Question 1

For the given expression   , find y'' and sketch the general shape of the graph of y = f(x).
-y' =   - 1

Accepted Answer

A)  
  
B)  
  
C)  
  
D)  
  
A)  
  
B)  
  
C)  
  
D)

Question 2

Choose the one alternative that best completes the statement or answers the question. 
-A small frictionless cart, attached to the wall by a spring, is pulled 10 cm back from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is   . What is the cart's maximum speed? When is the cart moving that fast? What is the magnitude of of the acceleration then?

Accepted Answer

A)  $\pi$ $\approx$ 3.14 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/   
B)  10$\pi$ $\approx$ 31.42 cm/sec; t = 0 sec, 1 sec, 2 sec, 3 sec; acceleration is 0 cm/   
C)  10$\pi$ $\approx$31.42 cm/sec; t = 0.5 sec, 2.5 sec; acceleration is 1 cm/   
D)  10$\pi$ $\approx$ 31.42 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/   
A)  $\pi$ $\approx$ 3.14 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/   
B)  10$\pi$ $\approx$ 31.42 cm/sec; t = 0 sec, 1 sec, 2 sec, 3 sec; acceleration is 0 cm/   
C)  10$\pi$ $\approx$31.42 cm/sec; t = 0.5 sec, 2.5 sec; acceleration is 1 cm/   
D)  10$\pi$ $\approx$ 31.42 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/

Question 3

Choose the one alternative that best completes the statement or answers the question. 
-The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a   cylindrical log. (Round answers to the nearest tenth.)

Accepted Answer

A)  w = 6.9 in.; d = 9.8 in. 
B)  w = 7.9 in.; d = 8.8 in. 
C)  w = 7.9 in.; d = 10.8 in. 
D)  w = 5.9 in.; d = 10.8 in. 
A)  w = 6.9 in.; d = 9.8 in. 
B)  w = 7.9 in.; d = 8.8 in. 
C)  w = 7.9 in.; d = 10.8 in. 
D)  w = 5.9 in.; d = 10.8 in.

Question 4

Find the largest open interval where the function is changing as requested.
-Decreasing y =   + 7

Accepted Answer

A)  (-7, 0) 
B)  (0, $\infty$) 
C)  (7, $\infty$) 
D)  (-7, 7) 
A)  (-7, 0) 
B)  (0, $\infty$) 
C)  (7, $\infty$) 
D)  (-7, 7)

Question 5

Find the linearization L(x) of f(x) at x = a.
-f(x) = 3   + 4x + 1, a = 3

Accepted Answer

A)  L(x) = 22x + 28 
B)  L(x) = 14x + 28 
C)  L(x) = 14x - 26 
D)  L(x) = 22x - 26 
A)  L(x) = 22x + 28 
B)  L(x) = 14x + 28 
C)  L(x) = 14x - 26 
D)  L(x) = 22x - 26

Question 6

Use l'Hopital's Rule to evaluate the limit.
-

Accepted Answer

A)  -4 
B)  0 
C)  4 
D)  8 
A)  -4 
B)  0 
C)  4 
D)  8

Question 7

Find the most general antiderivative.      
-  d$\theta$

Accepted Answer

A)  sin $\theta$ + C 
B)  cos $\theta$ + C 
C)  csc$\theta$ + cos $\theta$ + C 
D)  sin $\theta$ + $\theta$ + C 
A)  sin $\theta$ + C 
B)  cos $\theta$ + C 
C)  csc$\theta$ + cos $\theta$ + C 
D)  sin $\theta$ + $\theta$ + C

Question 8

Find the absolute extreme values of the function on the interval.
-g(x) = -   + 5x - 6, 2 $\le$ x $\le$ 3

Accepted Answer

A)  absolute maximum is 1/4 at x = 7/2; absolute minimum is 0 at 3 and 0 at x = 2 
B)  absolute maximum is 5/4 at x = 7/2; absolute minimum is 0 at 3 and 0 at x = 2 
C)  absolute maximum is 49/4 at x = 5/2; absolute minimum is 0 at 3 and 0 at x = 2 
D)  absolute maximum is 1/4 at x = 5/2; absolute minimum is 0 at 3 and 0 at x = 2 
A)  absolute maximum is 1/4 at x = 7/2; absolute minimum is 0 at 3 and 0 at x = 2 
B)  absolute maximum is 5/4 at x = 7/2; absolute minimum is 0 at 3 and 0 at x = 2 
C)  absolute maximum is 49/4 at x = 5/2; absolute minimum is 0 at 3 and 0 at x = 2 
D)  absolute maximum is 1/4 at x = 5/2; absolute minimum is 0 at 3 and 0 at x = 2

Question 9

Find the largest open interval where the function is changing as requested.
-Increasing f(x) =

Accepted Answer

A)  (1, $\infty$) 
B)  (-$\infty$, 0) 
C)  (-$\infty$, 1) 
D)  (0, $\infty$) 
A)  (1, $\infty$) 
B)  (-$\infty$, 0) 
C)  (-$\infty$, 1) 
D)  (0, $\infty$)

Question 10

Express the relationship between a small change in x and the corresponding change in y in the form   . 
-y =

Accepted Answer

A)  
  
B)  
  
C)  
  
D)  
  
A)  
  
B)  
  
C)  
  
D)

Question 11

Determine the indefinite integral. Check your work by differentiation.
-  d$\theta$

Accepted Answer

A)  -cot $\theta$+ C 
B)  cot $\theta$ + C 
C)  -cos$\theta$+ C 
D)  cos $\theta$ + C 
A)  -cot $\theta$+ C 
B)  cot $\theta$ + C 
C)  -cos$\theta$+ C 
D)  cos $\theta$ + C

Question 12

Solve the problem.
-V =   $\pi$   , where r is the radius, in centimeters. By approximately how much does the volume of a sphere increase when the radius is increased from 2.0 cm to 2.1 cm? (Use 3.14 for  $\pi$.)

Accepted Answer

A)  0.3   
B)  5.2   
C)  5.0   
D)  4.8   
A)  0.3   
B)  5.2   
C)  5.0   
D)  4.8

Question 13

Use l'Hopital's Rule to evaluate the limit.
-

Accepted Answer

A)  13 
B)  10 
C)  -7 
D)  8 
A)  13 
B)  10 
C)  -7 
D)  8

Question 14

A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the function. Use preliminary analysis and graphing to determine good initial approximations. Round approximations to six decimal places.
-f(x) = 3 sin x on [0, 2π]

Accepted Answer

A)  x = 0 and x $\approx$ 2.278863 
B)  x = 0 and x $\approx$ 2.389974 
C)  x = 0 and x $\approx$ 2.167752 
D)  x = 0 and x $\approx$ 1.756641 
A)  x = 0 and x $\approx$ 2.278863 
B)  x = 0 and x $\approx$ 2.389974 
C)  x = 0 and x $\approx$ 2.167752 
D)  x = 0 and x $\approx$ 1.756641

Question 15

Find the extreme values of the function and where they occur.
-y =

Accepted Answer

A)  Absolute minimum value is   at x =   ; no maximum value. 
B)  Absolute maximum value is   at x =   ; no minimum value. 
C)  Absolute maximum value is   at x =   ; absolute minimum value is 0 at x = 1. 
D)  None 
A)  Absolute minimum value is   at x =   ; no maximum value. 
B)  Absolute maximum value is   at x =   ; no minimum value. 
C)  Absolute maximum value is   at x =   ; absolute minimum value is 0 at x = 1. 
D)  None

Question 16

Use a calculator to compute the first 10 iterations of Newton's method when applied to the function with the given initial approximation. Make a table for the values. Round to six decimal places.
-f(x) =   + 2x + 4;   = 0

Accepted Answer

A)  
  
B)  
  
C)  
  
D)  
  
A)  
  
B)  
  
C)  
  
D)

Question 17

Find all the roots of the function. Use preliminary analysis and graphing to determine good initial approximations. Round to six decimal places.
-f(x) =   +

Accepted Answer

A)  x $\approx$ -2.217825 
B)  x $\approx$ -0.717825 
C)  x $\approx$ -1.717714 
D)  x$\approx$ -1.717825 
A)  x $\approx$ -2.217825 
B)  x $\approx$ -0.717825 
C)  x $\approx$ -1.717714 
D)  x$\approx$ -1.717825

Question 18

Choose the one alternative that best completes the statement or answers the question. 
-Suppose that c(x) = 3   - 29   + 8413x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Accepted Answer

A)  -83 items 
B)  102 items 
C)  There is not a production level that will minimize average cost. 
D)  8413 items 
A)  -83 items 
B)  102 items 
C)  There is not a production level that will minimize average cost. 
D)  8413 items

Question 19

Use Newton's method to find an approximate answer to the question. Round to six decimal places.
-Where is the local extremum of f(x) =   located?

Accepted Answer

A)  x $\approx$ 1.687547 
B)  x $\approx$ 1.102547 
C)  x $\approx$ 1.127450 
D)  x$\approx$1.000000 
A)  x $\approx$ 1.687547 
B)  x $\approx$ 1.102547 
C)  x $\approx$ 1.127450 
D)  x$\approx$1.000000

Question 20

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.   
-

Accepted Answer

A)  Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (-$\infty$, -3) and (3, $\infty$); concave down on (-3, 3) 
B)  Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (-$\infty$ -3) and (3,$\infty$); concave up on (-3, 3) 
C)  Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (0, $\infty$); concave down on (-$\infty$, 0) 
D)  Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (0, $\infty$); concave up on (-$\infty$, 0) 
A)  Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (-$\infty$, -3) and (3, $\infty$); concave down on (-3, 3) 
B)  Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (-$\infty$ -3) and (3,$\infty$); concave up on (-3, 3) 
C)  Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (0, $\infty$); concave down on (-$\infty$, 0) 
D)  Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (0, $\infty$); concave up on (-$\infty$, 0)

For the given expression , find y'' and sketch the general shape of the graph of y = f(x). -y' = - 1

Find the largest open interval where the function is changing as requested. -Decreasing y = + 7

Find the linearization L(x) of f(x) at x = a. -f(x) = 3 + 4x + 1, a = 3

Use l'Hopital's Rule to evaluate the limit. -

Find the most general antiderivative. - $Find the most general antiderivative. - d \theta$ d $\theta$

Find the absolute extreme values of the function on the interval. -g(x) = - $Find the absolute extreme values of the function on the interval. -g(x) = - + 5x - 6, 2 \le x \le 3$ + 5x - 6, 2 $\le$ x $\le$ 3

Find the largest open interval where the function is changing as requested. -Increasing f(x) =

Express the relationship between a small change in x and the corresponding change in y in the form . -y =

Determine the indefinite integral. Check your work by differentiation. - $Determine the indefinite integral. Check your work by differentiation. - d \theta$ d $\theta$

Use l'Hopital's Rule to evaluate the limit. -

Find the extreme values of the function and where they occur. -y =

Use a calculator to compute the first 10 iterations of Newton's method when applied to the function with the given initial approximation. Make a table for the values. Round to six decimal places. -f(x) = + 2x + 4; = 0

Find all the roots of the function. Use preliminary analysis and graphing to determine good initial approximations. Round to six decimal places. -f(x) = +

Choose the one alternative that best completes the statement or answers the question. -Suppose that c(x) = 3 - 29 + 8413x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Use Newton's method to find an approximate answer to the question. Round to six decimal places. -Where is the local extremum of f(x) = located?

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. -

Functions

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Derivatives

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

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Multiple Integration

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Exam 4: Applications of the Derivative

For the given expression , find y'' and sketch the general shape of the graph of y = f(x). -y' = - 1

Find the largest open interval where the function is changing as requested. -Decreasing y = + 7

Find the linearization L(x) of f(x) at x = a. -f(x) = 3 + 4x + 1, a = 3

Use l'Hopital's Rule to evaluate the limit. -

Find the most general antiderivative. - d θ\thetaθ

Find the absolute extreme values of the function on the interval. -g(x) = - + 5x - 6, 2 ≤\le≤ x ≤\le≤ 3

Find the largest open interval where the function is changing as requested. -Increasing f(x) =

Express the relationship between a small change in x and the corresponding change in y in the form . -y =

Determine the indefinite integral. Check your work by differentiation. - d θ\thetaθ

Solve the problem. -V = π\piπ , where r is the radius, in centimeters. By approximately how much does the volume of a sphere increase when the radius is increased from 2.0 cm to 2.1 cm? (Use 3.14 for π\piπ .)

Use l'Hopital's Rule to evaluate the limit. -

Find the extreme values of the function and where they occur. -y =

Use a calculator to compute the first 10 iterations of Newton's method when applied to the function with the given initial approximation. Make a table for the values. Round to six decimal places. -f(x) = + 2x + 4; = 0

Find all the roots of the function. Use preliminary analysis and graphing to determine good initial approximations. Round to six decimal places. -f(x) = +

Choose the one alternative that best completes the statement or answers the question. -Suppose that c(x) = 3 - 29 + 8413x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Use Newton's method to find an approximate answer to the question. Round to six decimal places. -Where is the local extremum of f(x) = located?

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. -

Functions

Limits

Derivatives

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Calculus

Filters

Find the most general antiderivative. - $Find the most general antiderivative. - d \theta$ d $\theta$

Find the absolute extreme values of the function on the interval. -g(x) = - $Find the absolute extreme values of the function on the interval. -g(x) = - + 5x - 6, 2 \le x \le 3$ + 5x - 6, 2 $\le$ x $\le$ 3

Determine the indefinite integral. Check your work by differentiation. - $Determine the indefinite integral. Check your work by differentiation. - d \theta$ d $\theta$