0
Section	2.8
         Linear	Approximation	and
                Differentials

               V63.0121.006/016, Calculus	I

...
Announcements




     Quiz	2	is	February	26, covering	§§1.5–2.3
     Midterm	is	March	4, covering	§§1.1–2.5




         ...
Outline


  The	linear	approximation	of	a	function	near	a	point
     Examples
     Questions


  Midterm	Review


  Differ...
The	Big	Idea


   Question
   Let f be	differentiable	at a. What	linear	function	best
   approximates f near a?




      ...
The	Big	Idea


   Question
   Let f be	differentiable	at a. What	linear	function	best
   approximates f near a?

   Answer...
The	Big	Idea


   Question
   Let f be	differentiable	at a. What	linear	function	best
   approximates f near a?

   Answer...
The	Big	Idea


   Question
   Let f be	differentiable	at a. What	linear	function	best
   approximates f near a?

   Answer...
The	tangent	line	is	a	linear	approximation



                                          y
                                ...
The	tangent	line	is	a	linear	approximation



                                          y
                                ...
Example
  Example
  Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
  approximation
  (i) about a = 0     (ii) about a...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Example
   Example
   Estimate sin(61◦ ) = sin(61π/180) by	using	a	linear
   approximation
   (i) about a = 0     (ii) abo...
Illustration

       y
       .




                      y
                      . = sin x




       .                  ...
Illustration

       y
       .
                      y
                      . = L1 (x) = x




                       y
...
Illustration

       y
       .
                                         y
                                         . = L1...
Illustration

       y
       .
                                y
                                . = L1 (x) = x


       ...
Illustration

       y
       .
                                  y
                                  . = L1 (x) = x


   ...
Another	Example

  Example
             √
  Estimate    10 using	the	fact	that 10 = 9 + 1.




                           ...
Another	Example

  Example
             √
  Estimate    10 using	the	fact	that 10 = 9 + 1.

  Solution                    ...
Another	Example

  Example
             √
  Estimate    10 using	the	fact	that 10 = 9 + 1.

  Solution                    ...
Another	Example

  Example
             √
  Estimate    10 using	the	fact	that 10 = 9 + 1.

  Solution                    ...
Another	Example

  Example
             √
  Estimate    10 using	the	fact	that 10 = 9 + 1.

  Solution                    ...
Dividing	without	dividing?
   Example
   Suppose	I have	an	irrational	fear	of	division	and	need	to	estimate
   577 ÷ 408. ...
Dividing	without	dividing?
   Example
   Suppose	I have	an	irrational	fear	of	division	and	need	to	estimate
   577 ÷ 408. ...
Questions

  Example
  Suppose	we	are	traveling	in	a	car	and	at	noon	our	speed	is
  50 mi/hr. How	far	will	we	have	travele...
Answers


  Example
  Suppose	we	are	traveling	in	a	car	and	at	noon	our	speed	is
  50 mi/hr. How	far	will	we	have	traveled...
Answers


  Example
  Suppose	we	are	traveling	in	a	car	and	at	noon	our	speed	is
  50 mi/hr. How	far	will	we	have	traveled...
Questions

  Example
  Suppose	we	are	traveling	in	a	car	and	at	noon	our	speed	is
  50 mi/hr. How	far	will	we	have	travele...
Answers



  Example
  Suppose	our	factory	makes	MP3	players	and	the	marginal	cost	is
  currently	$50/lot. How	much	will	i...
Questions

  Example
  Suppose	we	are	traveling	in	a	car	and	at	noon	our	speed	is
  50 mi/hr. How	far	will	we	have	travele...
Answers



  Example
  Suppose	a	line	goes	through	the	point (x0 , y0 ) and	has	slope m. If
  the	point	is	moved	horizonta...
Answers



  Example
  Suppose	a	line	goes	through	the	point (x0 , y0 ) and	has	slope m. If
  the	point	is	moved	horizonta...
Outline


  The	linear	approximation	of	a	function	near	a	point
     Examples
     Questions


  Midterm	Review


  Differ...
Midterm	Facts

     Covers	sections	1.1–2.5
     (Limits, Derivatives,
     Differentiation	up	to
     Chain	Rule)
     Ca...
Outline


  The	linear	approximation	of	a	function	near	a	point
     Examples
     Questions


  Midterm	Review


  Differ...
Differentials	are	another	way	to	express	derivatives


   f(x + ∆x) − f(x) ≈ f′ (x) ∆x   y
                               ...
Differentials	are	another	way	to	express	derivatives


   f(x + ∆x) − f(x) ≈ f′ (x) ∆x          y
                        ...
Using	differentials	to	estimate	error



                                 y
                                 .
  If y = f(...
Example
A sheet	of	plywood	measures 8 ft × 4 ft. Suppose	our
plywood-cutting	machine	will	cut	a	rectangle	whose	width	is
e...
Example
A sheet	of	plywood	measures 8 ft × 4 ft. Suppose	our
plywood-cutting	machine	will	cut	a	rectangle	whose	width	is
e...
Example
A sheet	of	plywood	measures 8 ft × 4 ft. Suppose	our
plywood-cutting	machine	will	cut	a	rectangle	whose	width	is
e...
Example
A sheet	of	plywood	measures 8 ft × 4 ft. Suppose	our
plywood-cutting	machine	will	cut	a	rectangle	whose	width	is
e...
Why?



  Why	use	linear	approximations dy when	the	actual	difference ∆y
  is	known?
       Linear	approximation	is	quick	...
Outline


  The	linear	approximation	of	a	function	near	a	point
     Examples
     Questions


  Midterm	Review


  Differ...
Gravitation
Pencils	down!
    Example
         Drop	a	1 kg	ball	off	the	roof	of	the	Silver	Center	(50m	high).
         We	...
Gravitation
Pencils	down!
    Example
         Drop	a	1 kg	ball	off	the	roof	of	the	Silver	Center	(50m	high).
         We	...
Solution
We	wonder	if ∆F = F(re + ∆r) − F(re ) is	small.
    Using	a	linear	approximation,

                              ...
Systematic	linear	approximation

      √                          √
          2 is	irrational, but       9/4   is	rational...
Systematic	linear	approximation

      √                          √
          2 is	irrational, but       9/4   is	rational...
Systematic	linear	approximation

      √                          √
          2 is	irrational, but       9/4   is	rational...
Systematic	linear	approximation

      √                            √
          2 is	irrational, but         9/4   is	rati...
Illustration	of	the	previous	example




                 .




                                       .   .   .   .   .  ...
Illustration	of	the	previous	example




                 .




                                       .   .   .   .   .  ...
Illustration	of	the	previous	example




                 .
                             2
                             .
...
Illustration	of	the	previous	example




                                 .




                 .
                       ...
Illustration	of	the	previous	example




                                 .




                 .
                       ...
Illustration	of	the	previous	example




                        . 2, 17 )
                        ( 12
                  ...
Illustration	of	the	previous	example




                        . 2, 17 )
                        ( 12
                  ...
Illustration	of	the	previous	example




                                       .
                     . 2, 17/12)
       ...
Illustration	of	the	previous	example




                                                 .
                     . 2, 17/1...
Illustration	of	the	previous	example




                                               .
                     . 2, 17/12)...
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Lesson 12: Linear Approximation

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Transcript of "Lesson 12: Linear Approximation"

  1. 1. Section 2.8 Linear Approximation and Differentials V63.0121.006/016, Calculus I February 26, 2010 Announcements Quiz 2 is February 26, covering §§1.5–2.3 Midterm is March 4, covering §§1.1–2.5 . . . . . .
  2. 2. Announcements Quiz 2 is February 26, covering §§1.5–2.3 Midterm is March 4, covering §§1.1–2.5 . . . . . .
  3. 3. Outline The linear approximation of a function near a point Examples Questions Midterm Review Differentials Using differentials to estimate error Advanced Examples . . . . . .
  4. 4. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? . . . . . .
  5. 5. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent line, of course! . . . . . .
  6. 6. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent line, of course! Question What is the equation for the line tangent to y = f(x) at (a, f(a))? . . . . . .
  7. 7. The Big Idea Question Let f be differentiable at a. What linear function best approximates f near a? Answer The tangent line, of course! Question What is the equation for the line tangent to y = f(x) at (a, f(a))? Answer L(x) = f(a) + f′ (a)(x − a) . . . . . .
  8. 8. The tangent line is a linear approximation y . L(x) = f(a) + f′ (a)(x − a) is a decent approximation to L . (x) . f near a. f .(x) . f .(a) . . x−a . x . a . x . . . . . . .
  9. 9. The tangent line is a linear approximation y . L(x) = f(a) + f′ (a)(x − a) is a decent approximation to L . (x) . f near a. f .(x) . How decent? The closer x is to a, the better the f .(a) . . x−a approxmation L(x) is to f(x) . x . a . x . . . . . . .
  10. 10. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. . . . . . .
  11. 11. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) If f(x) = sin x, then f(0) = 0 and f′ (0) = 1. So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180 . . . . . .
  12. 12. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) If f(x) = sin x, then f(0) = 0 We have f π = and and f′ (0) = 1. ( ) 3 f′ π = . 3 So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180 . . . . . .
  13. 13. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 3 and and f′ (0) = 1. ( ) 3 2 f′ π = . 3 So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180 . . . . . .
  14. 14. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 3 and and f′ (0) = 1. ( ) 3 2 f′ π = 1 . 3 2 So the linear approximation near 0 is L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180 . . . . . .
  15. 15. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 3 and and f′ (0) = 1. ( ) 3 2 f′ π = 1 . 3 2 So the linear approximation near 0 is So L(x) = L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180 . . . . . .
  16. 16. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 23 and and f′ (0) = 1. ( ) 3 f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) near 0 is So L(x) = + x− 2 2 3 L(x) = 0 + 1 · x = x. Thus ( ) 61π 61π sin ≈ ≈ 1.06465 180 180 . . . . . .
  17. 17. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 23 and and f′ (0) = 1. ( ) 3 f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) near 0 is So L(x) = + x− 2 2 3 L(x) = 0 + 1 · x = x. Thus Thus ( ) ( ) 61π 61π 61π sin ≈ sin ≈ ≈ 1.06465 180 180 180 . . . . . .
  18. 18. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 23 and and f′ (0) = 1. ( ) 3 f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) near 0 is So L(x) = + x− 2 2 3 L(x) = 0 + 1 · x = x. Thus Thus ( ) ( ) 61π 61π 61π sin ≈ 0.87475 sin ≈ ≈ 1.06465 180 180 180 . . . . . .
  19. 19. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 23 and and f′ (0) = 1. ( ) 3 f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) near 0 is So L(x) = + x− 2 2 3 L(x) = 0 + 1 · x = x. Thus Thus ( ) ( ) 61π 61π 61π sin ≈ 0.87475 sin ≈ ≈ 1.06465 180 180 180 Calculator check: sin(61◦ ) ≈ . . . . . .
  20. 20. Example Example Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation (i) about a = 0 (ii) about a = 60◦ = π/3. Solution (i) Solution (ii) ( ) √ If f(x) = sin x, then f(0) = 0 We have f π = 23 and and f′ (0) = 1. ( ) 3 f′ π = 1 . 3 2 √ So the linear approximation 3 1( π) near 0 is So L(x) = + x− 2 2 3 L(x) = 0 + 1 · x = x. Thus Thus ( ) ( ) 61π 61π 61π sin ≈ 0.87475 sin ≈ ≈ 1.06465 180 180 180 Calculator check: sin(61◦ ) ≈ 0.87462. . . . . . .
  21. 21. Illustration y . y . = sin x . x . . 1◦ 6 . . . . . .
  22. 22. Illustration y . y . = L1 (x) = x y . = sin x . x . 0 . . 1◦ 6 . . . . . .
  23. 23. Illustration y . y . = L1 (x) = x b . ig difference! y . = sin x . x . 0 . . 1◦ 6 . . . . . .
  24. 24. Illustration y . y . = L1 (x) = x √ 3 1 ( ) y . = L2 (x) = 2 + 2 x− π 3 y . = sin x . . . x . 0 . . π/3 . 1◦ 6 . . . . . .
  25. 25. Illustration y . y . = L1 (x) = x √ 3 1 ( ) y . = L2 (x) = 2 + 2 x− π 3 y . = sin x . . ery little difference! v . . x . 0 . . π/3 . 1◦ 6 . . . . . .
  26. 26. Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. . . . . . .
  27. 27. Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution √ The key step is to use a linear approximation to f(x) = √ x near a = 9 to estimate f(10) = 10. . . . . . .
  28. 28. Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution √ The key step is to use a linear approximation to f(x) = √ x near a = 9 to estimate f(10) = 10. √ √ d√ 10 ≈ 9 + x (1) dx x=9 1 19 =3+ (1 ) = ≈ 3.167 2·3 6 . . . . . .
  29. 29. Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution √ The key step is to use a linear approximation to f(x) = √ x near a = 9 to estimate f(10) = 10. √ √ d√ 10 ≈ 9 + x (1) dx x=9 1 19 =3+ (1 ) = ≈ 3.167 2·3 6 ( )2 19 Check: = 6 . . . . . .
  30. 30. Another Example Example √ Estimate 10 using the fact that 10 = 9 + 1. Solution √ The key step is to use a linear approximation to f(x) = √ x near a = 9 to estimate f(10) = 10. √ √ d√ 10 ≈ 9 + x (1) dx x=9 1 19 =3+ (1 ) = ≈ 3.167 2·3 6 ( )2 19 361 Check: = . 6 36 . . . . . .
  31. 31. Dividing without dividing? Example Suppose I have an irrational fear of division and need to estimate 577 ÷ 408. I write 577 1 1 1 = 1 + 169 = 1 + 169 × × . 408 408 4 102 1 But still I have to find . 102 . . . . . .
  32. 32. Dividing without dividing? Example Suppose I have an irrational fear of division and need to estimate 577 ÷ 408. I write 577 1 1 1 = 1 + 169 = 1 + 169 × × . 408 408 4 102 1 But still I have to find . 102 Solution 1 Let f(x) = . We know f(100) and we want to estimate f(102). x 1 1 f(102) ≈ f(100) + f′ (100)(2) = − (2) = 0.0098 100 1002 577 =⇒ ≈ 1.41405 408 577 Calculator check: ≈ 1.41422. . . . . . .
  33. 33. Questions Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:00pm? by 3:00pm? By midnight? . . . . . .
  34. 34. Answers Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:00pm? by 3:00pm? By midnight? . . . . . .
  35. 35. Answers Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:00pm? by 3:00pm? By midnight? Answer 100 mi 150 mi 600 mi (?) (Is it reasonable to assume 12 hours at the same speed?) . . . . . .
  36. 36. Questions Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:00pm? by 3:00pm? By midnight? Example Suppose our factory makes MP3 players and the marginal cost is currently $50/lot. How much will it cost to make 2 more lots? 3 more lots? 12 more lots? . . . . . .
  37. 37. Answers Example Suppose our factory makes MP3 players and the marginal cost is currently $50/lot. How much will it cost to make 2 more lots? 3 more lots? 12 more lots? Answer $100 $150 $600 (?) . . . . . .
  38. 38. Questions Example Suppose we are traveling in a car and at noon our speed is 50 mi/hr. How far will we have traveled by 2:00pm? by 3:00pm? By midnight? Example Suppose our factory makes MP3 players and the marginal cost is currently $50/lot. How much will it cost to make 2 more lots? 3 more lots? 12 more lots? Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx, while staying on the line, what is the corresponding vertical movement? . . . . . .
  39. 39. Answers Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx, while staying on the line, what is the corresponding vertical movement? . . . . . .
  40. 40. Answers Example Suppose a line goes through the point (x0 , y0 ) and has slope m. If the point is moved horizontally by dx, while staying on the line, what is the corresponding vertical movement? Answer The slope of the line is rise m= run We are given a “run” of dx, so the corresponding “rise” is m dx. . . . . . .
  41. 41. Outline The linear approximation of a function near a point Examples Questions Midterm Review Differentials Using differentials to estimate error Advanced Examples . . . . . .
  42. 42. Midterm Facts Covers sections 1.1–2.5 (Limits, Derivatives, Differentiation up to Chain Rule) Calculator free 20 multiple-choice questions and 4 free-response questions To study: outline do problems metacognition ask questions! (maybe in recitation?) . . . . . .
  43. 43. Outline The linear approximation of a function near a point Examples Questions Midterm Review Differentials Using differentials to estimate error Advanced Examples . . . . . .
  44. 44. Differentials are another way to express derivatives f(x + ∆x) − f(x) ≈ f′ (x) ∆x y . ∆y dy Rename ∆x = dx, so we can write this as . ∆y ≈ dy = f′ (x)dx. . dy . ∆y And this looks a lot like the . . dx = ∆x Leibniz-Newton identity dy . x . = f′ (x ) dx x x . . + ∆x . . . . . .
  45. 45. Differentials are another way to express derivatives f(x + ∆x) − f(x) ≈ f′ (x) ∆x y . ∆y dy Rename ∆x = dx, so we can write this as . ∆y ≈ dy = f′ (x)dx. . dy . ∆y And this looks a lot like the . . dx = ∆x Leibniz-Newton identity dy . x . = f′ (x ) dx x x . . + ∆x Linear approximation means ∆y ≈ dy = f′ (x0 ) dx near x0 . . . . . . .
  46. 46. Using differentials to estimate error y . If y = f(x), x0 and ∆x is known, and an estimate of ∆y is desired: Approximate: ∆y ≈ dy . Differentiate: . dy dy = f′ (x) dx . ∆y . Evaluate at x = x0 and . dx = ∆x dx = ∆x. . x . x x . . + ∆x . . . . . .
  47. 47. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? . . . . . .
  48. 48. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? Solution 1 Write A(ℓ) = ℓ2 . We want to know ∆A when ℓ = 8 ft and 2 ∆ℓ = 1 in. . . . . . .
  49. 49. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? Solution 1 Write A(ℓ) = ℓ2 . We want to know ∆A when ℓ = 8 ft and 2 ∆ℓ = 1 in. ( ) 97 9409 (I) A(ℓ + ∆ℓ) = A = So 12 288 9409 ∆A = − 32 ≈ 0.6701. 288 . . . . . .
  50. 50. Example A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting machine will cut a rectangle whose width is exactly half its length, but the length is prone to errors. If the length is off by 1 in, how bad can the area of the sheet be off by? Solution 1 Write A(ℓ) = ℓ2 . We want to know ∆A when ℓ = 8 ft and 2 ∆ℓ = 1 in. ( ) 97 9409 (I) A(ℓ + ∆ℓ) = A = So 12 288 9409 ∆A = − 32 ≈ 0.6701. 288 dA (II) = ℓ, so dA = ℓ dℓ, which should be a good estimate for dℓ 1 ∆ℓ. When ℓ = 8 and dℓ = 12 , we have 8 2 dA = 12 = 3 ≈ 0.667. So we get estimates close to the hundredth of a square foot. . . . . . .
  51. 51. Why? Why use linear approximations dy when the actual difference ∆y is known? Linear approximation is quick and reliable. Finding ∆y exactly depends on the function. These examples are overly simple. See the “Advanced Examples” later. In real life, sometimes only f(a) and f′ (a) are known, and not the general f(x). . . . . . .
  52. 52. Outline The linear approximation of a function near a point Examples Questions Midterm Review Differentials Using differentials to estimate error Advanced Examples . . . . . .
  53. 53. Gravitation Pencils down! Example Drop a 1 kg ball off the roof of the Silver Center (50m high). We usually say that a falling object feels a force F = −mg from gravity. . . . . . .
  54. 54. Gravitation Pencils down! Example Drop a 1 kg ball off the roof of the Silver Center (50m high). We usually say that a falling object feels a force F = −mg from gravity. In fact, the force felt is GMm F (r ) = − , r2 where M is the mass of the earth and r is the distance from the center of the earth to the object. G is a constant. GMm At r = re the force really is F(re ) = = −mg. r2 e What is the maximum error in replacing the actual force felt at the top of the building F(re + ∆r) by the force felt at ground level F(re )? The relative error? The percentage error? . . . . . .
  55. 55. Solution We wonder if ∆F = F(re + ∆r) − F(re ) is small. Using a linear approximation, dF GMm ∆F ≈ dF = dr = 2 3 dr dr r re ( e ) GMm dr ∆r = 2 = 2mg re re re ∆F ∆r The relative error is ≈ −2 F re re = 6378.1 km. If ∆r = 50 m, ∆F ∆r 50 ≈ −2 = −2 = −1.56 × 10−5 = −0.00156% F re 6378100 . . . . . .
  56. 56. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. . . . . . .
  57. 57. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. So √ √ √ 1 17 2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) = 2(3/2) 12 . . . . . .
  58. 58. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. So √ √ √ 1 17 2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) = 2(3/2) 12 This is a better approximation since (17/12)2 = 289/144 . . . . . .
  59. 59. Systematic linear approximation √ √ 2 is irrational, but 9/4 is rational and 9/4 is close to 2. So √ √ √ 1 17 2 = 9/4 − 1/4 ≈ 9/4 + (−1/4) = 2(3/2) 12 This is a better approximation since (17/12)2 = 289/144 Do it again! √ √ √ 1 2= 289/144 − 1/144 ≈ 289/144+ (−1/144) = 577/408 2(17/12) ( )2 577 332, 929 1 Now = which is away from 2. 408 166, 464 166, 464 . . . . . .
  60. 60. Illustration of the previous example . . . . . . .
  61. 61. Illustration of the previous example . . . . . . .
  62. 62. Illustration of the previous example . 2 . . . . . . .
  63. 63. Illustration of the previous example . . 2 . . . . . . .
  64. 64. Illustration of the previous example . . 2 . . . . . . .
  65. 65. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . .
  66. 66. Illustration of the previous example . 2, 17 ) ( 12 . . . 2 . . . . . . .
  67. 67. Illustration of the previous example . . 2, 17/12) ( . (9 2 . 4, 3) . . . . . .
  68. 68. Illustration of the previous example . . 2, 17/12) ( .. . 9, 3) ( ( 289 17 ) 4 2 . 144 , 12 . . . . . .
  69. 69. Illustration of the previous example . . 2, 17/12) ( .. . 9, 3) ( ( 577 ) ( 289 17 ) 4 2 . 2, 408 . 144 , 12 . . . . . .
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