Vector calculus

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 ( , ) tW 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 -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= + −



 

 and 4 3w i j k= − + 

 

 . Find each of the following: (a) A vector of length 5 parallel to w . (b) A vector perpendicular to v but not perpendicular to w . (c) The angle between v and w . (d) The component of v in the direction of w . (e) A vector perpendicular to both v and w . 12. Consider the vectors 2 3u i j k= − +



 

 and 2v ai aj k= − + − 

 

 . (a) For what value(s) of a are u and v perpendicular? (b) For what value(s) of a are u and v parallel? (c) Find an equation of the plane normal to u and containing the point (1, 2,3)− . (d) Find a parameterization for the line parallel to u and containing the point (1, 2,3)− .

13. Let 3 2u j k= − 



 . If v is a vector of length 12 in the yz-plane such that the angle between u and v is

3 π , find u v×  .

14. Find the following:

(a) ( ) ( )( )3 2 2ln 3 arctanx y x yx ∂

+ − + ∂

(b) Hf if ( )3 2( , ) 5 H Tf H T

H +

= −

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 )( , ) vs v

g αα = 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= − + +  for t in minutes, at what rate is the depth changing when 2t = ? 18. Calculate the following:

(a) 2

21 yzgrad

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. Suppose ( , ) 8uG a b = for some function ( , )G x y in the direction u  . If this is the greatest slope at the

point ( , )a b , find the value of ( , )vG a b in the direction v where the angle between u and v is 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 vy 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

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)

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∫ ∫ ∫

0 1

y x dxdy+∫ ∫ (d)

( )3 22 2 2 2 2 2 2 4 4 0 0 0

x x y x y ze 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= − − .

6−

32. A child is sliding down a helical slide. Her position at time t seconds after the start is given in feet by ( ) ( )3cos 3sin (10 )r t i t j t k= + + −



 

 . The ground is the xy-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−  

(d) yi 

(e) i xj+  

(f) 2x i xyj+  

(i) (ii) (iii)

(iv) (v) (vi)

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⋅∫

 where C is the square of side 2 centered at (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.

P• P•

39. Consider the flux of the vector field p rH

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 (1100 ) (1100 )xv xzi xe j z z k−= − + + − 

 

 . Use the Divergence Theorem and the flux through the open 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⋅∫ 

 is positive negative 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•

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). Assume that u and v are 3-D vectors, r xi yj zk= + +



 

 , S is a smooth surface, C is a smooth curve, G 

is a differentiable 3-D vector field, and f is a differentiable scalar function of x, y, and z.

(a) ( )curlG r× 

 (b) ( )div G r× 

 (c) ( , , )uf a b c (d) ( )divG r 

 (e) u v r ⋅  

(f) ( )curl fG 

(g) ( ) C

curlG dr⋅∫ 

 (h) ( ) 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: (a) (15,78)gradf (b) (15,76)uf  in the direction i j− +

 

f dr∇ ⋅∫  where C is the path from (15,76) to (24,76) .

(e) ( , ) R

f x y dA∫ where R is the rectangle 9 15x≤ ≤ , 76 80y≤ ≤ .