Sample Chapter

INSTANT DOWNLOAD COMPLETE TEST BANK WITH ANSWERS

 

Test Bank Of Calculus Concepts and Contexts 4th Edition By James Stewart

 

 

SAMPLE QUESTIONS

 

Section 1.3: New Functions From Old Functions

 

  1. , then f(2x) is equal to
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B                    PTS:   1

 

  1. Let f(x) =  and g(x) = . Find the domains of .
a. (¥, 0] e. [¥)
b. (2, ¥) f. (¥, )
c. (¥, -2) g. (¥, ] [2, ¥)
d. (¥, 2)  (2, ¥) h.

 

 

ANS:  H                    PTS:   1

 

  1. Let f(x) =  and g(x) = . Find the domains of
a. [1, ¥) e. [3, 3]
b. [, ] f. (¥, -1]  [1, ¥)
c. (¥, ]  [, ¥) g. (¥, -1]
d. (¥, -3]  [3, ¥) h. [, ¥)

 

 

ANS:  E                    PTS:   1

 

  1. Let h(x) = sin x  3 sin x  4 and g(x)  sin x. Find f(x) so that h(x)
a. f (x) = (3x + 2) e. f (x) = 3 4x
b. f (x) = x + 3 f. f (x) =  3x  4
c. f(x) = 3x  4 g. f (x) = 4
d. f(x) = x 3x  4 h. f (x) = (x  4)

 

 

ANS:  F                    PTS:   1

 

  1. Let f(x) = 3x  2 and g(x)  2  3x. Find the value of (f ° g)(x) when x = 3..
a. 23 e. 3
b. 9 f. 6
c. 6 g. 9
d. 3 h. 23

 

 

ANS:  A                    PTS:   1

 

  1. Let f(x) = 2  x and g(x)  3  x. Find the value of  when x = 5.
a. 510 e. 5
b. 5 f. 10
c. 2 g. 127
d. 0 h. 130

 

 

ANS:  F                    PTS:   1

 

  1. Let f(x) = x and . Find g(2).
a. 0 e. 8
b. 1 f. 16
c. 2 g. 32
d. 4 h. 64

 

 

ANS:  E                    PTS:   1

 

  1. Relative to the graph of y = x + 2, the graph of y = (x  2)  2 is changed in what way?
a. Shifted 2 units upward
b. Compressed vertically by a factor of 2
c. Compressed horizontally by a factor of 2
d. Shifted 2 units to the left
e. Shifted 2 units to the right
f. Shifted 2 units downward
g. Stretched vertically by a factor of 2
h. Stretched horizontally by a factor of 2

 

 

ANS:  E                    PTS:   1

 

  1. Relative to the graph of y = x, the graph of y = x  2 is changed in what way?
a. Shifted 2 units downward
b. Stretched horizontally by a factor of 2
c. Shifted 2 units to the right
d. Stretched vertically by a factor of 2
e. Compressed horizontally by a factor of 2
f. Compressed vertically by a factor of 2
g. Stretched vertically by a factor of 2
h. Stretched horizontally by a factor of 2

 

 

ANS:  A                    PTS:   1

 

  1. Relative to the graph of y = x , the graph of y = x is changed in what way?
a. Compressed horizontally by a factor of 2
b. Shifted 2 units downward
c. Stretched vertically by a factor of 2
d. Stretched horizontally by a factor of 2
e. Shifted 2 units upward
f. Compressed vertically by a factor of 2
g. Shifted 2 units to the right
h. Shifted 2 units to the left

 

 

ANS:  F                    PTS:   1

 

  1. Relative to the graph of y = x  2, the graph of y = 4x  2 is changed in what way?
a. Compressed vertically by a factor of 2
b. Stretched horizontally by a factor of 2
c. Compressed horizontally by a factor of 2
d. Shifted 2 units upward
e. Shifted 2 units to the right
f. Stretched vertically by a factor of 2
g. Shifted 2 units to the left
h. Shifted 2 units downward

 

 

ANS:  C                    PTS:   1

 

  1. Relative to the graph of y = sin x, the graph of y = 3 sin x is changed in what way?
a. Compressed horizontally by a factor of 3
b. Shifted 3 units to the right
c. Compressed vertically by a factor of 3
d. Shifted 3 units upward
e. Shifted 3 units to the left
f. Stretched vertically by a factor of 3
g. Shifted 3 units downward
h. Stretched horizontally by a factor of 3

 

 

ANS:  F                    PTS:   1

 

  1. Relative to the graph of y = e, the graph of y =  is changed in what way?
a. Shifted 5 units upward
b. Shifted 5 units downward
c. Shifted 5 units to the right
d. Shifted 5 units to the left
e. Stretched horizontally by a factor of 5
f. Stretched vertically by a factor of 5
g. Compressed horizontally by a factor of 5
h. Compressed vertically by a factor of 5

 

 

ANS:  D                    PTS:   1

 

  1. Relative to the graph of y = sin x, where x is in the radians, the graph of y = sin x, where x is in degrees, is changed in what way?
a. Compressed horizontally by a factor of
b. Stretched vertically by a factor of
c. Compressed horizontally by a factor of
d. Stretched horizontally by a factor of
e. Compressed vertically by a factor of
f. Stretched vertically by a factor of
g. Stretched horizontally by a factor of
h. Compressed vertically by a factor of

 

 

ANS:  G                    PTS:   1

 

  1. Let f (x) = 8 + x. Find each of the following:

 

(a)  f (2)  f ()

 

(b)  f (x  2)

 

(c)  [f (x)]

 

(d)  f (x)

 

ANS:

(a)  24

(b)

(c)

(d)

 

PTS:   1

 

  1. Let f (x) = . Find each of the following:

 

(a)  f (0)  f ()

 

(b)  f (x  2)

 

(c)  [f (x)]

 

(d)  f (x)

 

ANS:

(a)  f (0)  f () =  = 1

(b)  f (x  2) =  =

(c)  [ f (x)] = 2x  5, x

(d)  f (x) =

 

PTS:   1

 

  1. Let f (x) = . Find each of the following:

 

(a)  f (0)  f ()

 

(b)  f (x  2)

 

(c)  [f (x)]

 

(d)  f (x)

 

ANS:

(a)  f (0)  f () =  = 4 27.46

(b)  f (x  2) =  = = ,  x  2

(c)  [ f (x)] = 16x,

(d)  f (x) = = , 0  x  2

 

PTS:   1

 

  1. Let f (x) = , x  . Find each of the following:

 

(a)  f (1)  f ()

 

(b)  f (x 3)

 

(c)  f (x) 3

 

(d)  [f (x 3)]

 

ANS:

(a)  f (1)  f ()   1

(b)  f (x3)  =, x

(c)  f (x)  3

(d.)  , x  0

 

PTS:   1

 

  1. Evaluate the difference quotient  for f(x) .

 

ANS:

 

 

PTS:   1

 

  1. Given the graph of y = f(x):

 

Sketch the graph of each of the following functions:

 

(a)

 

(b)

 

(c)

 

(d)

 

(e)

 

(f)

 

(g)

 

(h)  f

 

(i)  1

 

(j)  1

 

ANS:

 

PTS:   1

 

  1. Use the graphs of f and g given below to estimate the values of f(g(x)) for x = , , , 0, 1, 2, and 3, and use these values to sketch a graph of y  f(g(x)).

 

 

ANS:

 

x 3 2 1 0 1 2 3
f(g(x)) 0.5 3.88 3.50 2.88 3.50 3.88 .05

 

 

PTS:   1

 

  1. f and g are functions defined by the following table.

 

x 3 2 1 0 1 2 3
f(x) 5 4 3 2 1 2 3
g(x) _4 1 1 2 1 1 4

 

Determine the following:

 

(a)  (f   g)(2)

 

(b)  (f  g)(1)

 

(c)

 

(d)

 

(e)

 

(f)

 

(g)

 

(h)

 

ANS:

(a)  1

(b)  2

(c)  4

(d)

(e)  1

(f)  4

(g)  4

(h)  1

 

PTS:   1

 

  1. Find functions f and g such that F(x)  1 2 cosx

 

ANS:

2x, g(x) = cos x is one possible answer. Answers will vary.

 

PTS:   1

 

  1. Find functions f and g such that F(x) = 1   =

 

ANS:

f (x) = , g (x) = 1  cos x is one possible answer. Answers will vary.

 

PTS:   1

 

  1. Find functions f and g such that F(x) = e =

 

ANS:

f (x) = e, g(x)  sin x is one possible answer. Answers will vary.

 

PTS:   1

 

 

Section 3.1: Derivatives of Polynomials and Exponential Functions

 

  1. If , find .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H                    PTS:   1

 

  1. Find the derivative of .
a. e.
b. f.
c. g.
d. h. None of the above

 

 

ANS:  A                    PTS:   1

 

  1. Find the derivative of .
a. e.
b. f.
c. g.
d. h. None of the above

 

 

ANS:  D                    PTS:   1

 

  1. Find the derivative of .
a. e.
b. f.
c. g.
d. h. None of the above

 

 

ANS:  C                    PTS:   1

 

  1. If , find .
a. e.
b. f.
c. g. 2
d. h.

 

 

ANS:  G                    PTS:   1

 

  1. If , find
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B                    PTS:   1

 

  1. If , find
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C                    PTS:   1

 

  1. Find the slope of the normal line to the curve  at the point (1, .
a. 8 e.
b. f.
c. 3 g.
d. 1 h. 0

 

 

ANS:  E                    PTS:   1

 

  1. Find the y-intercept of the tangent line to the curve y = xat the point (1, 1).
a. 4 e. 8
b. 4 f. 8
c. 16 g. 2
d. 2 h. 16

 

 

ANS:  G                    PTS:   1

 

  1. The curve y = x +x x has two horizontal tangents. Find the distance between these two horizontal lines.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C                    PTS:   1

 

  1. Passing through the origin (0, 0), there are two lines tangent to the curve , one with negative slope, the other with positive slope. Find the value of the positive slope.
a. e. 4
b. f. 1
c. g. 3
d. h. 2

 

 

ANS:  H                    PTS:   1

 

  1. At how many different values of x does the curve y = x 2x have a tangent line parallel to the line ?
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. 7

 

 

ANS:  C                    PTS:   1

 

  1. Given , ,  and , find the value of .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H                    PTS:   1

 

  1. Given , ,  and , find the value of .
a. 1.8 e. 0.9
b. 0.3 f.
c. g. 3.5
d. h.

 

 

ANS:  F                    PTS:   1

 

  1. If , find .
a. 1 e. 3
b. 4 f. 0
c. 2 g. 5
d. 6 h. 8

 

 

ANS:  D                    PTS:   1

 

  1. If , find .
a. 48 e. 21
b. 32 f. 12
c. 29 g. 4
d. 34 h. 0

 

 

ANS:  F                    PTS:   1

 

  1. If , then, .
a. 90 e. 20
b. 360 f. 72
c. 30 g. 120
d. 60 h. 240

 

 

ANS:  G                    PTS:   1

 

  1. If , find .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D                    PTS:   1

 

  1. Find

 

(a)

 

(b)

 

(c)

 

(d)

 

ANS:

(a)

(b)

(c)

(d)

 

PTS:   1

 

  1. Differentiate the following functions:

(a)

 

(b)

 

(c)

 

(d)

 

ANS:

(a)

(b)

(c)

(d)

 

PTS:   1

 

  1. Differentiate the following functions:

 

(a)

 

(b)

 

(c)

 

(d )

 

ANS:

(a)

(b)

(c)

(d)

 

PTS:   1

 

  1. Given , find  and .

 

ANS:

 

PTS:   1

 

  1. Find an equation of the tangent line to at (-1, 2).

 

ANS:

. At (-1, 2), . The point-slope equation of the tangent line is , or .

 

PTS:   1

 

  1. There are two tangent lines to the curve  that pass through (2, 5). Find an equation for each of the lines.

 

ANS:

. We seek points (x, y) on the curve  where . When  So

 

So the lines are  or , and or .

 

PTS:   1

 

  1. The tangent line to  when x = 1 is y = x1. Determine the equation of another tangent line to this curve that is parallel to the given tangent line.

 

ANS:

From the given line, m = 1. Note that , so setting , we find x = 1 or . Using this value we get another tangent line .

 

PTS:   1

 

  1. Given , find an equation of the line(s) tangent to the graph of  and parallel to .

 

ANS:

, so . The tangent lines must have slope 3. Combining this information,  The tangent lines are given by  and .

 

PTS:   1

 

  1. Given

 

(a)  Find an equation of the tangent line to the graph of  at

 

(i)  x = 0.

 

(ii)  x = 4.

 

(iii)  x = 9.

 

(b)  Sketch a graph of  and the tangent lines you found in parts (i), (ii) and (iii) on one set of coordinate axes.

 

ANS:

(a)

(i)  The slope of the tangent is undefined. The tangent line is the vertical line y = 0.

(ii)  The slope of the tangent is . An equation of the tangent line is  or

(iii)  The slope of the tangent is . An equation of the tangent line is .

(b)

 

PTS:   1

 

  1. The position function for a particle is , where s is measured in feet and t is measured in seconds.

 

(a)  Find the velocity at t = 2.

 

(b)  When does the velocity equal zero?

 

ANS:

(a)  –16 ft/s

(b)  1.5 s

 

PTS:   1

 

  1. The position function for a particle moving along the x-axis at time t > 0 (in seconds) has its x-coordinate given by .

 

(a)  Find a formula for the velocity and acceleration of the particle at any time t.

 

(b)  What is the velocity of the particle when t = 3? What does this value indicate?

 

ANS:

(a)   and .

(b)  v(3) = 32. When the time is at 3 seconds the particle is moving at a rate of 32 units/second in the negative direction.

 

PTS:   1

 

  1. Show that the curve  has no tangent line with slope 3.

 

ANS:

 

PTS:   1

 

  1. If , what are  and ?

 

ANS:

,

 

PTS:   1

 

  1. Let . For what value of x is

 

ANS:

 

PTS:   1

 

  1. Find an equation of the line tangent to the curve

 

ANS:

and , so an equation is or .

 

PTS:   1

 

  1. Suppose that  is a differentiable function. Find  for each of the following, in terms of  and .

 

(a)

 

(b)

 

(c)

 

(d)

 

ANS:

(a)

(b)

(c)

(d)

 

PTS:   1

 

  1. Below is a table containing information about the differentiable functions f and g.

 

Suppose also that , , and .

 

(a)  Find

 

(b)  Find

 

(c)  Find

 

ANS:

(a)

(b)

(c)

 

PTS:   1

 

  1. Given , find the value(s) of x where .

 

ANS:

So  for .

 

PTS:   1

 

 

Section 3.2: The Product and Quotient Rules

 

  1. Find the derivative of .
a. e.
b. f.
c. g.
d. h. None of these

 

 

ANS:  F                    PTS:   1

 

  1. Find the derivative of .
a. e.
b. f.
c. g.
d. h. None of these

 

 

ANS:  B                    PTS:   1

 

  1. Find the derivative of .
a. e.
b. f.
c. g.
d. h. None of these

 

 

ANS:  C                    PTS:   1

 

  1. Find the slope of the tangent line to the curve y = at the point (1, ).
a. e.
b. 2 f.
c. g. 1
d. h.

 

 

ANS:  E                    PTS:   1

 

a. 6 e. 8
b. 9 f. 16
c. 4 g. 5
d. 12 h. 20

 

 

ANS:  G                    PTS:   1

 

a. 8 e. 20
b. 5 f. 9
c. 10 g. 12
d. 4 h. 16

 

 

ANS:  B                    PTS:   1

 

a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D                    PTS:   1

 

a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. 7

 

 

ANS:  D                    PTS:   1

 

  1.  find
a. –3 e. 1
b. –2 f. 2
c. –1 g. 3
d. 0 h. 4

 

 

ANS:  G                    PTS:   1

 

  1. Given
a. 1.8 e. 0.9
b. 0.4 f.
c. g. 0.77
d. h.

 

 

ANS:  C                    PTS:   1

 

  1. Given
a. 0.025 e. 1.975
b. 0.49375 f.
c. g. 0.5625
d. h.

 

 

ANS:  C                    PTS:   1

 

  1. If
a. e. 2e
b. 0 f. 4e
c. 2 g. 6e
d. 4 h. 7e

 

 

ANS:  C                    PTS:   1

 

  1. If
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  G                    PTS:   1

 

  1. If
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  G                    PTS:   1

 

  1. Find an equation of the tangent line to the curve
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D                    PTS:   1

 

Refer to the following table:

 

 

  1. Find
a. 1 e.
b. 2 f.
c. 3 g.
d. 4 h.

 

 

ANS:  A                    PTS:   1

 

  1. Find
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B                    PTS:   1

 

  1. Find
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  G                    PTS:   1

 

  1. Find
a. 0 e. 6
b. 1 f. 7
c. 3 g. 8
d. 5 h. 9

 

 

ANS:  F                    PTS:   1

 

  1. Find
a. e. 7
b. f.
c. g.
d. 1 h. 20

 

 

ANS:  C                    PTS:   1

 

  1. Find
a. e. 24
b. f. 36
c. 0 g. 48
d. 12 h. 72

 

 

ANS:  E                    PTS:   1

 

  1. Find
a. e. 14
b. f. 16
c. 2 g. 20
d. 10 h. 24

 

 

ANS:  C                    PTS:   1

 

  1. Each of the following is a derivative of a function obtained by using the Product Rule. Determine the original function:

 

(a)

 

(b)

 

ANS:

(a)

(b)

 

PTS:   1

 

  1. Each of the following is a derivative of a function obtained by using the Quotient Rule. Determine the original function:

 

(a)

 

(b)

 

(c)

 

ANS:

(a)

(b)

(c)

 

PTS:   1

 

  1. Differentiate the following functions:

 

(a)

 

(b)

 

(c)

 

(d)

 

ANS:

(a)

(b)

(c)

(d)

 

PTS:   1

 

  1. Find the derivative of the following functions:

 

(a)

 

(b)

 

(c)

 

(d)

 

ANS:

(a)

(b)

(c)

(d)

 

PTS:   1

 

  1. The quantity q of Zeng Athletic Shoes which are sold depends on the selling price p. [That is, f(p).]

 

(a)  If you know that f (150) = 14,000, what can you say about the sale of these shoes?

 

(b)  If you know that , what does that tell you about the sale of these shoes?

 

(c)  The total revenue, R, earned through the sale of Zeng shoes is given by . Find

 

(d)  Suppose that the shoes are currently priced at $150. What effect will lowering the price likely have on the total revenue? Justify your answer.

 

ANS:

(a)  14,000 pairs of shoes would be sold if the price of the shoes were $150 per pair.

(b)  With the price at $150 per pair, a dollar increase in price will decrease the number of pairs sold by 100.

(c)

(d)  Since  is negative, lowering the price would increase the total revenue.

 

PTS:   1

 

  1. f and g are functions whose graphs are given below. Let

 

(a)  Find .

 

(b)  Find .

 

(c)  Find

 

ANS:

(a)

(b)

(c)

 

PTS:   1

 

  1. If

 

ANS:

 

PTS:   1

 

  1. Find an equation of the tangent line to the curve  at .

 

ANS:

 

PTS:   1

 

  1. Given  find all values of x where

 

ANS:

When

 

PTS:   1

 

  1. Find the point(s) where the tangent to the curve has zero slope.

 

ANS:

 

PTS:   1

 

  1. Find the x-coordinate(s) of the point(s) where the tangent to the curve has zero slope.

 

ANS:

 

PTS:   1

 

  1. (a)  Differentiate  by differentiating .

 

(b)  Differentiate  by differentiating and using the result of part (a).

 

(c)  Continue as above to find  using the results from above.

 

(d)  Based upon your answers to parts (a)-(c), make a conjecture about .

 

ANS:

(a)

(b)

(c)

(d)

 

PTS:   1

 

  1. Suppose you are given a function f where determine:

 

(a)

 

(b)

 

ANS:

(a)

(b)

 

PTS:   1

Section 11.1: Functions of Several Variables

Section 11.2: Limits and Continuity

 

  1. Evaluate
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. Does not exist

 

 

ANS:  F

 

  1. Evaluate
a. 1 e. 8
b. 2 f. 9
c. 3 g. 16
d. 4 h. Does not exist

 

 

ANS:  C

 

  1. Evaluate
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. Does not exist

 

 

ANS:  A

 

  1. Evaluate
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. Does not exist

 

 

ANS:  H

 

  1. Evaluate
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. Does not exist

 

 

ANS:  B

 

  1. Evaluate
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. Does not exist

 

 

ANS:  B

 

  1. Evaluate
a. 0 e.
b. 1 f.
c. g.
d. h. Does not exist

 

 

ANS:  H

 

  1. Evaluate
a. 0 e.
b. 1 f.
c. g.
d. h. Does not exist

 

 

ANS:  H

 

  1. Evaluate
a. 1 e. 2
b. 0 f. 4
c. g. 8
d. h. Does not exist

 

 

ANS:  B

 

  1. Evaluate
a. 0 e.
b. f.
c. g.
d. 1 h. Does not exist

 

 

ANS:  H

 

  1. Evaluate
a. e.
b. f. 2
c. 0 g.
d. 1 h. Does not exist

 

 

ANS:  E

 

  1. Evaluate
a. e.
b. f.
c. g.
d. 1 h. Does not exist

 

 

ANS:  A

 

  1. Evaluate
a. e.
b. f.
c. g.
d. 1 h. Does not exist

 

 

ANS:  D

 

  1. Evaluate
a. e.
b. f.
c. g.
d. 1 h. Does not exist

 

 

ANS:  H

 

  1. Let  and let  be the curve with the equation , where  is a constant. The value of the limit of  as  approaches  along  is
a. 0 d.
b. e.
c. 1

 

 

ANS:  D

 

  1. Let  Does this function have a limit at the origin?

If so, prove it. If not, demonstrate why not.

 

ANS:

Approaching the origin along , the limit equals 1; approaching the origin along , the limit equals . Thus, this function does not have a limit at the origin.

 

  1. Prove that the following limit does not exist:  where .

 

ANS:

is defined everywhere in  except at . Let ; then , and taking the limit of as  gives 0. Let ; then  and taking the limit of  as  gives 1. Since these limits are not equal, the limit does not exist.

 

  1. Show that the limit  does not exist.

 

ANS:

Approaching  along the -axis, we have , so the limit (if it exists) equals 2. Approaching  along the -axis we have , so the limit (if it exists) equals . Since these limits are different, the limit does not exist.

 

  1. If  then
a. exists d. is equal to
b. does not exist e. is equal to 1
c. is equal to 0

 

 

ANS:  B

 

  1. Determine if  is everywhere continuous, and if not, locate the point(s) of discontinuity.

 

ANS:

is continuous everywhere its denominator does not equal zero. The limit does not exist at (0,0) and thus  is discontinuous at .

 

  1. Determine whether  is continuous at .

 

ANS:

The function is discontinuous at .

 

  1. Consider  Where is  continuous?

 

ANS:

Continuous everywhere except at .

 

  1. Let  Where is  continuous?

 

ANS:

Continuous everywhere

 

  1. Let  Where is  continuous?

 

ANS:

Continuous everywhere

 

  1. Find  if it exists, or show that the limit does not exist.

 

ANS:

Along , (if the limit exists). Along ,  (if the limit exists). Because these limits are different, does not exist.

 

  1. Consider . Where  is continuous?

 

ANS:

Continuous inside and on the sphere center  and radius 1.

 

  1. Consider . Where  is continuous?

 

ANS:

Continuous everywhere except the points on the cone whose equation is  or

 

  1. Let . Find .
a. e.
b. f.
c. g. 1
d. h. 0

 

 

ANS:  A

 

  1. Let . If , find .
a. 12 e. 32
b. 16 f. 36
c. 24 g. 42
d. 28 h. 48

 

 

ANS:  F

 

  1. Let . If , find .
a. 3 e. –3
b. 2 f. –2
c. 4 g. 1
d. –1 h. 0

 

 

ANS:  H

 

  1. Let . If , find .
a. 1 e. 8
b. 2 f. 9
c. 3 g. 16
d. 4 h. 27

 

 

ANS:  H

 

  1. Find the domain of the function .
a. All points on or to the left of e. All points on or to the left of
b. All points on or to the right of f. All points on or to the right of
c. All points to the left of g. All points to the left of
d. All points to the right of h. All points in the -plane

 

 

ANS:  B

 

  1. Find the range of the function .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Find the domain of the function .
a. All points on or to the left of e. All points on or to the left of
b. All points on or to the right of f. All points on or to the right of
c. All points to the left of g. All points to the left of
d. All points to the right of h. All points in the -plane

 

 

ANS:  H

 

  1. Find the range of the function .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Find the domain of the function .
a. All points on or to the left of e. All points on or to the left of
b. All points on or to the right of f. All points on or to the right of
c. All points to the left of g. All points to the left of
d. All points to the right of h. All points in the -plane

 

 

ANS:  D

 

  1. Find the range of the function .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C

 

  1. Describe the level curves of the function .
a. Concentric circles e. Parabolas with the same vertex
b. Non-concentric circles f. Parabolas with the same focus
c. Concentric ellipses (not circles) g. Hyperbolas with the same vertices
d. Non-concentric ellipses (not circles) h. Hyperbolas with the same foci

 

 

ANS:  A

 

  1. Describe the level curves of the function .
a. Concentric circles e. Parabolas with the same vertex
b. Non-concentric circles f. Parabolas with the same focus
c. Concentric ellipses (not circles) g. Hyperbolas with the same vertices
d. Non-concentric ellipses (not circles) h. Hyperbolas with the same foci

 

 

ANS:  C

 

  1. Identify the graph of the function .
a. Cone e. Hyperboloid of two sheets
b. Paraboloid f. Hyperbolic cylinder
c. Ellipsoid g. Elliptic cylinder
d. Hyperboloid of one sheet h. Parabolic cylinder

 

 

ANS:  B

 

  1. Describe the level curves of the function .
a. Concentric circles e. Parabolas with the same vertex
b. Non-concentric circles f. Parabolas with the same focus
c. Concentric ellipses (not circles) g. Hyperbolas with the same vertices
d. Non-concentric ellipses (not circles) h. Hyperbolas with the same foci

 

 

ANS:  E

 

  1. Describe the level curves of the function .
a. Concentric circles e. Parabolas with the same vertex
b. Non-concentric circles f. Parabolas with the same focus
c. Concentric ellipses (not circles) g. Hyperbolas with the same vertices
d. Non-concentric ellipses (not circles) h. Hyperbolas with the same foci

 

 

ANS:  F

 

  1. Identify the graph of the function .
a. Cone e. Hyperboloid of two sheets
b. Paraboloid f. Hyperbolic cylinder
c. Ellipsoid g. Elliptic cylinder
d. Hyperboloid of one sheet h. Parabolic cylinder

 

 

ANS:  D

 

  1. Find the domain of the function .
a. All points above
b. All points below
c. All points above
d. All points below
e. All points below  but not above the plane
f. All points above  but not above the plane
g. All points below  or between  and
h. All points in the -plane

 

 

ANS:  E

 

  1. Sketch the domain of the function .

 

ANS:

 

  1. Let .

 

(a)  Evaluate .

 

(b)  Sketch the domain of .

 

(c)  What is the range of the function ?

 

ANS:

(a)

(b)   and  is any real number

(c)

 

  1. Let .

 

(a)  Evaluate .

 

(b)  Sketch the domain of .

 

(c)  What is the range of the function ?

 

ANS:

(a)

(b)

(c)

 

  1. Let .

 

(a)  Evaluate .

 

(b)  Sketch the domain of .

 

(c)  What is the range of the function ?

 

ANS:

(a)

(b)  All the points in the -plane except

(c)  All real except 0.

 

  1. Let .

 

(a)  Evaluate .

 

(b)  Sketch the domain of .

 

(c)  What is the range of the function ?

 

ANS:

(a)

(b)   and  is any real number.

(c)

 

  1. Let .

 

(a)  Evaluate .

 

(b)  Sketch the domain of .

 

(c)  What is the range of the function ?

 

ANS:

(a)

(b)

(c)

 

  1. Let .

(a)  Sketch the intersection of  and  in the -plane.

(b)  Sketch the intersection of  and  in the -plane.

(c)  Sketch the intersection of  and  in the -plane.

(d)  Sketch the graph of  in .

 

ANS:

(a)

 

(b)

 

(c)

 

(d)

 

  1. Let .

 

(a)  Sketch the intersection of  and  in the -plane.

 

(b)  Sketch the intersection of  and  in the -plane.

 

(c)  Sketch the intersection of  and  in the -plane.

 

(d)  Sketch the graph of  in .

 

ANS:

(a)

 

(b)

 

(c)

 

(d)

 

  1. Let .

 

(a)  Sketch the intersection of  and  in the -plane.

 

(b)  Sketch the intersection of  and  in the -plane.

 

(c)  Sketch the intersection of  and  in the -plane.

 

(d)  Sketch the graph of  in .

 

ANS:

(a)

 

(b)

 

(c)

 

(d)

 

  1. The temperature at a point  of a flat metal plate is  where  is measured in degrees. Draw the isothermals for  0, 9, 16, and 144 degrees.

 

ANS:

 

  1. For the function , sketch the level curves  for  0, 1, 2, and 3.

 

ANS:

 

  1. Suppose the point  is on a curve  which is a level curve of the surface . Can it be concluded that the point  is also on ? Explain.

 

ANS:

Yes, given that  is a level curve of  means  has equation  for some constant . The fact that  lies on  gives , so  has equation  which does not contain the point

 

  1. Describe the level surfaces of the function .

 

ANS:

Paraboloids with vertices along the -axis

 

  1. Describe the level surfaces of the function .

 

ANS:

Parallel planes

 

  1. Describe the difference between the horizontal trace in  for the function  and the contour curve

 

ANS:

The contour curve  is the horizontal trace of  in  projected down to the -plane.

 

  1. Describe the vertical traces  and  and the horizontal traces  for the function .

 

ANS:

: , parabola; : , parabola; : , hyperbola

 

  1. Describe the level surfaces , ,  for the function .

 

ANS:

: point ; : ellipsoid ; : ellipsoid

 

  1. Describe how the graph of  is obtained from the graph of .

 

ANS:

The graph of  is the graph of  shifted 1 unit in the positive -direction, shifted 1 unit in the positive -direction, and stretched vertically (that is, in the -direction) by a factor of 2

 

  1. Let .

 

(a)  Sketch the level curves for .

 

(b)  Find a formula for the level curve that passes through the point .

 

ANS:

(a)

(b)

 

  1. The graph of level curves of  is given. Find a possible formula for  and stretch the surface .

 

ANS:

. There are other possible answers

 

  1. The graph of level curves of  is given. Find a possible formula for  and sketch the surface .

 

ANS:

. There are other possible answers.

 

  1. The graph of level curves of  is given. Find a possible formula for  and sketch the surface .

 

ANS:

Section 11.3: Partial Derivatives

 

  1. Let . Find the value of the partial derivative .
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. 7

 

 

ANS:  C

 

  1. Let . Find the value of the partial derivative .
a. e. 4
b. 1 f.
c. 2 g.
d. 3 h.

 

 

ANS:  F

 

  1. Let . Find .
a. 0 e. 4
b. 1 f.
c. g.
d. 3 h.

 

 

ANS:  B

 

  1. Let . Find .
a. 0 e. 4
b. f.
c. g.
d. 3 h.

 

 

ANS:  C

 

  1. Let . Find .
a. 0 e. 4
b. f.
c. 12 g. 16
d. 2 h. 8

 

 

ANS:  D

 

  1. Let . Find the value of the partial derivative.
a. e.
b. f.
c. g. 2
d. h. 0

 

 

ANS:  F

 

  1. Let . Find the value of the partial derivative.
a. 1 e.
b. 2 f.
c. 4 g.
d. h.

 

 

ANS:  C

 

  1. How many third-order partial derivatives does a function  have?
a. 3 e. 7
b. 4 f. 8
c. 5 g. 9
d. 6 h. 10

 

 

ANS:  F

 

  1. If all the third-order partial derivatives of  are continuous, what is the largest number of them that can be distinct?
a. 3 e. 7
b. 4 f. 8
c. 5 g. 9
d. 6 h. 10

 

 

ANS:  B

 

  1. Let . Find the value of  at the point .
a. e. 0
b. f. 4
c. g. 8
d. h. 16

 

 

ANS:  E

 

  1. Let . Find the value of the partial derivative .
a. e.
b. f.
c. 0 g.
d. h.

 

 

ANS:  B

 

  1. Let . Find the value of the partial derivative .
a. e.
b. f.
c. g.
d. h. 1

 

 

ANS:  D

 

  1. Let , . Find the value of the partial derivative .
a. 0 e.
b. 1 f.
c. g.
d. h. 2

 

 

ANS:  E

 

  1. Let , . Find the value of the partial derivative .
a. 0 e.
b. 2 f.
c. 3 g.
d. 8 h.

 

 

ANS:  H

 

  1. If , find the partial derivative of  with respect to  and the partial derivative with respect to , both at the point .

 

ANS:

;

 

  1. Let

 

(a)  Find  and .

 

(b)  Find the value of the above derivatives at , if they exist.

 

ANS:

(a)

 

(b)  ,

 

  1. Find both partial derivatives if

 

ANS:

 

  1. If , find .

 

ANS:

 

  1. Given , find  and evaluate each at .

 

ANS:

, ; ,

 

  1. Let . Find, , and .

 

ANS:

, ,

 

  1. Given , find , , and .

 

ANS:

; ,

 

  1. Let . Find all second-order partial derivatives of .

 

ANS:

, , , , , , , ,

 

  1. Find , , and  if .

 

ANS:

, ,

 

  1. If , find , , and .

 

ANS:

, ,

 

  1. If , find

 

ANS:

 

  1. Find  for

 

ANS:

 

  1. Find  for

 

ANS:

 

  1. Find  for

 

ANS:

 

  1. Find  for

 

ANS:

 

  1. Show that  satisfies the heat equation

 

ANS:

 

  1. Show that there does not exist any function  with continuous second partial derivatives such that  and

 

ANS:

which contradicts Clairaut’s Theorem.

 

  1. Show that there does not exist any function  with continuous second partial derivatives in such that  and .

 

ANS:

for  which contradicts Clairaut’s Theorem.

 

  1. Let . Compute .

 

ANS:

 

 

 

 

  1. Show that  satisfies Laplace Equation .

 

ANS:

,

 

  1. If  find  and .

 

ANS:

 

  1. If  is a differentiable function and , show that

 

ANS:

 

  1. If , find  and  and interpret these numbers as slopes. Illustrate with sketches.

 

ANS:

 

 

  1. If , find  and  and interpret these numbers as slopes. Illustrate with sketches.

 

ANS:

 

 

  1. Consider a function of three variables , where  is the monthly mortgage payment in dollars,  is the amount borrowed in dollars,  is the annual interest rate, and  is the number of years before the mortgage is paid off.

 

(a)  Suppose . What does this tell you in financial terms?

 

(b)  Suppose . What does this tell you in financial terms?

 

(c)  Suppose . What does this tell you in financial terms?

 

ANS:

(a)  If you borrow $180,000 with annual interest rate 6% and 30 year amortization, then your monthly payment is $1080.

 

(b)  If you borrow $180,000 with annual interest rate 6% and 30 year amortization, then your monthly payment increases by $115.73 per 1% increase in interest rate.

 

(c)  If you borrow $180,000 with annual interest rate 6% and 30 year amortization, then your monthly payment decreases by $12.86 for each additional year of amortization.

 

  1. Find all solutions to the partial differential equation .

 

ANS:

where  and  are any differentiable functions.

 

  1. Find all solutions to the partial differential equation .

 

ANS:

where  and  are any differentiable functions.

 

Section 11.4: Tangent Planes and Linear Approximations

 

  1. Find an equation of the tangent plant to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D

 

  1. Find an equation of the tangent plant to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  F

 

  1. Find an equation of the tangent plant to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Find an equation of the tangent plant to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  F

 

  1. Find an equation of the tangent plant to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Find an equation of the tangent plant to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Find an equation of the tangent plant to the parametric surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H

 

  1. Find an equation of the tangent plant to the parametric surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Find the differential of
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H

 

  1. Find the differential of
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Find the differential of  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Find the differential of  at the point .
a. e.
b. f.
c. g.
d. h. 0

 

 

ANS:  E

 

  1. Use a linear approximation to estimate .
a. 4.996 e. 5.022
b. 5.02 f. 5.01
c. 4.99 g. 5.11
d. 5.021 h. 5.002

 

 

ANS:  B

 

  1. Use differentials to approximate .
a. 6.85 e. 7.05
b. 6.90 f. 7.10
c. 6.95 g. 7.15
d. 7.00 h. 7.20

 

 

ANS:  G

 

  1. Use differentials to approximate .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. A boundary stripe 3 inches wide is painted around a rectangle whose dimensions are 100 feet by 200 feet. Use differentials to approximate the number of square feet of paint in the stripe.
a. 120 e. 160
b. 130 f. 170
c. 140 g. 180
d. 150 h. 190

 

 

ANS:  D

 

  1. Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the wall is 0.05 cm thick and the metal in the top and bottom is 0.1 cm thick.
a. 2.0 e. 3.6
b. 2.4 f. 4.0
c. 2.8 g. 4.4
d. 3.2 h. 4.8

 

 

ANS:  C

 

  1. A right triangle as leg  wit length 4, leg  with length 3, and hypotenuse with length 5. Use a total differential to approximate the length of the hypotenuse if leg  had length 4.2 and leg  had length 2.9.

 

ANS:

5.10

 

  1. Find the total differential of  if

 

ANS:

 

  1. Find an equation of the plane tangent to the surface  at .

 

ANS:

 

  1. If , find the values  for which , and then find an equation of the plane tangent to the graph of  at .

 

ANS:

 

  1. Find the linear approximation to the function  at  and use it to approximate .

 

ANS:

 

  1. Use a linear approximation to estimate .

 

ANS:

 

  1. Use a linear approximation to estimate .

 

ANS:

 

  1. Find the linear approximation to the function  at  and use it to approximate .

 

ANS:

 

  1. Suppose you want to give a closed cylindrical tank of radius 20 feet and height 15 feet a cost of paint 0.01 inch thick. Use the differential of the volume of the tank to estimate how many gallons of paint will be required. (1 gallon is approximately 231 cubic inches.)

 

ANS:

27.42 gallons

 

  1. The dimensions of a closed rectangular box are measured to be 60 cm, 40 cm, and 30 cm. The ruler that is used has a possible error in measurement of at most 0.1 cm. Use differentials to estimate the maximum error in the calculated volume of the box.

 

ANS:

540 cm

 

  1. Find an equation of the tangent plane to the parametric surface  at the point .

 

ANS:

 

  1. Find an equation of the tangent plane to the surface given by  at the point .

 

ANS:

 

  1. Find an equation of the tangent plane to the surface with parametric equations , ,  at the point .

 

ANS:

 

  1. Find an equation of the tangent plane to the surface with parametric equations , ,  at the point .

 

ANS:

 

  1. Find a normal vector to the surface with parametric equations  at the point .

 

ANS:

Section 11.5: The Chain Rule

 

  1. Let , and let  and  be functions of  with . Find  when .
a. 5 e. 9
b. 6 f. 10
c. 7 g. 11
d. 8 h. 12

 

 

ANS:  F

 

  1. Let , and let  and  be functions of  with . Find  when .
a. 3 e. 9
b. 4 f. 10
c. 6 g. 12
d. 8 h. 16

 

 

ANS:  B

 

  1. Let , and let  and  be functions of  with . Find  when .
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. 12

 

 

ANS:  C

 

  1. Let , and let  and  be functions of  with , and . Find  when .
a. 0 e. 2
b. f. 1
c. 4 g.
d. h.

 

 

ANS:  A

 

  1. Let , and let  and  be functions of  with , and . Find  when .
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. 7

 

 

ANS:  E

 

  1. Let , and let  and  be functions of  and  with , , and  at . Find  when .
a. 5 e. 9
b. 6 f. 10
c. 7 g. 11
d. 8 h. 12

 

 

ANS:  C

 

  1. Let , and let  and  be functions of  and  with , , and  at . Find  when .
a. 21 e. 25
b. 22 f. 26
c. 23 g. 27
d. 24 h. 28

 

 

ANS:  F

 

  1. Let , and let  and  be functions of  and  with , , and  at . Find  when .
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. 7

 

 

ANS:  E

 

  1. Let . Use implicit differentiation to find  when .
a. e.
b. f. 1
c. g.
d. h.

 

 

ANS:  B

 

  1. Let . Use implicit differentiation to find  when .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Let . Use implicit differentiation to find  when .
a. e. 1
b. f. 2
c. g. 3
d. 0 h. 4

 

 

ANS:  E

 

  1. Let , find .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  E

 

  1. Let , find .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Let , find .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C

 

  1. Let , find .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Suppose that , and . Find .

 

ANS:

 

  1. If is a function of  and , and  is a function of , then indirectly depends only on . If  and , calculate .

 

ANS:

 

  1. One side of a rectangle is increasing at 4 ft/min and another at 7 ft/min. At the time when the first side is 24 ft long and the second is 32 ft long, find

 

(a)  how fast the area is changing.

 

(b)  how fast the diagonal is changing.

 

ANS:

(a)  296 ft/min

 

(b)  8 ft/min

 

  1. The radius of a right circular cylinder is increasing at a rate of 2 cm/min and its height is decreasing at 4cm/min. At what rate is the volume changing at the instant when the radius is 4 cm and the height is 10 cm?

 

ANS:

cm/min

 

  1. Suppose that , where both  and  are changing with time. At a certain instant when  and ,  is decreasing at the rate of 2 units/s and  is increasing at the rate of 3 units/s. How fast is  changing at this instant? Is  increasing or decreasing?

 

ANS:

is decreasing

 

  1. The pressure  (in kilopascals), volume  (in liters), and temperature  (in K) of a mole of an ideal gas are related by the equation . Find the rate at which the pressure is changing when the temperature is 300K and increasing at a rate of 0.1K/s and the volume is 100 L and increasing at a rate of 0.2 L/s.

 

ANS:

kPa/s

 

  1. Suppose that , and that  and . Find  at .

 

ANS:

180

 

  1. If  is a function of  and , and  is a function of , then indirectly  depends only on : . Use the Chain Rule to write an expression for  in terms of  and .

 

ANS:

 

  1. Let , , and . Use the chain rule to find  and .

 

ANS:

, .

 

  1. Let , , and . Use the chain rule to find  and .

 

ANS:

, .

 

  1. Let , ,  and . Use the chain rule to show that .

 

ANS:

Answers may vary

 

  1. Let , ,  and . Use the chain rule to show that .

 

ANS:

Answers may vary

 

  1. Let , , and . Show that

 

ANS:

Answers may vary

 

  1. Let  and , and . Show that .

 

ANS:

Answers may vary

 

  1. Show that at , the equation  defines  implicitly as a function of  and , and then compute  and .

 

ANS:

for , which is true at . ,

 

  1. Find  and  given that  is defined implicitly as a function of  and  by the equation .

 

ANS:

,

 

  1. Use implicit differentiation to find  on the surface given by .

 

ANS:

 

  1. If , find  in terms of , , and .

 

ANS:

 

  1. If  has continuous partial derivatives, , and , show that

 

ANS:

Answers may vary

 

  1. If  has continuous partial derivatives, , , and , show that

 

ANS:

Answers may vary

 

  1. If  has continuous second partial derivatives, , and , express  in terms of , , and .

 

ANS:

 

Section 11.6: Directional Derivatives and the Gradient Vector

 

  1. Find the directional derivative of the function  at the point  in the direction .
a. 0 e. 4
b. 1 f. 5
c. 2 g. 6
d. 3 h. 7

 

 

ANS:  E

 

  1. Find the directional derivative of the function  at the point  in the direction .
a. e. 1
b. f. 2
c. g. 4
d. h.

 

 

ANS:  B

 

  1. Find the directional derivative of the function  at the point  in the direction .
a. e. 1
b. 0 f.
c. g.
d. 12 h.

 

 

ANS:  C

 

  1. Let . Find the gradient vector .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H

 

  1. Let . Find the gradient vector .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Let . Find the gradient vector  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  B

 

  1. Find the largest value of the directional derivative of the function  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  F

 

  1. Find the direction  in which the directional derivative of the function  at the point  is maximum.
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H

 

  1. Find the second directional derivative of  at the point  in the direction .
a. e.
b. f.
c. 2 g.
d. 6 h.

 

 

ANS:  F

 

  1. Find the second directional derivative of  at the point  in the direction .
a. e.
b. f.
c. g.
d. 14 h.

 

 

ANS:  G

 

  1. Find the directional derivative of the function  at the point  in the direction of the vector .
a. e.
b. f.
c. g.
d. 0 h.

 

 

ANS:  G

 

  1. Find the directional derivative of the function  at the point  in the direction from  toward the point ..
a. e. 7
b. f.
c. g.
d. h. 8

 

 

ANS:  A

 

  1. Let . Find the gradient vector  at the point .
a. i e.
b. j f.
c. k g.
d. h. 0

 

 

ANS:  E

 

  1. Find the direction of maximum increase of the function  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  G

 

  1. Find the direction of maximum increase of the function  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  D

 

  1. Find an equation of the tangent plane to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C

 

  1. Find an equation of the tangent plane to the hyperboloid  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  A

 

  1. Find an equation of the tangent plane to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  E

 

  1. Find a normal vector to the surface  at the point .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  C

 

  1. Find a point on the surface  such that the normal vector at the point is parallel to the vector .
a. e.
b. f.
c. g.
d. h.

 

 

ANS:  H

 

  1. Find a normal vector to the surface  at the point .
a. e.
b. f.
c. g.
d. h. None of these

 

 

ANS:  E

 

  1. Find the directional derivative of  at  in the direction of

 

ANS:

 

  1. Find the directional derivative of  at the point  in the direction toward the origin.

 

ANS:

 

  1. Find the directional derivative of  in the direction .

 

ANS:

 

  1. Let .

 

(a)  In which direction does  increase most rapidly at the point ?

 

(b)  What is the maximum rate of change of  at the point ?

 

(c)  Find a unit vector  such that  at .

 

ANS:

(a)

 

(b)

 

(c)

 

  1. Let  and let .

 

(a)  Find the directional derivative at  in the direction of .

 

(b)  In what direction does  increase most rapidly?

 

(c)  What is the maximum rate of change of  at the point ?

 

ANS:

(a)

 

(b)

 

(c)

 

  1. Find the directional derivative of  at the point  in the direction of

 

ANS:

 

  1. Let . Find the directional derivative of  at  in the direction of the vector .

 

ANS:

 

  1. Let . Find the rate of change of  at  in the direction from  toward .

 

ANS:

 

  1. Let . Find the rate of change of  at the point  in the direction  where  is a unit vector making an angle  with .

 

ANS:

 

  1. Let . Find the rate of change of  at the point  in the direction  where  is a unit vector making an angle  with .

 

ANS:

 

  1. Let the temperature in a flat plate be given by the function . What is the value of the directional derivative of this function at the point  in the direction ? In what direction is the plate cooling most rapidly at ?

 

ANS:

; the plate is cooling most rapidly in the direction

 

  1. A bug is crawling on the surface . When he reaches the point  he wants to avoid vertical change. In which direction should he head? (He wants the directional derivative in the -direction to be zero.)

 

ANS:

 

  1. Suppose that the equation  defines  implicitly as a function of  and . Let  be a point such that  and . Find  and .

 

ANS:

,

 

  1. Find

 

ANS:

 

  1. Given , , let  be the unit vector for which the directional derivative  has maximum value. This maximum value is
a. d.
b. 1 e. 15
c. 6

 

 

ANS:  E

 

  1. Let . Find the direction at the origin in which  is decreasing the fastest.

 

ANS:

 

  1. Given , at the point . Find

 

(a)  the maximum value of the directional derivative and

 

(b)  the unit vector in the direction in which the directional derivative takes on its maximum value.

 

ANS:

(a)

 

(b)

 

  1. If , find the gradient at the point . Also find the rate of change of  in the direction  at .

 

ANS:

;

 

  1. The surface of a certain lake is represented by a region in the -plane such that the depth under the point corresponding to  is . Zeke the dog is at the point .

 

(a)  In what direction should Zeke swim in order for the depth to decrease most rapidly?

 

(b)  In what direction would the depth remain the same?

 

ANS:

(a)

 

(b)   and

 

  1. Find an equation of the tangent plane to the surface  at the point .

 

ANS:

 

  1. Find an equation of the tangent plane to the surface  at the point .

 

ANS:

 

  1. Given , at the point  find the equation of the tangent plane to the graph of . Also, find a normal vector to the graph of .

 

ANS:

; two normal vectors are  and

 

  1. Consider the surface given by . Find an equation for the tangent plane to the surface at the point . Also, find parametric equations for the normal line to the surface at the point

 

ANS:

tangent plane: ; normal line: , ,

 

  1. The level curves of  are sketched below.

 

(a)  Find

 

(b)  Find

 

(c)  Sketch the gradient vector .

 

ANS:

(a)  About 2.5

 

(b)  About

 

(c)

 

  1. Find the directions in which the directional derivative of  at the point  has the value 1.

 

ANS:

,

 

  1. Consider the equation .

 

(a)  Sketch this surface.

 

 

(b)  Find an equation of the tangent plane to the surface at the point .

 

(c)  Find a symmetric equation of the line perpendicular to the tangent plane at the point .

 

ANS:

(a)

 

The surface is a sphere with radius 7, centered at the origin.

 

(b)

 

(c)

 

  1. Find the unit vectors  and  for  which describes the direction of maximal and minimal increase at  on the level curve .

 

ANS:

,

 

  1. In which direction does the directional derivative for  at  have value 1? Value ? Is there a direction for which the value is 4?

 

ANS:

Value 1 in direction of  and ; value  in direction of  and ; no

 

  1. What is the direction of maximal decrease for  at ?

 

ANS:

Any vector  perpendicular to , for example,

 

  1. Find the points on the hyperboloid of one sheet  where the tangent plane is parallel to the plane .

 

ANS:

,

 

  1. Given that the directional derivative of  at the point  in the direction of  is  and that , find .

 

ANS:

 

  1. Let  such that . Find an equation of the tangent line to the level curve of  that passes through point .

 

ANS:

 

  1. Find the point(s) on the surface  such that the normal vector at the point is parallel to the vector .

 

ANS:

and

 

  1. Find the point(s) on the surface  such that the tangent plane at the point is parallel to the plane

 

ANS:

and

 

  1. Let , , and .

 

(a)  Show that

 

(b)  Let  compute  by using the formula in part (a).

 

ANS:

(a)  Answers may vary

 

(b)

 

  1. Let  and .

 

(a)  Find the rate of change in the direction of .

 

(b)  Calculate  where  is a unit vector making an angle  with .

 

ANS:

(a)

 

(b)

 

  1. Let  and . Calculate  where  is a unit vector making an angle  with .

 

ANS: