately 2.2283. 23. Let P( x, y, z ) be any point on z = x 2 + y 2 . The square of the distance between the point (1, 2, 0) and P can be expressed as d 2 = f ( x, y ) = ( x − 1)2 + ( y − 2)2 + z 2 . To find the critical points, set f x ( x, y ) = 4 x3 + 2 x + 4 xy 2 − 2 = 0 and f y ( x, y ) = 4 y 3 + 2 y + 4 x 2 y − 4 = 0. Multiplying the first equation by y and the second equation by x and summing the results leads to the equation −2 y + 4 x = 0. Thus, y = 2 x . Substituting into the first equation yields 10 x3 + x − 1 = 0, whose solution is x ≈ 0.393. Consequently, y ≈ 0.786. f xx ( x, y ) = 2 + 12 x 2 + 4 y 2 , f yy ( x, y ) = 2 + 12 y 2 + 4 x 2 , and f xy ( x, y ) = 8 xy. The value of D for the critical point (0.393, 0.786) is approximately 57 and since f xx (0.393, 0.786) > 0, (0.393, 0.786) yields a minimum distance. The point on the surface z = x 2 + y 2 is (0.393, 0.786, 0.772) and this minimum distance is approximately 1.56. 778 Section 12.8 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 36. 24. Let (x, y, z) denote a point on the cone, and s denote the distance between (x, y, z) and (1, 2, 0). s 2 = ( x − 1)2 + ( y − 2) 2 + z 2 and z 2 = x 2 + y 2 . Minimize s 2 = f ( x, y ) = ( x – 1)2 + ( y – 2)2 + ( x 2 + y 2 ), x, y in R. ⎛1 ⎞ ⎛1 ⎞ ∇f ( x, y ) = 2 2 x − 1, 2 y − 2 = 0, 0 at ⎜ , 1⎟ . At ⎜ , 1⎟ , D > 0 and f xx > 0, so local minimum. ⎝2 ⎠ ⎝2 ⎠ 5 Conclusion: Minimum distance is s = ≈ 1.5811. 2 ⎛1⎞ 25. A = ⎜ ⎟ [ y + ( y + 2 x sin α )]( x cos α ) and ⎝2⎠ ⎛1⎞ ⎛ π⎞ 2x + y = 12. Maximize A( x, α ) = 12 x cos α – 2 x 2 cos α + ⎜ ⎟ x 2 sin 2α , x in (0, 6], a in ⎜ 0, ⎟ . ⎝2⎠ ⎝ 2⎠ ⎛ π⎞ A( x, α ) = 12 cos α – 4 x cos α + 2 x sin α cos α , – 12 x sin α + 2 x 2 sin α + x 2 cos 2α = 0, 0 at ⎜ 4, ⎟ . ⎝ 6⎠ ⎛ π⎞ At ⎜ 4, ⎟ , D > 0 and Axx < 0, so local maximum, and A = 12 3 ≈ 20.78. At the boundary point of x = 6, we get ⎝ 6⎠ π π π 2π α = , A = 18. Thus, the maximum occurs for width of turned-up sides = 4”, and base angle = + = . 4 2 6 3 26. The lines are skew since there are no values of s and t that simultaneously satisfy t – 1 = 3s, 2t = s + 2, and t + 3 = 2s – 1. Minimize f, the square of the distance between points on the two lines. f ( s, t ) = (3s – t + 1)2 + ( s + 2 – 2t )2 + (2s – 1 – t – 3)2 Let ∇f ( s, t ) = 2(3s – t + 1)(3) + 2( s – 2t + 2)(1) + 2(2 s – t – 4)(2), 2(3s – t + 1)(–1) + 2( s – 2t + 2)(–2) + 2(2 s – t – 4)(–1) = 28s − 14t − 6, −14 s + 12t − 28 = 0, 0 . 5 Solve 28s – 14t – 6 = 0, –14s + 12t – 2 = 0, obtaining s = , t = 1. 7 D = f ss ftt – f st = (28)(12) – (–14)2 > 0; f ss = 28 > 0. (local minimum) 2 The nature of the problem indicates the global minimum occurs here. 2 2 2 ⎛ 5 ⎞ ⎛ 15 ⎞ ⎛ 5 ⎞ ⎛ 25 ⎞ 875 f ⎜ , 1⎟ = ⎜ ⎟ + ⎜ ⎟ + ⎜ – ⎟ = ⎝7 ⎠ ⎝ 7 ⎠ ⎝7⎠ ⎝ 7 ⎠ 49 Conclusion: The minimum distance between the lines is 875 / 7 ≈ 4.2258. 27. Let M be the maximum value of f(x, y) on the b. x y –3x + 2y + 1 polygonal region, P. Then ax + by + (c – M) = 0 –3 0 10 is a line that either contains a vertex of P or divides P into two subregions. In the latter case 0 5 11 ax + by + (c – M) is positive in one of the regions 2 3 1 and negative in the other. ax + by + (c – M) > 0 4 0 –11 contradicts that M is the maximum value of 1 –4 –10 ax + by + c on P. (Similar argument for Minimum at (4, 0) minimum.) a. x y 2x + 3y + 4 28. x y 2x + y –1 2 8 0 0 0 0 1 7 2 0 4 1 0 6 1 4 6 –3 0 –2 0 14 / 3 14 / 3 0 –4 –8 Maximum of 6 occurs at (1, 4 ) Maximum at ( −1, 2 ) Instructor’s Resource Manual Section 12.8 779 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 37. ∂f n ∂ 30. a. =∑ ( yi − mxi − b )2 ∂m i =1 ∂m n = 2∑ ( yi − mxi − b )(− xi ) i =1 ( ) n = −2∑ xi yi − mxi 2 − bxi i =1 Setting this result equal to zero yields ( ) n 0 = −2∑ xi yi − mxi 2 − bxi The edges of P are segments of the lines: i =1 1. y = 0 ( ) n 2. 4x + y = 8 0 = ∑ xi yi − mxi 2 − bxi 3. 2x + 3y = 14, and i =1 4. x = 0 or equivalently, n n n 29. z ( x, y ) = y 2 – x 2 ∑ xi yi = m∑ xi 2 + b∑ xi i =1 i =1 i =1 z ( x, y ) = –2 x, 2 y = 0, 0 at (0, 0). There are no stationary points and no singular ∂f n ∂ = ∑ ( yi − mxi − b ) points, so consider boundary points. 2 ∂b i =1 ∂b n = 2∑ ( yi − mxi − b )(−1) i =1 n = −2∑ ( yi − mxi − b ) i =1 Setting this result equal to zero yields n 0 = −2∑ ( yi − mxi − b ) i =1 On side 1: n y = 2x, so z = 4 x 2 – x 2 = 3x 2 0 = ∑ ( yi − mxi − b ) z ′( x) = 6 x = 0 if x = 0. i =1 Therefore, (0, 0) is a candidate. or equivalently, n n On side 2: y = –4x + 6, so m∑ xi + nb = ∑ yi i =1 i =1 z = (–4 x + 6)2 – x 2 = 15 x 2 – 48 x + 36. z ′( x) = 30 x – 48 = 0 if x = 1.6. n n b. nb = ∑ yi − m∑ xi Therefore, (1.6, –0.4) is a candidate. i =1 i =1 On side 3: Therefore, y = –x, so z = (– x) 2 – x 2 = 0. n n Also, all vertices are candidates. ∑ yi − m∑ xi i =1 i =1 b= x y z n 0 0 0 n n n 1.6 –0.4 –2.4 2 –2 0 ∑ xi yi = m∑ xi 2 + b∑ xi i =1 i =1 i =1 1 2 3 ⎛ n n ⎞ n Minimum value of –2.4; maximum value of 3 n ⎜ ∑ yi − m∑ xi ⎟ ∑ xi ⎜ ⎟ n ∑ xi yi = m∑ xi 2 + ⎝ i =1 i =1 ⎠ i =1 n i =1 i =1 780 Section 12.8 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 38. This simplifies into z ( x ) = 2 x 2 – 4 x + 5, x in [0, 4]. z ′( x) = 4 x – 4, n n n 1 ∑ xi yi − n ∑ xi ∑ yi so z ′( x) = 0 if x = 1. Hence, an additional critical i =1 i =1 i =1 point is (1, 0). m= 2 On hypotenuse: x = 4 – 4y n 1⎛ n ⎞ ∑ xi 2 − n ⎜ ∑ xi ⎟ ⎜ ⎟ z ( y ) = 2(4 – 4 y ) 2 + y 2 – 4(4 – 4 y ) – 2 y + 5 i =1 ⎝ i =1 ⎠ = 33 y 2 – 50 y + 21, y in [0, 1]. ∂2 f n 25 c. = 2∑ xi 2 z ′( y ) = 66 y – 50, so z ′( y ) = 0 if y = . ∂m 2 i =1 33 ⎛ 32 25 ⎞ ∂ f 2 Hence, an additional critical point is ⎜ , ⎟ . = 2n ⎝ 33 33 ⎠ ∂b 2 x y z ∂2 f n = 2∑ xi 0 0 5 ∂m∂b i =1 4 0 21 Then, by Theorem C, we have 0 1 4 ⎛ n 2⎞ 1⎛ n ⎞ D = 4n ⎜ ∑ xi 2 − ⎜ ∑ xi ⎟ ⎟ . 1 0 3 ⎜ ⎜ i =1 n ⎜ i =1 ⎟ ⎟ ⎝ ⎠ ⎟ 32 / 33 25 / 33 2.06 ⎝ ⎠ Assuming that all the xi are not the same, we Maximum value of z is 21; it occurs at (4, 0). Minimum value of z is about 2.06; it occurs at ∂ f2 find that D > 0 and >0. ⎛ 32 25 ⎞ ∂m2 ⎜ , ⎟. ⎝ 33 33 ⎠ Thus, f (m, b) is minimized. 33. Let x and y be defined as shown in Figure 4 from 31. xi yi xi2 xi yi Section 12.8. The total cost is given by 3 2 9 6 C ( x, y ) = 400 x 2 + 502 + 200(200 − x − y ) 4 3 16 12 5 4 25 20 +300 y 2 + 1002 6 4 36 24 Taking partial derivatives and setting them equal 7 5 49 35 to 0 gives ∑ 5=1 25 18 135 97 C x ( x, y ) = 200( x 2 + 502 )−1 2 (2 x) − 200 = 0 i m(135) + b(25) = (97) and m(25) + (5)b = (18). C y ( x, y ) = 150( y 2 + 1002 )−1 2 (2 y ) − 200 = 0 Solve simultaneously and obtain m = 0.7, b = 0.1. The solution of these equations is The least-squares line is y = 0.7x + 0.1. 50 100 x= ≈ 28.8675 and y = ≈ 89.4427 32. z = 2 x 2 + y 2 – 4 x – 2 y + 5, so 3 1.25 We now apply the second derivative test: ∇z = 4 x − 4, 2 y − 2 = 0. ∇z = 0 at (1, 1) which is outside the region. 400 x 2 + 502 − 400 x 2 x 2 + 502 C xx ( x, y ) = Therefore, extreme values occur on the boundary. x 2 + 502 Three critical points are the vertices of the 300 y 2 + 1002 − 300 y 2 y 2 + 1002 triangle, (0, 0), (0, 1), and (4, 0). Others may C yy ( x, y ) = occur on the interior of a side of the triangle. y 2 + 1002 C xy ( x, y ) = 0 Evaluated at x = 50 3 and y = 100 1.25 , D ≈ (5.196)(1.24) − 0 > 0 2 Thus, a local minimum occurs with ( ) On vertical side: x = 0 C 50 3 ,100 1.25 ≈ \$79, 681 z ( y ) = y 2 – 2 y + 5, y = [0, 1]. z ′( y ) = 2 y – 2, so We must also check the boundary. When x = 0, z ′( y ) = 0 if y = 1. Hence, no additional critical point. C1 ( y ) = C (0, y ) = 200(200 − y ) + 300 y 2 + 1002 On horizontal side: y = 0 and when y = 0, Instructor’s Resource Manual Section 12.8 781 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 40. 38. The cost function in three variables is 1 C (l , w, h) = 4lw + 2lh + 2wh + 6l + 6 w + 4h, T ′( y ) = –1 – 2 y = 0 if y = – , so on the 2 60 boundary, critical points occur where y is where lwh = 60. Substituting h = yields lw 1 –1, – , 1. 120 240 2 C (l , w) = 4lw + + 6l + 6 w + with w lw ⎛ 1⎞ Thus, points to consider are ⎜ 0, ⎟ , (0, –1), l > 0 and w > 0. ⎝ 2⎠ 120 240 ⎛ 3 1⎞ ⎛ 3 1 ⎞ Cl (l , w) = 4w − +6− =0 ⎜ 2 wl 2 ⎜ 2 , – 2 ⎟ , ⎜ – 2 , – 2 ⎟ and (0, 1). Substituting ⎟ ⎜ ⎟ l ⎝ ⎠ ⎝ ⎠ 120 240 these into T(x, y) yields that the coldest spot is Cw (l , w) = 4l − +6− =0 w 2 lw2 ⎛ 1⎞ 1 ⎜ 0, ⎟ where the temperature is – , and there Multiplying both sides of the first equation by ⎝ 2⎠ 4 wl 2 , multiplying both sides of the second ⎛ 3 1⎞ 2 is a tie for the hottest spot at ⎜ ± ⎜ 2 , – 2 ⎟ where ⎟ equation by lw , and subtracting the resulting ⎝ ⎠ equations produces −120w + 120l = 0 or l = w. 9 120 240 the temperature is . Consequently, 4w − +6− = 0 or 4 2 w w3 2w4 + 3w3 − 60 w − 120 = 0 Using a CAS, this 40. Let x 2 , y 2 , z 2 denote the areas enclosed by the equation yields w ≈ 3.2134 circle, and the two squares, respectively. Then 240 480 240 480 x Cll (l , w) = + ; Cww (l , w) = + ; the radius of the circle is , and the edges of l 3 wl 3 w3 lw3 π 240 the two squares are y and z, respectively. We Clw (l , w) = 4 + ; Using the critical point l 2 w2 wish to optimize A(x, y, z) = x 2 + y 2 + z 2 , (3.2134, 3.2134), D ≈ 131.44 > 0 ⎛ x ⎞ subject to 2π ⎜ ⎟ + 4 y + 4 z = k , or Thus, w = l ≈ 3.2 yields a minimum. The ⎝ π⎠ minimum cost involved with making this box is approximately \$177.79. This minimum cost equivalently 2 π x + 4 y + 4 z = k , with each of x, occurs when the length and width are y, and z nonnegative. Geometrically: we seek the approximately 3.2 feet and the height is smallest and largest of all spheres with center at approximately 5.8 feet. the origin and some point in common with the triangular region indicated. 39. T ( x, y ) = 2 x 2 + y 2 – y ∇T = 4 x , 2 y – 1 = 0 1 ⎛ 1⎞ If x = 0 and y = , so ⎜ 0, ⎟ is the only interior 2 ⎝ 2⎠ critical point. k k Since > , the largest sphere will intersect 2 π 4 ⎛ k ⎞ the region only at point ⎜ , 0, 0 ⎟ and will ⎝2 π ⎠ k thus have radius . Thus A will be maximum On the boundary x 2 = 1 – y 2 , so T is a function 2 π k of y there. if x = , y = z = 0 (all of the wire goes into T ( y ) = 2(1 – y 2 ) + y 2 – y = 2 – y – y 2 , 2 π the circle). The smallest sphere will be tangent to y = [–1, 1] the triangle. The point of tangency is on the normal line through the origin, Instructor’s Resource Manual Section 12.8 783 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 41. x, y , z = t π , 2, 2 . Substituting x = π , y = 2, z = 2 into the equation of the plane yields k the value t = , so the minimum value of 2(π + 8) k π A is obtained for the values of x = , 2(π + 8) k y=z= . Thus the circle will have radius π+8 ⎡ k π ⎤ ⎢ 2( π+8) ⎥ ⎣ ⎦= k , and the squares will each π 2(π + 8) k have sides . Therefore, the circle will use (π + 8) πk units and the squares will each use (π + 8) 4k units. (π + 8) [Note: sum of the three lengths is k.] 41. Without loss of generality we will assume that α ≤ β ≤ γ . We will consider it intuitively clear that for a triangle of maximum area the center of the circle will be inside or on the boundary of the triangle; i.e., α , β , and γ are in the interval [0,π ] . Along with α + β + γ = 2π, this implies that α + β ≥ π. 1 2 The area of an isosceles triangle with congruent sides of length r and included angle θ is r sin θ . 2 1 2 1 1 Area(ΔABC ) = r sin α + r 2 sin β + r 2 sin γ 2 2 2 1 2 = r (sin α + sin β + sin[2π − (α + β )] 2 1 2 = r [sin α + sin β − sin(α + β )] 2 Area(ΔABC ) will be maximum if (*) A(α , β ) = sin α + sin β − sin(α + β ) is maximum. Restrictions are 0 ≤ α ≤ β ≤ π, and α + β ≥ π. ⎛π π⎞ Three critical points are the vertices of the triangular domain of A : ⎜ , ⎟ , (0, π), and (π, π). We will now search ⎝2 2⎠ for others. ΔA(α , β ) = cos α − cos(α + β ), cos β − cos(α + β ) = 0 if cos α = cos(α + β ) = cos β . Therefore, cos α = cos β , so α = β [due to the restrictions stated]. Then cos α = cos(α + α ) = cos 2α = 2 cos 2 α − 1, so cos α = 2 cos 2 α − 1. Solve for α : 2 cos 2 α − cos α − 1 = 0; (2 cos α + 1)(cos α − 1) = 0; 1 2π cos α = − or cos α = 1; α = or α = 0. 2 3 784 Section 12.8 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 42. ⎛ 2π 2π ⎞ (We are still in the case where α = β .) ⎜ , ⎟ is a new critical point, but (0, 0) is out of the domain of A. ⎝ 3 3 ⎠ There are no critical points in the interior of the domain of A. On the β = π edge of the domain of A; A(α ) = sin α − sin(α − π) = 2sin α so A′(α ) = 2 cos α . π ⎛π ⎞ A′(α ) = 0 if α =. ⎜ , π ⎟ is a new critical point. 2 ⎝2 ⎠ On the β = π − α edge of the domain of A: A(α ) = sin α + sin(π − α ) − sin(2α − π) = 2sin α + sin 2α , so A′(α ) = 2 cos α + 2 cos 2α = 2[cos α + (2 cos 2 α − 1)] = 2(2 cos α − 1)(cos α + 1) . 1 π A′(α ) = 0 if cos α = or cos α = −1, so α = or α = π . 2 3 ⎛ π 2π ⎞ ⎜ , ⎟ and (π ,0) are outside the domain of A. ⎝3 3 ⎠ (The critical points are indicated on the graph of the domain of A.) α β A π π 2 2 2 0 π 0 π π 0 2π 2π 3 3 Maximum value of A. The triangle is equilateral. 3 3 2 π π 2 2 42. If the plane through (a, b, c) is expressed as 1 1 1 Ax + By + Cz = 1, then the intercepts are , , ; volume A B C ⎛ 1 ⎞ ⎡⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎤ ⎛ 1 ⎞ 1 of tetrahedron is V = ⎜ ⎟ ⎢⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎥ ⎜ ⎟ = . ⎝ 3 ⎠ ⎣⎝ 2 ⎠ ⎝ A ⎠ ⎝ B ⎠ ⎦ ⎝ C ⎠ 6 ABC To maximize V subject to Aa + Bb + Cc = 1 is equivalent to maximizing z = ABC subject to Aa + Bb + Cc = 1. 1 − aA − bB AB(1 − aA − bB) C= , so z = . c c ⎛1⎞ 1 1 ⎡ 1⎤ ∇z = ⎜ ⎟ B − 2aAB − bB 2 , A − 2bAB − aA2 = 0 if A = , B = ⎢C = 3c ⎥ . ⎝c⎠ 3a 3b ⎣ ⎦ ⎛ 1 1 ⎞ ⎜ , ⎟ is the only critical point in the first quadrant. The second partials test yields that z is maximum at this ⎝ 3a 3b ⎠ 1 1 1 x y z point. The plane is x + y + z = 1, or + + = 3. 3a 3b 3c a b c 1 9abc The volume of the first quadrant tetrahedron formed by the plane is = . ⎡6 ⎣ 1 1 3a 3b 3c ⎦ 1 ⎤ 2 ( )( )( ) 43. Local max: f(1.75, 0) = 1.15 46. Global max: f (0, 0) = 1 Global max: f(–3.8, 0) = 2.30 Global min: f (2, −2) = f (−2, 2) = e−9 44. Global max: f(0, 1) = 0.5 ≈ 0.00012341 Global min: f(0, –1) = –0.5 47. Global max: f(1.13, 0.79) = f(1.13, –0.79) = 0.53 45. Global min: f(0, 1) = f(0, –1) = –0.12 Global min: f(–1.13, 0.79) = f(–1.13, –0.79) = –0.53 Instructor’s Resource Manual Section 12.8 785 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 43. 48. No global maximum or global minimum 52. Global max: f(–5.12, –4.92) = 1071 Global min: f(5.24, –4.96) = –658 49. Global max: f (3,3) = f (−3,3) ≈ 74.9225 Global min: f (1.5708, 0) = f ( −1.5708, 0) = −8 53. Global max: f(2.1, 2.1) = 3.5 Global min: f(4.2, 4.2) = –3.5 50. Global max: f(1, 43, 0) = 0.13 Global min: f(–1.82, 0) = –0.23 51. Global max: f(0.67, 0) = 5.06 Global min: f(–0.75, 0) = –3.54 54. a. 1 k (α , β ) = [80sin α + 60sin β + 48sin(2π − α − β )] 2 = 40sin α + 30sin β − 24sin(α + β ) L(α , β ) = (164 − 160 cos α )1/ 2 + (136 − 120 cos β )1/ 2 + (100 − 96 cos(α + β ))1/ 2 b. (1.95, 2.04) c. (2.26, 2.07) 12.9 Concepts Review 3. Let ∇f ( x, y ) = λ∇g ( x, y ), where 1. free; constrained g ( x, y ) = x 2 + y 2 – 1 = 0. 8 x - 4 y , -4 x + 2 y = λ 2 x, 2 y 2. parallel 1. 4x – 2y = λx 3. g(x, y) = 0 2. –2x + y = λy 3. x 2 + y 2 = 1 4. (2, 2) 4. 0 = λx + 2λy (From equations 1 and 2) 5. λ = 0 or x + 2y = 0 (4) Problem Set 12.9 λ = 0: 6. y = 2x (1) 1 7. x = ± (6, 3) 1. 2 x, 2 y = λ y , x 5 2x = λy, 2y = λx, xy = 3 2 Critical points are 8. y = ± (7, 6) 5 (± ) ( 3, ± 3 , f ± 3, ± 3 = 6. ) x + 2y = 0: 9. x = –2y It is not clear whether 6 is the minimum or 1 10. y = ± (9, 3) maximum, so take any other point on xy = 3, for 5 example (1, 3). f(1, 3) = 10, so 6 is the minimum 2 value. 11. x = (10, 9) 5 2. y, x = λ 8 x, 18 y ⎛ 1 2 ⎞ ⎛ 1 2 ⎞ Critical points: ⎜ , ⎟, ⎜ – ,– ⎟, y = 8λx, x = 18λy, 4 x 2 + 9 y 2 = 36 ⎝ 5 5⎠ ⎝ 5 5⎠ ⎛ 2 1 ⎞ ⎛ 2 1 ⎞ ⎛ 3 2 ⎞ ⎛ 3 2 ⎞ ⎜ ,– ⎟, ⎜ – , ⎟ Critical points are ⎜ ,± ⎟, ⎜ – ,± ⎟. ⎝ 5 5⎠ ⎝ 5 5⎠ ⎝ 2 2⎠ ⎝ 2 2⎠ f(x, y) is 0 at the first two critical points and 5 at ⎛ 3 2 ⎞ the last two. Therefore, the maximum value of Maximum value of 3 occurs at ⎜ ± ,± ⎟. ⎝ 2 2⎠ f(x, y) is 5. 786 Section 12.9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 44. 4. 2 x + 4 y, 4 x + 2 y = λ 1, -1 9. Let l and w denote the dimensions of the base, h the depth. Maximize V(l, w, h) = lwh subject to 2x + 4y = λ, 4x + 2y = –λ, x – y = 6 0.601w + 0.20(lw + 2lh + 2wh) = 12, which Critical point is (3, –3). simplifies to 21w + lh + wh = 30, or g(l, w, h) = 2lw + lh + wh – 30. 5. 2 x, 2 y, 2 z = λ 1,3, -2 Let ∇V (l , w, h) = λ∇g (l , w, h); 2x = λ, 2y = 3λ, 2z = –2λ, x + 3y – 2z = 12 wh, lh, lw = λ 2 w + h, 2l + h, l + w . ⎛ 6 18 12 ⎞ Critical point is ⎜ , , – ⎟ . 1. wh = λ(2w + h) ⎝7 7 7⎠ 2. lh = λ(2l + h) ⎛ 6 18 12 ⎞ 72 3. lw = λ(l + w) f ⎜ , , – ⎟= is the minimum. ⎝7 7 7⎠ 7 4. 2lw + lh + wh = 30 5. (w – l)h = 2λ(w – l) (1, 2) 6. Let ∇f ( x, y, z ) = λ∇g ( x, y, z ), where 6. w = l or h = 2λ g ( x, y, z ) = 2 x 2 + y 2 – 3z = 0. w = 1: 7. l = 2λ = w (3) Note: w ≠ 0 , for then V = 0 . 4, −2,3 = λ 4 x, 2 y, −3 8. h = 4λ (7, 2) 1. 4 = 4λx 5 2. –2 = 2λy 9. λ = (7, 8, 4) 2 3. 3 = –3λ 10. l = w = 5, h = 2 5 (9, 7, 8) 4. 2 x 2 + y 2 – 3 z = 0 h = 2λ: 5. λ = –1 (3) 11. λ = 0 (2) 6. x = –1, y = 1 (5, 1, 2) 12. l = w = h = 0 (11, 1 – 3) 7. z = 1 (6, 4) (Not possible since this does not satisfy 4.) Therefore, (–1, 1, 1) is a critical point, and f(–1, 1, 1) = –3. (–3 is the minimum rather than ( ) 5, 5, 2 5 is a critical point and maximum since other points satisfying g = 0 have larger values of f. For example, g(1, 1, 1) = 0, and V ( ) 5, 5, 2 5 = 10 5 ≈ 22.36 ft3 is the f(1, 1, 1) = 5.) maximum volume (rather than the minimum volume since, for example, g(1, 1, 14) = 30 and 7. Let l and w denote the dimensions of the base, h V(1, 1, 14) = 14 which is less than 22.36). denote the depth. Maximize V(l, w, h) = lwh subject to g(l, w, h) = lw + 2lh + 2wh = 48. 10. Minimize the square of the distance, wh, lh, lw = λ w + 2h, l + 2h, 2l + 2 w f ( x, y , z ) = x 2 + y 2 + z 2 , subject to wh = λ(w + 2h), lh = λ(l + 2h), lw = λ(2l + 2w), g ( x, y, z ) = x 2 y – z 2 + 9 = 0. lw + 2lh + 2wh = 48 Critical point is (4, 4, 2). 2 x, 2 y, 2 z = λ 2 xy, x 2 , – 2 z V(4, 4, 2) = 32 is the maximum. (V(11, 2, 1) = 22, for example.) 2x = 2λxy, 2 y = λ x 2 , 2z = –2λz, 8. Minimize the square of the distance to the plane, x2 y – z 2 + 9 = 0 Critical points are (0, 0, ±3) [case x = 0]; f ( x, y , z ) = x 2 + y 2 + z 2 , subject to x + 3y – 2z – 4 = 0. (± ) 2, – 1, ± 7 [case x ≠ 0 , λ = –1]; and 2 x, 2 y, 2 z = λ 1, 3, -2 2x = λ, 2y = 3λ, 2z = –2λ, x + 3y – 2z = 4 ( ±3 6 2 / 9, − 3 9 / 2, 0) [case x ≠ 0, λ ≠ −1 ]. Evaluating f at each of these eight points yields 9 ⎛2 6 4⎞ Critical point is ⎜ , , – ⎟ . The nature of the (case x = 0), 10 (case x ≠ 0 , λ = –1), and ⎝7 7 7⎠ problem indicates that this will give a minimum 33 3 2 2 ( ) 2 9 (case x ≠ 0, λ ≠ −1 ). The latter is rather than a maximum. The least distance to the 1/ 2 1/ 2 the smallest, so the least distance between the ⎡ ⎛2 6 4 ⎞⎤ ⎛8⎞ 3 plane is ⎢ f ⎜ , , – ⎟ ⎥ =⎜ ⎟ ≈ 1.0690. origin and the surface is 36 ≈ 2.8596. ⎣ ⎝7 7 7 ⎠⎦ ⎝7⎠ 4 Instructor’s Resource Manual Section 12.9 787 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 45. 11. Maximize f(x, y, z) = xyz, subject to 15. Minimize d 2 = f ( x, y, z ) 2 2 2 2 2 2 2 2 2 2 2 2 g ( x, y , z ) = b c x + a c y + a b z – a b c = ( x − 1) 2 + ( y − 2) 2 + z 2 with the constraint =0 g ( x, y, z ) = x 2 + y 2 − z = 0; yz , xz , xy = λ 2b 2 c 2 x, 2a 2 c 2 y, 2a 2 b 2 z 2 x − 2, 2 y − 4, 2 z = λ 2 x, 2 y, −1 yz = 2b 2 c 2 x, xz = 2a 2 c 2 y, xy = 2a 2 b 2 z , Setting up, solving each equation for λ , and 2 2 2 2 2 2 2 2 2 2 2 2 b c x +a c y +a b z = a b c substituting into equation x 2 + y 2 − z = 0 ⎛ a b c ⎞ produces λ ≈ −1.5445; The resulting critical Critical point is ⎜ , , ⎟. ⎝ 3 3 3⎠ point is approximately (0.393, 0.786, 0.772). The nature of this problem indicates this will give a ⎛ a b c ⎞ 8abc V⎜ , , ⎟= , which is the minimum value rather than a maximum. The ⎝ 3 3 3⎠ 3 3 minimum distance is approximately 1.5616. maximum. 16. Minimize d 2 = f ( x, y, z ) 12. Maximize V(x, y, z) = xyz, subject to x y z = ( x − 1) 2 + ( y − 2) 2 + z 2 with the constraint g ( x, y, z ) = + + –1 = 0. Let a b c g ( x, y , z ) = x 2 + y 2 − z 2 = 0 ∇V ( x, y, z ) = λ∇g ( x, y, z ), so 2 x − 2, 2 y − 4, 2 z = λ 2 x, 2 y, −2 z 1 1 1 yz , xz , xy = λ , , . Then λ = −1, x = 1 , y = 1, z = ± 5 ; The critical points a b c 2 2 λx λ y λz a = b = c (each equals xyz). are ( 1 ,1, 5 2 2 ) and ( 1 ,1, − 5 2 2 ) which both lead to λ ≠ 0 since λ = 0 would imply x = y = z = 0 10 a minimum distance of . which would not satisfy the constraint. 2 x y z Thus, = = . These along with the 17. (See problem 37, section 12.8). Let the a b c dimensions of the box be l, w, and h . Then the a b c cost of the box is constraints yield x = , y = , z = . 3 3 3 .25(2hl + 2hw + lw) + .4(lw) or The maximum value of V = abc . C (l , w, h) = .5hl + .5hw + .65lw . 27 We want to minimize C subject to the constraint lhw = 2 ; set V (l , h, w) = lwh − 2 . 13. Maximize f(x,y,z) = x + y + z with the constraint Now: g ( x, y, z ) = x 2 + y 2 + z 2 − 81 = 0. Let ∇C (l , w, h) = (.5h + .65w)i + (.5h + .65l ) j + .5(l + w)k ∇f ( x, y, z ) = λ∇g ( x, y, z ) , so and 1,1,1 = λ 2 x, 2 y, 2 z ; Thus, x = y = z ∇V (l , w, h) = wh i + lh j + lw k and 3x2 = 81 or x = y = z = ±3 3. Thus the Lagrange equations are .5h + .65w = λ wh (1) The maximum value of f is 9 3 when .5h + .65l = λ lh (2) x, y, z = 3 3,3 3,3 3 .5(l + w) = λ lw (3) lwh = 2 (4) 14. Minimize d 2 = f ( x, y, z ) = x 2 + y 2 + z 2 with Solving (4) for h and putting the result in (1) and (2), we get the constraint g ( x, y, z ) = 2 x + 4 y + 3z − 12 = 0 1 2λ ∇f ( x , y , z ) = λ ∇ g ( x , y , z ) + .65w = (5) lw l 2 x, 2 y, 2 z = λ 2,3, 4 ; 1 2λ + .65l = (6) 2 x = 2λ ; 2 y = 4λ ; 2 z = 3λ leads to a critical lw w point of ( 24 , 29 , 36 ) The nature of the problem 29 48 29 Multiply (5) by l and (6) by w to get 1 indicates this will give a minimum rather than a + .65lw = 2λ (7) maximum value. The minimum distance is w 1 24 2 2 2 + 29 + 36 ≈ 2.2283 48 + .65lw = 2λ (8) 29 29 l 788 Section 12.9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 46. from which we conclude that l = w . Putting this equation (3), we have result into (3) we have ⎛ w +1⎞ 2w + 2 w + 4 = λ w2 or λ = 4 ⎜ ⎟. l = λl 2 (9) ⎝ w2 ⎠ 1 Therefore, from equation (1), we have Since V ≠ 0, l ≠ 0 and (9) tells us that l = ; λ 120 ⎛ w + 1 ⎞ ⎛ 60 ⎞ + 4w + 6 = 4 ⎜ ⎟ w⎜ ⎟ or 2 thus l = 1 , w=l = 1 2 = 2λ 2 . , h= w ⎝ w2 ⎠ ⎝ w 2 ⎠ λ λ lw (multiplying through by w3 and simplifying) Putting these results into equation (1), we conclude 2w4 + 3w3 − 60 w − 120 = 0 ⎛1⎞ ⎛1⎞ Using one of several techniques available to .5(2λ 2 ) + .65 ⎜ ⎟ = λ ⎜ ⎟ (2λ 2 ) or solve, we conclude that w = l = 3.213 and ⎝λ⎠ ⎝λ⎠ 60 ⎛1⎞ h= ≈ 5.812 . .65 ⎜ ⎟ = λ 2 . (3.213)2 ⎝λ⎠ Hence: λ = 3 .65 ≈ .866 , so the minimum cost is 19. (See problem 40, section 12.8) obtained when: Let 1 c = circumference of circle l = w = ≈ 1.154 and h = 2λ 2 ≈ 1.5 p = perimeter of first square λ q = perimeter of second square 18. (See problem 38, section 12.8). Let the Then the sum of the areas is dimensions of the box be l, w, and h . Then the c2 p2 q2 1 ⎡ c2 p2 q2 ⎤ cost of the box is A(c, p, q ) = + + = ⎢ + + ⎥ 4π 16 16 4 ⎢ π ⎣ 4 4 ⎥ ⎦ 1(2hl + 2hw) + 4(lw) + 3(2l + 2 w) + 4h or so we wish to maximize and minimize C (l , w, h) = 2hl + 2hw + 4lw + 6l + 6w + 4h . c2 p2 q2 We want to minimize C subject to the constraint A(c, p, q ) = + + subject to the lhw = 60 ; set V (l , h, w) = lwh − 60 . π 4 4 constraint L(c, p, q) = c + p + q − k = 0 . Now: ∇C ( l , w, h ) = ( 2h + 4 w + 6 ) i + ( 2h + 4l + 6 ) j and Now 2c p q + ( 2l + 2w + 4 ) k ∇A(c, p, q ) = i + j + k π 2 2 ∇V (l , w, h) = wh i + lh j + lw k ∇L ( c , p , q ) = i + j + k Thus the Lagrange equations are so the Lagrange equations are 2h + 4 w + 6 = λ wh (1) 2c 2h + 4l + 6 = λ lh (2) =λ (1) π 2l + 2 w + 4 = λ lw (3) p lwh = 60 (4) =λ (2) 2 60 Solving (4) for h = and putting the result in q =λ (3) lw 2 (1) and (2), we get c+ p+q = k (4) 120 60λ + 4w + 6 = (5) Putting (1), (2) and (3) into (4) we get lw l π 2k 120 60λ (4 + )λ = k or λ = + 4l + 6 = (6) 2 8+π lw w Therefore: Multiply (5) by l and (6) by w to get πk 120 c0 = ≈ 0.282k + 4lw + 6l = 60λ (7) 8+π w 4k 120 p0 = ≈ 0.359k + 4lw + 6 w = 60λ (8) 8+π l 4k from which we conclude that q0 = ≈ 0.359k 8+π 120 120 + 6w = + 6l or (l − w)(lw + 20) = 0 . Now A(c0 , p0 , q0 ) ≈ 0.0224k 2 while l w Since lw cannot be negative (= −20 ), we A(k , 0, 0) ≈ .079k 2 , so we conclude that conclude that l = w ; putting this result into Instructor’s Resource Manual Section 12.9 789 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 47. A(c0 , p0 , q0 ) is a minimum value. There is also −a −b −c and ∇g ( A, B, C ) = i+ j+ k . Thus 2 2 a maximum value (see problem 40, section 12.8) A B C2 but our Lagrange approach does not capture this. the Lagrange equations are The reason is that the maximum exists because c, A −λ a p, and q must all be ≥ 0 . Our constraint, = (1) however, does not require this and allows ABC A2 negative values for any or all of the variables. B −λ b = (2) Under these conditions, there is no global ABC B2 maximum. C −λ c = (3) 20. (See problem 42, section 12.8). Let P be the ABC C2 x y z plane + + = 1 . This plane will cross the a b c + + =1 (4) A B C A B C first octant, forming a triangle, T, in P; the From (1) – (3) we have vertices of this triangle occur where P intersects the coordinate axes. They are: − A3 − B3 −C 3 λ ABC = = = (5) Vx = ( A, 0, 0), V y = (0, B, 0), Vz = (0, 0, C ) . a b c Solving in pairs we get a. Define the vectors g = − A, B, 0 and ⎛ b⎞ ⎛ c⎞ B = ⎜ 3 ⎟ A, C = ⎜ 3 ⎟ A ⎜ a⎟ ⎜ a⎟ (6) h = − A, 0, C . From example 3 in 11.4, we ⎝ ⎠ ⎝ ⎠ and putting these results into (4) we obtain know the area of T is 3 3 1 1 A = a + ab 2 + ac 2 g×h = ( BC ) 2 + ( AC ) 2 + ( AB)2 . 2 2 = 3 a ⎛ a 2 + b2 + c2 ⎞ 3 3 3 ⎜ ⎟ ⎝ ⎠ b. The height of the tetrahedron in question is Similarly, we have the distance is the distance between (0,0,0) 3 3 and P . By example 10 in 11.3, this distance B = a 2 b + b + bc 2 is = 3 b ⎛ a 2 + b2 + c2 ⎞ 3 3 3 ⎜ ⎟ 1 ( ABC ) 2 ⎝ ⎠ h= = 1 + 1 + 1 ( BC ) + ( AC )2 + ( AB) 2 2 3 C = a 2 c + b2 c + c 3 A2 B 2 C 2 = 3 c ⎛ a 2 + b2 + c2 ⎞ 3 3 3 ⎜ ⎟ c. Finally, the volume of the tetrahedron is ⎝ ⎠ 1 Finally, the volume of the tetrahedron is h(area of T ) , or 3 3 ⎛3 3 3 ⎞ 3 abc ⎜ a 2 + b 2 + c 2 ⎟ 1 ABC ⎝ ⎠ . V ( A, B, C ) = ( BC )2 + ( AC )2 + ( AB )2 = 6 6 6 ⎡ ( ABC ) 2 ⎤ ⋅⎢ ⎥ 21. Finding critical points on the interior first: ⎢ ( BC ) 2 + ( AC ) 2 + ( AB )2 ⎥ ∂f ∂f ⎣ ⎦ =1≠ 0 = 1 ≠ 0; There are no critical ∂x ∂y 1 1 That is, V ( A, B, C ) = ( ABC ) 2 = ABC . points on the interior. Finding critical points on 6 6 the boundary: ∇f ( x, y ) = λ∇g ( x, y ); Hence we want to minimize V ( A, B, C ) = ABC subject to the constraint 1,1 = λ 2 x, 2 y ; The solution to the system a b c 1 = λ ⋅ 2 x, 1 = λ ⋅ 2 y , x 2 + y 2 = 1 is λ = ± 1 , + + = 1 ; define 2 A B C x=± 1 , y=± 1 The four critical points are a b c 2 2 g ( A, B, C ) = + + − 1 . Now A B C ( 1 ,± 1 2 ) and ( − , ± ). 2 1 2 1 2 ∇V ( A, B, C ) = A i+ B j+ C k f ( , ) = 10 + 2 is the maximum value. 1 1 2 2 ABC ABC ABC f ( − , − ) = 10 − 2 is the minimum value. 1 1 2 2 790 Section 12.9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 48. 22. Finding critical points on the interior: ∂f = 1 − y = 0 ⇒ y = 1; { boundary S = ( x, y ): x4 2 + y2 9 } = 1 ; this is done ∂x using Lagrange multipliers. Let ∂f x2 y 2 = 1 − x = 0 ⇒ x = 1; The only critical point on g ( x, y ) = + − 1 ; then ∂y 4 9 the interior is c1 = (1,1). Finding critical points 1 −2 xy ∇f ( x, y ) = i+ j and 2 on the boundary: Solve the system of equations 1+ y (1 + y 2 )2 1 − y = λ ⋅ 2 x; 1 − x = λ ⋅ 2 y; x 2 + y 2 = 9 x 2y ∇g ( x , y ) = i + j Using substitution, it can be found that the 2 9 critical points on the boundary are The Lagrange equations are ⎛ 3 3 ⎞ ⎛ 3 3 ⎞ 1 λx c2 = ⎜ , ⎟ , c3 = ⎜ − ,− ⎟, = (1) ⎝ 2 2⎠ ⎝ 2 2⎠ 2 2 1+ y c4 = (2.56155, -1,56155), −2 xy 2λ y = (2) c5 = (-1.56155, 2.56155) 2 2 (1 + y ) 9 The maximum value of 5 is obtained substituting 9 x 2 + 4 y 2 = 36 (3) either c4 or c5 into f. The minimum value of Putting (1) into (2) yields about -8.7426 is obtained by substituting c3 into −λ 2 x3 y 2λ y f. = (4) 2 9 23. Finding critical points on the interior: One solution to (4) is y = 0 which yields, from ∂f ∂f (3), x = ±2 . Thus (2, 0) and (−2, 0) are = 2 x + 3 − y = 0; = 2y − x = 0 ∂x ∂y candidates for optimization points. The solution to this system is the only critical If y ≠ 0 , (4) can be reduced to point on the interior, c1 = (-2,-1). − λ 2 x 3 2λ Critical points on the boundary will come from = (5) 2 9 the solutions to the following system of −4 equations: so that λ = . Putting this result into (1) 2 x + 3 − y = λ ⋅ 2 x, 2 y − x = λ ⋅ 2 y , 9 x3 1 2 x 2 + y 2 = 9 . From the solutions to this system, yields =− , which has no solutions 2 the critical points are c2 = (0,3), 1+ y 9 x2 (left side always +, right side always -). ⎛3 3 3⎞ ⎛ 3 3 3⎞ Therefore the only two candidates for max/min c3 = ⎜⎜ 2 , − ⎟ , c4 = ⎜ − ⎜ 2 ,− 2 ⎟ ⎝ 2⎟⎠ ⎝ ⎟ ⎠ are (2, 0) and (−2, 0) . Since f (2, 0) = 2 and f (c1 ) = −3, f (c2 ) = 9, f (c3 ) ≈ 20.6913, f (−2, 0) = −2 we conclude that the max value of f (c4 ) ≈ −2.6913 The max value of f is f on S is 2 and the min value is −2 . ≈ 20.6913 and the min value is -3. ∂f ∂f = = 2(1 + x + y ) = 0 ⇒ x + y = −1 { } 2 2 25. 24. f ( x, y ) = x on the set S = ( x, ≤1 y ): x4 + y9 ∂x ∂y 1 + y2 There is no minimum or maximum value on the We first find the max and min for f on the set interior since there are an infinite number of { } 2 critical points. The critical points on the y2 S = ( x, y ): x4 + 9 < 1 using the methods of boundary will come from the solutions to the following system of equations: section 12.8: 1 1 −2 xy 2(1 + x + y ) = λ ⋅ x ∇f ( x, y ) = i+ j so setting 2 2 1+ y (1 + y 2 )2 1 2(1 + x + y ) = λ ⋅ y 1 8 ∇f ( x, y ) = 0 we have = 0 (impossible). 1 + y2 Solving these two equations for λ leads to y = − x − 1 or y = 4 x . Together with the Thus f has no max or min on S . x2 y2 We now look for the max and min of f on the constraint + − 1 = 0 leads to the critical 4 16 Instructor’s Resource Manual Section 12.9 791 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 49. ⎛ −1 − 2 19 −4 + 2 19 ⎞ ⎛θ ⎞ points on the boundary: ⎜ , ⎟, 2r sin ⎜ ⎟ . Then we wish to maximize ⎜ 5 5 ⎟ ⎝2⎠ ⎝ ⎠ ⎡ ⎛α ⎞ ⎛β⎞ ⎛ γ ⎞⎤ ⎛ −1 + 2 19 −4 − 2 19 ⎞ ⎛ 2 8 ⎞ P (α , β , γ ) = 2r ⎢sin ⎜ ⎟ + sin ⎜ ⎟ + sin ⎜ ⎟ ⎥ ⎜ , ⎟ , ⎜− ,− ⎟ and ⎣ ⎝2⎠ ⎝2⎠ ⎝ 2 ⎠⎦ ⎜ 5 5 ⎟ ⎝ 5 5⎠ ⎝ ⎠ subject to g(α, β, γ) = α + β + γ − 2π = 0 = 0. ⎛ 2 8 ⎞ ⎜ , ⎟ . Respectively, the maximum value is ⎛α ⎞ ⎛β⎞ ⎛γ ⎞ Let r cos ⎜ ⎟ , cos ⎜ ⎟ , cos ⎜ ⎟ = λ 1, 1, 1 . ⎝ 5 5⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ≈ 29.9443 and the minimum value is 0. ⎛α ⎞ ⎛β⎞ ⎛γ ⎞ Then λ = r cos ⎜ ⎟ = r cos ⎜ ⎟ = r cos ⎜ ⎟ , so 26. It is clear that the maximum will occur for a ⎝2⎠ ⎝2⎠ ⎝2⎠ triangle which contains the center of the circle. ⎛ α β γ ⎞ (With this observation in mind, there are α = β = γ ⎜ since + + = π ⎟ . ⎝ 2 2 2 ⎠ additional constraints: 0 < α < π , 0 < β < π , 2π 2π 0 > γ < π .) 3αa = 2 π , so α = ; then β = γ = . Note that in an isosceles triangle, the side 3 3 opposite the angle θ which is between the congruent sides of length r has length 27. Let α + β + γ = 1, α > 0, β > 0, and γ > 0. Maximize P ( x, y, z ) = kxα y β zγ , subject to g(x, y, z) = ax + by + cz – d = 0. Let ∇P ( x, y, z ) = λ∇g ( x, y, z ). Then kα xα –1 y β zγ , k β xα y β –1 zγ , kγ xα y β zγ –1 = λ a, b, c . λ ax λ by λ cz Therefore, = = (since each equals kxα y β zγ ). α β γ λ ≠ 0 since λ = 0 would imply x = y = z = 0 which would imply P = 0. ax by cz Therefore, = = (*). α β γ ⎛ ax ⎞ ⎛ by ⎞ ⎛ cz ⎞ The constraints ax + by + cz = d in the form α ⎜ ⎟ + β ⎜ ⎟ + γ ⎜ ⎟ = d becomes ⎝α ⎠ ⎝β ⎠ ⎝γ ⎠ ⎛ ax ⎞ ⎛ ax ⎞ ⎛ ax ⎞ α ⎜ ⎟ + β ⎜ ⎟ + γ ⎜ ⎟ = d , using (*). ⎝α ⎠ ⎝α ⎠ ⎝α ⎠ ⎛ ax ⎞ ax Then (α + β + γ ) ⎜ ⎟ = d , or = d (since α + β + γ = 1). ⎝ α ⎠ α αd βd γd x= (**); y = and z = then following using (*) and (**). a b c Since there is only one interior critical point, and since P is 0 on the boundary, P is maximum when αd βd γd x= , y= ,z= . a b c 28. Let (x, y, z) denote a point of intersection. Let 5. 2x – y + 3z = 28 f(x, y, z) be the square of the distance to the 6. 3λ + 4μ = 16 (1, 2, 3, 4) origin. Minimize f ( x, y , z ) = x 2 + y 2 + z 2 7. 2λ + 7μ = 28 (1, 2, 3, 5) 8. λ = 0, μ = 4 (6, 7) subject to g ( x, y, z ) = x + y + z - 8 = 0 and 9. x = 4, y = –2, z = 6 (8, 1-3) h(x, y, z) = 2x – y + 3z – 28 = 0. f(4, –2, 6) = 56, and the nature of the problem Let ∇f ( x, y, z ) = λ∇g ( x, y, z ) + μ∇h( x, y, z ). indicates this is the minimum rather than the 2 x, 2 y, 2 z = λ 1,1,1 + μ 2, -1, 3 maximum. 1. 2x = λ + 2μ Conclusion: The least distance is 56 ≈ 7.4833. 2. 2y = λ – μ 3. 2z = λ + 3μ 4. x + y + z = 8 792 Section 12.9 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 50. 29. -1, 2, 2 = λ 2 x, 2 y, 0 + μ 0,1, 2 The function to be maximized in a hyperplane with positive coefficients and constant (so –1 = 2λx, 2 = 2λy + μ, 2 = 2μ, x 2 + y 2 = 2, intercepts on all axes are positive), and the y + 2z = 1 constraint is a hypersphere of radius 1, so the Critical points are (–1, 1, 0) and (1, –1, 1). maximum will occur where each xi is positive. f(–1, 1, 0) = 3, the maximum value; There is only one such critical point, the one f(1, –1, 1) = –1, the minimum value. obtained from the above by taking the principal square root to solve for xi . 30. a. Maximize w( x1 , x2 , …, xn ) = x1 x2 , …, xn , ( xi > 0) Then the maximum value of w is ⎛ a ⎞ ⎛ a ⎞ ⎛ a ⎞ A subject to the constraint a1 ⎜ 1 ⎟ + a2 ⎜ 2 ⎟ +…+ an ⎜ n ⎟ = = A g ( x1 , x2 , …, xn ) = x1 + x2 +…+ xn – 1 = 0. Le ⎝ A⎠ ⎝ A⎠ ⎝ A⎠ A 2 2 2 t ∇w( x1 , x2 , …, xn ) = λ∇g ( x1 , x2 , …, xn ). where A = a1 + a2 + … + an . x2 … xn , x1 x3 … xn , x1 … xn –1 = λ 1, 1, …, 1 . 32. Max: f(–0.71, 0.71) = f(–0.71, – 0.71) = 0.71 Therefore, λ x1 = λ x2 =…= λ xn (since each equals x1 x2 … xn ). Then x1 = x2 = … = xn . 33. Min: f(4, 0) = –4 (If λ = 0, some xi = 0, so w = 0.) 34. Max: f(1.41, 1.41) = f(–1.41, –1.44) = 0.037 1 Therefore, nxi = 1; xi = . 35. Min: f(0, 3) = f(0, –3) = –0.99 n n ⎛1⎞ The maximum value of w is ⎜ ⎟ , and 12.10 Chapter Review ⎝n⎠ 1 occurs when each xi = . Concepts Test n n 1. True: Except for the trivial case of z = 0, ⎛1⎞ which gives a point. b. From part a we have that x1 x2 … xn ≤ ⎜ ⎟ . ⎝n⎠ 1 xy Therefore, n x1 x2 … xn ≤ . 2. False: Use f(0, 0) = 0; f ( x, y ) = n x + y2 2 ai a elsewhere for counterexample. If xi = = i for each i, then a1 + … + an A 3. True: Since g ′(0) = f x (0, 0) a a a 1 A n 1 2 … n ≤ , so n a a … a ≤ , or 1 2 n A A A n n 4. True: It is the limit along the path, y = x. n a a … a ≤ a1 + a2 + … an . 5. True: Use “Continuity of a Product” 1 2 n n Theorem. 31. Let a1 , a2 , … an = λ 2 x1 , 2 x2 , …, 2 xn . 6. True: Straight forward calculation of partial Therefore, ai = 2λ xi , for each i = 1, 2, …, n derivatives (since λ = 0 implies ai = 0, contrary to the 7. False: See Problem 25, Section 12.4. hypothesis). xi x j ⎛ 1 ⎞ 8. False: It is perpendicular to the level curves = for all i, j ⎜ since each equals ⎟. of f. The gradient of ai a j ⎝ 2λ ⎠ F(x, y, z) = f(x, y) – z is perpendicular The constraint equation can be expressed to the graph of z = f(x, y). 2 2 2 2⎛x ⎞ 2⎛x ⎞ 2⎛x ⎞ a1 ⎜ 1 ⎟ + a2 ⎜ 2 ⎟ + … + an ⎜ n ⎟ = 1. 9. True: Since 0, 0, −1 is normal to the ⎝ a1 ⎠ ⎝ a2 ⎠ ⎝ an ⎠ tangent plane 2 Therefore, ( 2 a1 2 + a2 + 2 … + an ) ⎛ x1 ⎞ ⎜ ⎟ = 1. ⎝ a1 ⎠ 10. False: C ex : For the cylindrical surface 2 f ( x, y ) = y 3 , f(p) = 0 for every p on a1 2 x1 = ; similar for each other xi2 . the x-axis, but f(p) is not an extreme 2 2 a1 + … + an value. Instructor’s Resource Manual Section 12.10 793 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 51. 11. True: It will point in the direction of greatest b. 2x − y −1 ≥ 0 increase of heat, and at the origin, ∇T (0, 0) = 1, 0 is that direction. 12. True: It is nonnegative for all x, y, and it has a value of 0 at (0, 0). 13. True: Along the x-axis, f ( x, 0) → ±∞ as x → ±∞. 14. False: Du f ( x, y ) = 4, 4 ⋅ u ≤ 4 2 ⎛ 1 ⎞ ⎞ 2. x + y 2 = k (equality if u = ⎜ ⎟ 1,1 ⎟ ⎝ 2⎠ ⎠ x – k = – y2 15. True: – Du f ( x, y ) = –[∇f ( x, y ) ⋅ u] = ∇f ( x, y ) ⋅ (– u) = D– u f ( x, y ) 16. True: The set (call it S, a line segment) contains all of its boundary points because for every point P not in S (i.e., not on the line segment), there is an open neighborhood of P (i.e., a circle with P as center) that contains no point of S. 3. f x ( x, y ) = 12 x3 y 2 + 14 xy 7 17. True: By the Min-Max Existence Theorem f xx ( x, y ) = 36 x 2 y 2 + 14 y 7 18. False: ( x0 , y0 ) could be a singular point. f xy ( x, y ) = 24 x3 y + 98 xy 6 ⎛π ⎞ ⎛π⎞ 4. f x ( x, y ) = –2 cos x sin x = – sin 2 x 19. False: f ⎜ , 1⎟ = sin ⎜ ⎟ = 1, the maximum ⎝2 ⎠ ⎝2⎠ f xx ( x, y ) = –2 x cos 2 x value of f, and (π / 2, 1) is in the set. f xy ( x, y ) = 0 20. False: The same function used in Problem 2 5. f x ( x, y ) = e – y sec2 x provides a counterexample. f xx ( x, y ) = 2e – y sec 2 x tan x f xy ( x, y ) = – e – y sec2 x Sample Test Problems 1. a. x 2 + 4 y 2 − 100 ≥ 0 6. f x ( x, y ) = – e – x sin y x2 y 2 f xx ( x, y ) = e – x sin y + ≥1 100 25 f xy ( x, y ) = – e – x cos y 7. Fy ( x, y ) = 30 x3 y5 – 7 xy 6 Fyy ( x, y ) = 150 x3 y 4 – 42 xy 5 Fyyx ( x, y ) = 450 x 2 y 4 – 42 y 5 8. f x ( x, y, z ) = y 3 – 10 xyz 4 f y ( x, y, z ) = 3 xy 2 – 5 x 2 z 4 f z ( x, y, z ) = −20 x 2 yz 3 Therefore, f x (2, – 1, 1) = 19 ; f y (2, – 1, 1) = –14 ; f z (2, – 1, 1) = 80 794 Section 12.10 Instructor's Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 52. y 2 16. In the direction of ∇f (1, 2) = 4 9, 4 9. z y ( x, y ) = ; z y (2, 2) = = 1 2 2 10. Everywhere in the plane except on the parabola x2 x2 y 2 17. a. f(4, 1) = 9, so + y 2 = 9, or + = 1. x 2 = y. 2 18 9 b. ∇f ( x, y ) = x, 2 y , so f (4,1) = 4, 2 . x–x 11. No. On the path y = x, lim = 0. On the x →0 x + x c. x–0 path y = 0, lim = 1. x →0 x + 0 x2 – 2 y 4–4 12. a. lim = =0 ( x, y )→(2, 2) x 2 + 2y 4+4 b. Does not exist since the numerator lends to 4 and the denominator to 0. 18. Fx = Fu u x + Fv vx ( x 2 + 2 y 2 )( x 2 – 2 y 2 ) c. lim v y u 1 ( x, y )→(0, 0) x2 + 2 y 2 = + 2 2 2 2 1 + u v 2 xy 1+ u v 2 x = lim ( x2 – y 2 ) = 0 ( x, y )→(0, 0) v y +u = 2(1 + u 2 v 2 ) x 13. a. ∇f ( x, y, z ) = 2 xyz 3 , x 2 z 3 , 3 x 2 yz 2 Fy = Fu u y + Fv v y f (1, 2, −1) = −4, −1, 6 v x u −1 = + 2 2 2 2 b. ∇f ( x , y , z ) 1 + u v 2 xy 1+ u v 2 y v x –u = y 2 z cos xz , 2 y sin xz , xy 2 cos xz = 2(1 + u 2 v 2 ) y ∇f (1, 2, – 1) = –4 cos(1), sin(1), – cos(1) ≈ −2.1612, −3.3659, 2.1612 ⎛1⎞ ⎛ u ⎞ 19. f x = fu u x + fv u y = ⎜ ⎟ (2 x) + ⎜ – ⎟ ( yz ) ⎝ v⎠ ⎝ v2 ⎠ 14. Du f ( x, y ) = 3 y (1 + 9 x 2 y 2 ) –1 , 3x(1 + 9 x 2 y 2 ) –1 ⋅ u = x –2 y –1 z –1 ( x 2 + 3 y – 4 z ) Du f (4, 2) = 6 12 , ⋅ 3 ,− 1 = (3 3−6 ) ⎛1⎞ ⎛ u ⎞ f y = fu u y + f v v y = ⎜ ⎟ (–3) + ⎜ – ⎟ ( xz ) 577 577 2 2 577 ⎝v⎠ ⎝ v2 ⎠ ≈ −0.001393 = − x −1 y − 2 z − 1 ( x 2 + 4 z ) ⎛1⎞ ⎛ u ⎞ 15. z = f ( x, y ) = x 2 + y 2 f z = fu u z + f v vz = ⎜ ⎟ (4) + ⎜ – ⎟ ( xy ) ⎝v⎠ ⎝ v2 ⎠ 1, – 3, 0 is horizontal and is normal to the = x −1 y −1 z −2 (3 y − x 2 ) vertical plane that is given. By inspection, 3, 1, 0 is also a horizontal vector and is perpendicular to 1, – 3, 0 and therefore is 3 1 parallel to the vertical plane. Then u = , 2 2 is the corresponding 2-dimensional unit vector. Du f ( x, y ) = ∇f ( x, y ) ⋅ u 3 1 = 2 x, 2 y ⋅ , = 3x + y 2 2 Du f (1, 2) = 3 + 2 ≈ 3.7321 is the slope of the tangent to the curve. Instructor’s Resource Manual Section 12.10 795 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 53. dF dF dx dF dy 20. = + dt dx dt dy dt = (3 x 2 − y 2 )(−6sin 3t ) + (−2 xy − 4 y 3 )(3cos t ) ⎛ dF ⎞ t = 0 ⇒ x = 2 and y = 0, so ⎜ ⎟ = 0. ⎝ dt ⎠ t =0 21. Ft = Fx xt + Fy yt + Fz zt ⎛ 10 xy ⎞ ⎛ 3t1/ 2 ⎞ ⎛ 5x2 ⎞ ⎛ 1 ⎞ ⎛ 15 x 2 y ⎞ 3t =⎜ ⎟⎜ ⎟+⎜ 3 ⎟ ⎜ ⎟ + ⎜ – 4 ⎟ (3e ) ⎝ z3 ⎠ ⎜ 2⎝ ⎟ ⎜ z ⎠ ⎝ ⎟⎝ t ⎠ ⎜ ⎠ ⎝ z ⎟ ⎠ 15 xy t 5x2 45 x 2 ye3t = + – z3 z 3t z4 dc db dα 22. = 3, = –2, = 0.1 dt dt dt ⎛1⎞ Area = A(b, c, α ) = ⎜ ⎟ c(b sin α ) ⎝2⎠ dA ⎡⎛ b ⎞ ⎛ dc ⎞ ⎛ c ⎞ ⎛ db ⎞ ⎛ b ⎞ ⎛ dα ⎞ ⎤ = ⎜ ⎟ (sin α ) ⎜ ⎟ + ⎜ ⎟ (sin α ) ⎜ ⎟ + ⎜ ⎟ (bc cos α ) ⎜ ⎟⎥ dt ⎢⎝ 2 ⎠ ⎣ ⎝ dt ⎠ ⎝ 2 ⎠ ⎝ dt ⎠ ⎝ 2 ⎠ ⎝ dt ⎠ ⎦ ⎛ dA ⎞ = ( 7 + 4 3 ) ≈ 6.9641 in.2/s ⎜ ⎟⎛ π⎞ ⎝ dt ⎠ ⎜ 8, 10, ⎟ 2 ⎝ 6⎠ 23. Let F ( x, y , z ) = 9 x 2 + 4 y 2 + 9 z 2 – 34 = 0 ∇F ( x, y, z ) = 18 x, 8 y , 18 z , so ∇f (1, 2, – 1) = 2 9, 8, – 9 . Tangent plane is 9(x – 1) + 8(y – 2) – 9(z + 1) = 0, or 9x + 8y – 9z = 34. 24. V = πr 2 h; dV = Vr dr + Vh dh = 2πrh dr + πr 2 dh If r = 10, dr ≤ 0.02, h = 6, dh = 0.01, then dV ≤ 2πrh dr + πr 2 dh ≤ 2 π (10)(6)(0.02) + π (100)(0.01) = 3.4 π V(10, 6) = π (100)(6) = 600 π Volume is 600 π ± 3.4 π ≈ 1884.96 ± 10.68 25. df = y 2 (1 + z 2 ) –1 dx + 2 xy (1 + z 2 ) –1 dy – 2 xy 2 z (1 + z 2 ) –2 dz If x = 1, y = 2, z = 2, dx = 0.01, dy = –0.02, dz = 0.03, then df = –0.0272. Therefore, f(1.01, 1.98, 2.03) ≈ f(1, 2, 2) + df = 0.8 – 0.0272 = 0.7728 26. ∇f ( x, y ) = 2 xy – 6 x, x 2 – 12 y = 0, 0 at (0, 0) and (±6, 3). D = f xx f yy – f xy = (2 y – 6)(–12) – (2 x)2 2 = 4(18 − 6 y − x 2 ); f xx = 2( y – 3) At (0, 0): D = 72 > 0 and f xx < 0, so local maximum at (0, 0). At (±6, 3): D < 0, so (±6, 3) are saddle points. 796 Section 12.10 Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 54. 27. Let (x, y, z) denote the coordinates of the 1st Review and Preview Problems octant vertex of the box. Maximize f(x, y, z) = xyz subject to 1. 2 2 2 g ( x, y, z ) = 36 x + 4 y + 9 z – 36 = 0 (where x, y, z > 0 and the box’s volume is V(x, y, z) = f(x, y, z). Let ∇f ( x, y, z ) = λ∇g ( x, y, z ). yz , xz , xy 8 = λ 72 x,8 y,18 z 1. 8yz = 72λx 2. 8xz = 8λy 3. 8xy = 18λz 4. 36 x 2 + 4 y 2 + 9 z 2 = 36 yz 72λ x 5. = , so y 2 = 9 x 2 . (1, 2) xz 8λ y 2. yz 72λ x 6. = , so z 2 = 4 x 2 . (1, 3) xz 18λ y 1 7. 36 x 2 + 36 x 2 + 36 x 2 = 36, so x = . 3 (5, 6, 4) 3 2 8. y = ,z= (7, 5, 6) 3 3 ⎛ 1 3 2 ⎞ ⎛ 1 ⎞⎛ 3 ⎞⎛ 2 ⎞ V⎜ , , ⎟ = 8⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ 3 3 3⎠ ⎝ 3 ⎠⎝ 3 ⎠⎝ 3 ⎠ 16 = ≈ 9.2376 3 The nature of the problem indicates that the critical point yields a maximum value rather than 3. a minimum value. 28. y , x = λ 2 x, 2 y y = 2λx, x = 2λy, x 2 + y 2 = 1 ⎛ 1 1 ⎞ Critical points are ⎜ ,± ⎟ and ⎝ 2 2⎠ ⎛ 1 1 ⎞ 1 ⎜– ,± ⎟ . Maximum of at ⎝ 2 2⎠ 2 ⎛ 1 1 ⎞ 1 ⎜± ,± ⎟ ; minimum of – at ⎝ 2 2⎠ 2 4. ⎛ 1 1 ⎞ ⎜± , ⎟. ⎝ 2 2⎠ 29. Maximize V (r , h) = πr 2 h, subject to S (r , h) = 2πr 2 + 2πrh – 24π = 0. 2π rh, π r 2 = λ 4π r + 2π h, 2π r rh = λ(2r + h), r = 2λ, r 2 + rh = 12 Critical point is (2, 4). The nature of the problem indicates that the critical point yields a maximum value rather than a minimum value. Conclusion: The dimensions are radius of 2 and height of 4. Instructor’s Resource Manual Review and Preview 797 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 55. 5. 9. 6. 10. 7. 11. 8. 798 Review and Preview Instructor’s Resource Manual © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
• 56. 12. 2 ⎡ 1 1 ⎤ ∫ ( a + bx + c x ) dx = ⎢ax + 2 bx 2 18. 2 2 2 + c 2 x3 ⎥ 0 ⎣ 3 ⎦0 8 = 2a + 2b + c 2 3 π π ⎡1 1 ⎤ π 19. ∫0 sin 2 x dx = ⎢ x − sin 2 x ⎥ = ⎣2 4 ⎦0 2 3/ 4 3/ 4 1 1 x +1 1 ⎛ 21 ⎞ 20. ∫1/ 4 1 − x2 dx = ln = ln ⎜ ⎟ 2 x − 1 1/ 4 2 ⎝ 5 ⎠ 3/ 4 1 1 3/ 4 1 13. 21. ∫ x =1/ 4 1+ u du = ln 1 + x 2 2 1/ 4 = ln 2 2 4 1 4 22. ∫ x=0 1+ u 2 du = ⎡ tan −1 e x ⎤ ≈ 0.7671 ⎣ ⎦0 u = 4r 2 + 1; du = 8r dr 3 1 3 ∫0 r 4r 2 + 1 dr = 8 ∫r =0 23. u du 3 −1 + 373 / 2 ⎡1 2 ( ) 3/ 2 ⎤ = ⎢ i 4r 2 + 1 ⎥ = ⎣8 3 ⎦0 12 24. u = a 2 − r 2 ; du = −2r dr a/2 ar a a/2 1 ∫0 dr = − ∫ 2 r =0 u du a2 − r 2 14. a ⎡1 2 2 ⎤ =− ⎢ a −r ⎥ a/2 = ( a2 2 − 3 ) 2 ⎣2 ⎦0 8 π / 2⎛ 1 1 ⎞ 25. ∫0 ⎜ 2 + 2 cos 2θ ⎟ dθ ⎝ ⎠ π /2 ⎡1 1 ⎤ π = ⎢ θ + sin 2θ ⎥ = ⎣2 4 ⎦0 4 2 π / 2⎛ 1 1 ⎞ 26. ∫0 ⎜2 ⎝ + 2 cos 2θ ⎟ dθ ⎠ 2 π / 2⎛ 1 1 1 ⎞ =∫ ⎜4 + cos 2θ + cos 2 2θ ⎟ dθ 0 ⎝ 2 4 ⎠ 1 ∫e −2 x 15. dx = − e −2 x + C π / 2⎛ 3 1 1 ⎞ 2 =∫ ⎜ 8 + 2 cos 2θ + 8 cos 4θ ⎟ dθ 0 ⎝ ⎠ 1 1 π /2 ⎡3 1 1 ⎤ 3π ∫ xe dx = − xe −2 x + ∫ e−2 x dx + C −2 x 16. = ⎢ θ + sin 2θ + sin 4θ ⎥ = 2 2 ⎣8 4 32 ⎦0 16 1 −2 x 1 − 2 x = − xe − e + C 2 4 27. 2π ⎛ a 2 − b 2 − a 2 − c 2 ⎞ ⎜ ⎟ a/2 ⎝ ⎠ a/2 ⎛ xπ ⎞ a ⎛ xπ ⎞ 2a θ is not part of the integrand. 17. ∫− a / 2 cos ⎜ a ⎝ ⎟dx = sin ⎜ ⎠ π ⎟ = ⎝ a ⎠− a / 2 π Instructor’s Resource Manual Review and Preview 799 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.