xy2.5 3.0 3.5-1.0 6 7 81.0 0 1 23.0 -6 -5 .docx

x
y
2.5 3.0 3.5
-1.0 6 7 8
1.0 0 1 2
3.0 -6 -5 -4
MATH 223
FINAL EXAM REVIEW PACKET ANSWERS
(Fall 2012)
1. (a) increasing (b) decreasing
2. (a) 2 2( 3) 25y z− + = This is a cylinder parallel to the x-axis
with radius 5.
(b) 3x = , 3x = − . These are vertical planes parallel to the yz-
plane.
(c) 2 2 2z x y= + . This is a cone (one opening up and one
opening down) centered on the z-axis.
3. There are many possible answers.
(a) 0x = produces the curve 23y z= − .

(b) 1y = produces the curves 23 cosz x= − and 23 cosz x= − −
.
(c)
2
x
π
= produces the curves 3z = and 3z = − .
4. (a) (b) (i) 1 (ii) Increase (iii) Decrease
5. (a) Paraboloids centered on the x-axis, opening up in the
positive x direction. 2 2x y z c= + +
(b) Spheres centered at the origin with radius 1 ln c− for 0 c e<
≤ . 2 2 2 1 lnx y z c+ + = −
6. (a) 6 am 11:30 am
(b) Temperature as a function of time at a depth of 20 cm.
(c) Temperature as a function of depth at noon.

7. ( , ) 2 3 2z f x y x y= = − −
8. (a) II, III, IV, VI (b) I (c) I, III, VI (d) VI
(e) I, V
9. (a)
12
4 12
5
z x y= − + (b) There are many possible answers.
12
4
5
i j k+ −
(c)
3 569
2

10. (a) iii, vii (b) iv (c) viii (d) ii (e) v, vi (f)
i, ix
(a) ( )5 4 3
26
i j k− +
or ( )5 4 3
26
i j k− − +
(b) 2 3i j− +
(c)
4
cos
442
θ = , 1.38θ ≈ radians (d) ( )4 4 3

26
i j k− +
(e) 4 11 17i j k− − −
12. (a)
3
5
a = − (b)
1
3
a = (c) 2( 1) ( 2) 3( 3) 0x y z− − + + − = (d)
1 2 , 2 , 3 3x t y t z t= + = − − = +
13. 6 39i
or 6 39i−
14. (a)

( )
2
23 2 2
3 2
3 1
z x y x
x x y x y
∂
= −
∂ + + +
(b)
( )4
10 4 3
5
H
H T
f
H
+ +
=
−
(c)
2

2 2
1 1z
x y y x
∂
= − −
∂ ∂
15. (a) 2 2 24 ( 1) 3 ( 2) 2z e x e y e= − + − + (b) 4( 3)
8( 3) 6( 6) 0x y z− + − + − =
16. (a)
2sin(2 ) cos(2 )
5 5
v v
ds dv d
α α
α= +
(b) The distance s decreases if the angle α increases and the
initial speed v remains constant.
(c) 0.0886α∆ ≈ − . The angle decreases by about 0.089 radians.
17. (a) The water is getting shallower.
4

( 1, 2)
17
(b) There are many possible answers. 3i j+
(c) 72 ft/min
18. (a)
( )
2 2 2
22 2 22
2 2
1 1 11
yz xyz z yz
grad i j k
x x xx

(b) ( ) ( ) ( )( )2 2 2 2curl x y z i y z j xz k i zj yk+ + − + + = +
−
(c) ( ) ( ) ( )( )2 3 3cos sec 2 cos sin sec tan 3z zdiv x i x y j e k
x x x y y e+ + = − + +
(d)
37
3
(e) ( , , ) sin zg x y z xy e c= + +
20. (a) positive (b) negative (c) negative (d) negative
(e) positive (f) zero
21. (a)
(1, 1 2)

15
8
w
u
π
∂
=
∂
and
(1, 1 2)
15
4
w
v
π
∂
= −
∂
(b)
1
3
t
dw

dt e=
= −
22. ( )1, 2 37rz π =
23. ( 1, 3)− − local maximum, (2,1) local minimum, ( 1,1)−
and (2, 3)− saddle points
24. (b) 4K < , saddle point, 4K > local minimum, no values of
K for local maximum.
25. The minimum distance from the surface to the origin is 6 .
This occurs at the points (2, 1,1)− and
(2, 1, 1)− − .
26. (a) 8r = (b)
4
π
θ = (c)
5
4
π
φ = (d)
10
cos
ρ

φ
=
27. (a) positive (b) positive (c) negative (d) negative
28. (a) ( )1 sin18 cos 5 cos11
6
− (b) 3
2
(5)
3
π volume of a half sphere
(c) ( )3 22 28 1
9
− change the order (d) 8
1 1
2 3 3
e
convert to spherical

(e)
7
3
− (f)
25
2
area of a triangle
29. (a)
2 2 2
2 2 2
2 2 4
2 2
x x y
x x y
dzdydx
− − −
− − − +∫ ∫ ∫
2 2 2 4
0 0
r
r

rdzdrd
π
θ
−
∫ ∫ ∫
2 4 2 2
0 0 0
sin d d d
π π
ρ φ ρ φ θ∫ ∫ ∫
(b)
2 2 2
2 2 2
3 9 20
3 9 2
z y z
z y z
dxdydz
− − −

− − − + +∫ ∫ ∫
2
2
2 3 20
0 0 2
r
r
rdxdrd
π
θ
−
+∫ ∫ ∫
(c)
2 4 12 16
0 0 0
z
dxdydz
−
∫ ∫ ∫
2 4 12

0 0 0
rdydrd
π
θ∫ ∫ ∫
(d)
0 1 3 2 2 6 12
6 0 0
x x y
dzdydx
+ − +
−∫ ∫ ∫
30.
2 4 4
0 0 0
(4 )
r
k z rdzdrd
π
θ
−

−∫ ∫ ∫
(a) 3 cos 2, 1, 3sin 0 2x t y z t t π= + = = − ≤ ≤
(b) 1 2 , 2 3 , 3 x t y t z t t= + = − − = − −∞ < < ∞
(c) ( )3, 2 2 0x t y t t= − = − + − ≤ ≤
(d) 2 cos , 2 sin , 2 0 2x t y t z t π= = = ≤ ≤
32. (a) 4t = seconds (b) 10 feet per second
(c) There are many possible answers. 3, 3( 2 ), 10 2 ( 2 )
2x y t z t tπ π π π= = − = − − − ≥
33. (a) There are many possible answers. 1 2 , 1 6 , 7+ 0x
t y t z t t= + = + = ≥ (b) No
34. (a) iii (b) v (c) vi (d) i (e) ii (f) iv
35.
1 2 3C C C
F dr F dr F dr⋅ < ⋅ < ⋅ ∫ ∫ ∫

36. (a)
21
2
− (b) 10 cos 2− (c) 18π (d)
4
1
16
π
− (e)
1875
2
π
37. (a)
12
2
−
(b) 320π (c) 450π (d)
1875
2
π
38. (a) 0 (b) 0

39. 0p = , 500flux π=
40. 22, 000, 000 40, 000, 000 18, 000, 000π π π− = −
41. (a) (i) 0 (ii) 0 (iii) zero k
component (iv) could be a gradient field
(b) (i) positive (ii) 0 (iii) positive k
component (iv) could not be a gradient field
(c) (i) 0 (ii) positive (iii) zero k
component (iv) could be a gradient field
42. (a) On a sphere of radius 5. (b)
6 5 1S S S
F dA F dA F dA⋅ < ⋅ < ⋅ ∫ ∫ ∫
43. 75 12 63π π π− =

44. (a) 10− (b) 29
45. (a) V (b) S (c) S (d) V (e) S (f) V
(g) S (h) ND (i) ND
46. (a) false (b) true (c) true (d) true (e) true
(a)
10 10
3 2
i j−
(b)
8
2
− (c) (18.5, 74.5) (d) 10 (e) 6(60 80 50 70)
1560+ + + =
MATH 223
FINAL EXAM REVIEW PACKET

(Fall 2012)
The following questions can be used as a review for Math 223.
These questions are not actual samples
of questions that will appear on the final exam, but they will
provide additional practice for the
material that will be covered on the final exam. When solving
these problems keep the following in
mind: Full credit for correct answers will only be awarded if all
work is shown. Exact values must be
given unless an approximation is required. Credit will not be
given for an approximation when an
exact value can be found by techniques covered in the course.
The answers, along with comments, are
posted as a separate file on http://math.arizona.edu/~calc.
1. A sonic boom carpet is a region on the ground where the
sonic boom is heard directly from the
airplane and not as a reflection. The width of the carpet, W, can
be expressed as a function of the air
temperature on the ground directly below the airplane, t, and the
vertical temperature gradient at the
airplane’s altitude, d. Suppose ( , )
t
W t d k
d
= for some positive constant k.
(a) If d is fixed, is the width of the carpet an increasing or
decreasing function of t.

(b) If t is fixed, is the width of the carpet an increasing or
decreasing function of d.
2. Describe the following sets of points in words, write an
equation, and sketch a graph:
(a) The set of points whose distance from the line L is five. The
line L is the intersection of the
plane 3y = and the xy-plane.
(b) The set of points whose distance from the yz-plane is three.
(c) The set of points whose distance from the z-axis and the xy-
plane are equal.
3. By setting one variable constant, find a plane that intersects
2 2cos 3y x z+ = in a:
(a) parabola (b) waves (related to cosine curves) (c) line(s)
4. Consider the function 2( , )f x y y x= − .
(a) Plot the level curves of the function for 2, 1, 0,1, 2z = − − .
(b) Imagine the surface whose height above any point ( , )x y is
given by ( , )f x y . Suppose you are
standing on the surface at the point where 1, 2x y= = .
(i) What is your height?
(ii) If you start to move on the surface parallel to the y-axis in
the direction of increasing y, does
your height increase or decrease?
(iii) Does your height increase or decrease if you start to move
on the surface parallel to the x-
axis in the direction of increasing x?
5. Describe the level surfaces of each:
(a) 2 2( , , )f x y z x y z= − − (b)

2 2 21( , , ) x y zg x y z e − − −=
http://math.arizona.edu/~calc�
2.5 3.0 3.5
-1.0 6 8
1.0 1 2
3.0 -6
6. The figure at the right shows the level curves of the
temperature T in degrees Celsius as a function
of t hours and depth h in centimeters beneath the surface of the
ground from midnight ( 0t = ) one day
to midnight ( 24t = ) the next.
(a) Approximately what time did the sunrise?
When do you think the sun is directly overhead?
(b) Sketch a graph of the temperature as a
function of time at 20 centimeters.
(c) Sketch a graph of the temperature as a
function of the depth at noon.
from S. J. Williamson, Fundamentals of Air Pollution,
(Reading: Addison-Wesley, 1973)
7. Given the table of some values of a linear function, complete

the table and find a formula for the
function.
8. Consider the planes:
I. 3 5 2x y z− − = II. 5 3x y= + III. 5 3 2x y+ =
IV. 3 5 2x y+ = V. 3 5 2x y z+ + = VI. 1 0y + =
List all of the planes which:
(a) Are parallel to the z-axis.
(b) Are parallel to 3 5 7x y z= + + .
(c) Contain the point (1, 1, 6)− .
(d) Are normal to ( ) ( )2 3 3i k i k+ × −
.
(e) Could be the tangent plane to a surface ( , )z f x y= , where f
is some function which has finite
partial derivatives everywhere.
9. A portion of the graph of a linear function is shown.
(a) Find an equation for the linear function.
(b) Find a vector perpendicular to the plane.
(c) Find the area of the shaded triangular region.

x
-2 -1 0 1 2
2 0.111 0.167 0.200 0.167 0.111
1 0.167 0.333 0.500 0.333 0.167
y 0 0.200 0.500 1.000 0.500 0.200
-1 0.167 0.333 0.500 0.333 0.167
-2 0.111 0.167 0.200 0.167 0.111
x
-2 -1 0 1 2
2 0.00 -3.00 -4.00 -3.00 0.00
1 3.00 0.00 -1.00 0.00 3.00
y 0 4.00 1.00 0.00 1.00 4.00
-1 3.00 0.00 -1.00 0.00 3.00
-2 0.00 -3.00 -4.00 -3.00 0.00
x
y
x
y
x
y
x

-2 -1 0 1 2
2 2.828 2.236 2.000 2.236 2.828
1 2.236 1.414 1.000 1.414 2.236
y 0 2.000 1.000 0.000 1.000 2.000
-1 2.236 1.414 1.000 1.414 2.236
-2 2.828 2.236 2.000 2.236 2.828
10. Match each of the following functions (a) – (f), given by a
formula, to the corresponding tables,
graphs, and/or contour diagrams (i) – (ix). There may be more
than one representation or no
representations for a formula.
(a) 2 2( , )f x y x y= − (b) ( , ) 6 2 3f x y x y= − + (c) 2 2( , )
1f x y x y= − −
(d) 2 2
1
( , )
1
f x y
x y
=
+ +
(e) ( , ) 6 2 3f x y x y= − − (f) 2 2( , )f x y x y= +
(i) Table 1 (ii) Table 2 (iii) Table 3

(iv) (v) (vi)
(vii) (viii) (ix)
11. Let 3 2 2v i j k= + −

12. Consider the vectors 2 3u i j k= − +
(a) For what value(s) o
the point (1, 2, 3)− .
containing the point (1, 2, 3)− .
13. Let 3 2u j k= −
-plane such that the
3
π

14. Find the following:
(a) ( ) ( )( )3 2 2ln 3 arctanx y x y
x
∂
+ − +
∂
(b) Hf if
( )3
2
( , )
5
H T
f H T
H
+
=
−
(c)
2 x y
x y y x
15. Find an equation for the tangent plane to:

(a) ( , ) xyf x y ye= at ( , ) (1, 2)x y = (b) 2 2 2( 1) 4( 2) ( 3)
17x y z− + − + − = at (3, 3, 6)
16. A ball is thrown from ground level with initial speed v
(m/sec) and at an angle of α with the
horizontal. It hits the ground at a distance
2 sin(2 )
( , )
v
s v
g
α
α = where 10 g ≈ m/sec2.
(a) Find the differential ds .
(b) What does the sign of (20, 3)sα π tell you?
(c) Use the linearization of s about (20, 3)π to estimate the
change in α that is needed to get
approximately the same distance if the initial speed changes to
19 m/sec.
17. The depth of a lake at the point ( , )x y is given by 2 2( , ) 2
3h x y x y= + feet. A boat is at (-1,2).
(a) If the boat sails in the direction of the point (3, 3) , is the
water getting deeper or shallower?
(b) In which direction should the boat sail for the depth to
remain constant? Give your answer as a
vector.
(c) If the boat moves on the curve ( ) ( )2( ) 1 2r t t i t j= − + +

changing when 2t = ?
18. Calculate the following:
(a)
2
21
yz
grad
x
(b) ( ) ( ) ( )( )2 2 2curl x y z i y z j xz k+ + − + +
(c) ( ) ( ) ( )( )2 3cos sec zdiv x i x y j e k+ +
(d) The greatest rate of change of 3( , , ) tanf x y z x z= + at (
4 , 3,1)π .

(e) The potential function for cos( )z zG yi xj e e k= + +
.
19. Sup
direction u
5
6
π
.
20. The contour plot for ( , )f x y is shown at the right.
Determine if each quantity is positive, negative, or zero.
(a) (1,1)xf (b) ( 1,1)xf −
(c) ( 2, 2)yf − − (d) (1,1) (1, 2)x xf f−
(e) (1,1)yyf (f) ( 2, 2)xyf − −
21. Let ( , ) 3 cos( )w x y x yπ= .

(a) Find
(1, 1 2)
w
u
∂
∂
and
(1, 1 2)
w
v
∂
∂
if 2 2x u v= + and
v
y
u
= .
(b) Find
1t
dw
dt =
if tx e−= and lny t= .

22. Suppose ( , )z f x y= , ( , )x g r θ= , and ( , )y h r θ= . Find (
)1, 2rz π if given the following:
( )1, 2 0g π = , ( )1, 2 1h π = , (0,1) 2xf = , (0,1) 3yf = ,
( )1, 2 5rg π = , ( )1, 2 7gθ π = , ( )1, 2 9rh π = , ( )1, 2 11hθ
π =
23. Find and classify all of the critical points for 3 2 3 2( , ) 2 3
12 3 9f x y x x x y y y= − − + + − .
24. Let 2 2( , ) 4f x y Kx y xy= + − .
(a) Verify that the point (0, 0) is a critical point.
(b) Determine the values of K, if any, for which (0, 0) can be
classified as the following.
(i) a saddle point (ii) a local minimum (iii) a local maximum
25. Find the minimum distance from the surface 2 3 9z x xy+ −
= to the origin.
26. Find an equation for each surface:
(a) 2 2 8x y+ = in cylindrical coordinates (b) y x= in
cylindrical coordinates
(c) 2 2z x y= − + in spherical coordinates (d) 10z = in
spherical coordinates
x
2
27. Determine (without calculation) whether the integrals are

positive, negative, or zero. Let D be the
region inside the unit circle centered at the origin, T be the top
half of the region, B be the bottom half
of the region, L be the left half of the region, and R be the right
half of the region.
(a) x
T
e dA−∫ (b) cosB ydA∫ (c) ( )L x y dA+∫ (d)
y
R
ye dA−∫
28. Evaluate each of the integrals:
(a)
3 6
0 0
cos(3 ) sin(2 5)y x dydx+∫ ∫ (b)
5 2
0 0 0
sin d d d
π π
ρ φ ρ φ θ∫ ∫ ∫
(c)
9 3 3

0
1
y
x dxdy+∫ ∫ (d)
( )3 22 2 2 2 2 2 2 4 4
0 0 0
x x y x y z
e dzdydx
− − − − + +
∫ ∫ ∫
(e)
R
xdA∫ where R is shown below. (f)
2 5 sin
4 0
rdrd
π θ
π
θ∫ ∫ Hint: sketch R first.

29. Set up integrals needed to find the following:
(a) The volume between the sphere 2ρ = and the cone z r= .
(Cartesian, cylindrical, and spherical)
(b) The volume between 2 220x y z= − − and 2 2 2x y z= + + .
(Cartesian and cylindrical)
(c) The volume of the solid in the first octant bounded from
above by 2 2 16x z+ = and 12y = .
(Cartesian in the order dxdydz and Cylindrical in the order
rdydrdθ )
(d) The volume of the tetrahedron under the portion of the plane
shown at the right, bounded by the planes 0y = , 0x = , and
0z = . (Cartesian)
30. A pile of dirt is approximately in the shape of 2 24z x y= −
+ , where x, y, and z are in meters. The
density (kg/m3) of the dirt is proportional to the distance from
the top of the pile of dirt. Set up an
integral for the mass of the pile of dirt.
31. Give parametric equations for the following curves:
(a) A circle of radius 3 on the plane 1y = centered at (2,1, 0)
oriented clockwise when viewed from
the origin.
(b) A line perpendicular to 2 3 7z x y= − + and through the
point (1, 2, 3)− .
(c) The curve ( )32y x= + oriented from (2, 64) to (0, 8) .
(d) The intersection of the surfaces 2 2 2z x y= + and 2 26z x y=
− − .

y
2
6−
12
1
32. A child is sliding down a helical slide. Her position at time t
seconds after the start is given in feet
by ( ) ( )3 cos 3sin (10 )r t i t j t k= + + −
-plane.
(a) When is the child 6 feet from the ground?
(b) How fast is the child traveling at 2 seconds?
(c) At time 2t π= seconds, the child leaves the slide tangent to
the slide at that point. What is the
equation of the tangent line?
33. The surface of a hill is represented by 2 212 3z x y= − − ,
where x and y are measured horizontally.
A projectile is launched from the point (1,1, 7) and travels in a
line perpendicular to the surface at that
point.
(a) Find parametric equations for the path.
(b) Does the projectile pass through the point (6,16,10) ?

34. Match the vector field to its sketch.
(a) xi yj+
(b) xi yj−
(c) yi xj+
(d) yi
(e) i xj+
(f) 2x i xyj+
(i) (ii) (iii)
(iv) (v) (vi)
2C
35. Given the plot of the vector field, F

, list the following quantities in increasing order. Also give a
possible formula for F
.
(i)
1C
F dr⋅ ∫
(ii)
2C
F dr⋅ ∫
(iii)
3C
F dr⋅ ∫
36. Evaluate

C
F dr⋅ ∫
(a) ( )F x z i zj yk= + + +
. C is the line from (2, 4, 4) to (1, 5, 2) .
(b) 2 sin( ) sin( )F x i z yz j y yz k= + +
. C is the curve from (0, 0,1)A to (3,1, 2)B as shown below.
(c) F yi xj zk= − +
. C is the circle of radius 3 centered on the z-axis in the plane

4z = oriented
clockwise when viewed from above.
(d) 34 ( )F x i x y j= + +
. C is the curve sin(2 )y x= from (0, 0) to ( 2 , 0)π .
(e) ( ) ( )3 2 3 2sin( ) ln( 1)F y x i x y j= − + + − +
. C is the circle of radius 5 centered at (0, 0) in the xy-
plane oriented counterclockwise.
37. Evaluate
S
F dA⋅ ∫
:
(a) 3 4 ( )F i j z x k= + + −
. S is a square of side 2 on the plane z x= oriented upward.
(b) 5F i zj yk= − + −

. S is 2 2x y z= + for 0 8x≤ ≤ , oriented in the negative x-
direction.
(c) 3 52 ( ) ( 7 )F xi z y j x z k= − − + +
. S is the closed cylinder centered on the y-axis with radius 3,
length 5, oriented outward.
(d) F xi yj zk= + +
. S is the part of the surface 2 225 ( )z x y= − + above the disk
of radius 5 centered
at the origin, oriented upward.
38. (a) Evaluate 2 3( )
C
grad x yz dr⋅ ∫
(1,1) in the xy-
plane,

oriented counterclockwise.
(b) Evaluate 2( ( ) )
S
curl x i y z j xzk dA− + + ⋅ ∫
where S is the cube of side 4 centered at (2,1, 3) , oriented
outward.
1C
3C
P• P•
39. Consider the flux of the vector field p
r
H
r
=
for 0p ≥ out of the sphere of radius 5 centered at

the origin. For what value of p is the flux a maximum? What is
that maximum value?
40. An oceanographic vessel suspends a paraboloid-shaped net
below the ocean at a depth of 1000 feet,
held open at the top by a circular metal ring of radius 20 feet,
with bottom 100 feet below the ring and
just touching the ocean floor. Set up coordinates with the origin
at the point where the net touches the
ocean floor and with z measured upward. Water is flowing with
velocity
2
2 (1100 ) (1100 )xv xzi xe j z z k−= − + + −
top of the paraboloid shape to find the flux of water through the
net (oriented from inside to outside).
41. The vector fields below have the form 1 2F F i F j= +
. Assume 1F and 2F depend only on x and y.
For each vector field, circle the best answers.
(a) (b) (c)

(i)
C
F dr⋅ ∫
zero
(ii) ( )divF P
is positive negative zero
(iii) curlF
at P has positive k
component negative k
component zero k
component
(iv) F

could be a gradient field could not be a gradient field
42. Let 3 3 3(75 )F x x i y j z k= − − −
and let 1S , 5S , and 6S be spheres of radius 1, 5, and 6
respectively,
centered at the origin.
(a) Where is 0divF =
?
(b) Without computing the flux, order the flux out of the
spheres from smallest to largest.
43. In the region between the circles 2 21 : 4C x y+ = and
2 2
2 : 25C x y+ = in the xy-plane, the vector
field F
has 3curlF k=

. If 1C and 2C are both oriented counterclockwise when viewed
from above,
find the value of
2 1C C
F dr F dr⋅ − ⋅ ∫ ∫
P•
y
44. Let ( )27 7F xyi x y j= + +
.
(a) Find the circulation density of F
around k
at (2,1, 3)
(b) Find the flux density of F

at ( 1, 4, 2)−
45. Determine if each of the following quantities is a vector
(V), a scalar (S), or is not defined (ND).
-D vectors, r xi yj zk= + +
is
a differentiable 3-D vector field, and f is a differentiable scalar
function of x, y, and z.
(a) ( )curlG r×
u v
r
⋅

(f) ( )curl fG
(g) ( )
C
curlG dr⋅ ∫
S
divG dA⋅ ∫
(i) gradG
46. True or False?
(a) If all of the contours of a function ( , )g x y are parallel
lines, then the function must be linear.
(b) If curlF
is parallel to the x-axis for all x, y, and z and if C is a circle in
the xy-plane, then the
circulation of F

around C must be zero.
(c) If f is a differentiable function, then ( , ) ( , )uf a b f a b≥ −
∇
(d) If F
is a divergence free vector field defined everywhere and S is a
closed surface oriented inward,
then 0
S
F dA⋅ =∫
.
(e) If G
is a curl free vector field defined everywhere and C is a simple
closed path, then 0
C
G dr⋅ =∫
47. Use the portion of the contour diagram of ( , )f x y shown
below to estimate the following:

he direction i j− +
(c) A critical point of ( , )f x y . (d)
C
f dr∇ ⋅ ∫
(e) ( , )
R
f x y dA∫ where R is the rectangle 9 15x≤ ≤ , 76 80y≤ ≤ .
x

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