Question 1

Find the absolute extreme values of the function on the interval.
-$$f ( x ) = 2 x - 3 , - 3 \leq x \leq 4$$

Accepted Answer

A)  absolute maximum is 5 at x = 4; absolute minimum is - 9 at x = -3 
B)  absolute maximum is 11 at x = -4; absolute minimum is - 9 at x = 3 
C)  absolute maximum is 11 at x = 4; absolute minimum is - 3 at x = -3 
D)  absolute maximum is 5 at x = -3; absolute minimum is - 3 at x = 4 
A)  absolute maximum is 5 at x = 4; absolute minimum is - 9 at x = -3 
B)  absolute maximum is 11 at x = -4; absolute minimum is - 9 at x = 3 
C)  absolute maximum is 11 at x = 4; absolute minimum is - 3 at x = -3 
D)  absolute maximum is 5 at x = -3; absolute minimum is - 3 at x = 4

Question 2

Find all possible functions with the given derivative.
-$y ^ { \prime } = x ^ { 5 }$

Accepted Answer

A)  $\frac { 1 } { 6 } x ^ { 6 } + C$ 
B)  $5 x ^ { 4 } + C$ 
C)  $\frac { 1 } { 5 } x ^ { 4 } + C$ 
D)  $6 x ^ { 6 } + C$ 
A)  $\frac { 1 } { 6 } x ^ { 6 } + C$ 
B)  $5 x ^ { 4 } + C$ 
C)  $\frac { 1 } { 5 } x ^ { 4 } + C$ 
D)  $6 x ^ { 6 } + C$

Question 3

Solve the problem.
-An object is dropped from $9 \mathrm { ft }$ above the surface of the moon. How long will it take the object to hit the surface of the moon if $\mathrm { d } ^ { 2 } \mathrm {~s} / \mathrm { dt } ^ { 2 } = - 5.2 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ ?

Accepted Answer

A)  3.46 sec 
B)  1.32 sec 
C)  0.87 sec 
D)  1.86 sec 
A)  3.46 sec 
B)  1.32 sec 
C)  0.87 sec 
D)  1.86 sec

Question 4

Graph the rational function.
-$\begin{array} { l } 
\frac { 6 x } { x ^ { 2 } + 9 } \
\end{array}$

Accepted Answer

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

Question 5

Answer each question appropriately.
-Suppose the velocity of a body moving along the s-axis is $\frac { \mathrm { ds } } { \mathrm { dt } } = 9.8 \mathrm { t } - 5$.
Is it necessary to know the initial position of the body to find the body's displacement over some time interval? J your answer. ify

Accepted Answer

A)  No, displacement has nothing to do with the position of the body. 
B)  Yes, knowing the initial position is the only way to find the exact positions at the beginning and end of the time interval. Those positions are needed to find the displacement. 
C)  No, the initial position is necessary to find the curve s= f(t) but not necessary to find the displacement. The initial position determines the integration constant. When finding the displacement the integration
Constant is subtracted out. 
D)  Yes, integration is not possible without knowing the initial position. 
A)  No, displacement has nothing to do with the position of the body. 
B)  Yes, knowing the initial position is the only way to find the exact positions at the beginning and end of the time interval. Those positions are needed to find the displacement. 
C)  No, the initial position is necessary to find the curve s= f(t) but not necessary to find the displacement. The initial position determines the integration constant. When finding the displacement the integration
Constant is subtracted out. 
D)  Yes, integration is not possible without knowing the initial position.

Question 6

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

Question 7

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme
values, if any, are absolute.
-$$f ( x ) = x ^ { 2 } - 4 x , - \infty < x \leq 4$$

Accepted Answer

A)  local and absolute minimum: -4 at x = 2; local and absolute maximum: 0 at x = 4 
B)  local minimum: -4 at x = 2; local and absolute maximum: 0 at x = 4 
C)  local minimum: 0 at x = 4; local and absolute maximum:-4 at x = 2 
D)  local and absolute minimum: -4 at x = 2; local maximum: 0 at x = 4 
A)  local and absolute minimum: -4 at x = 2; local and absolute maximum: 0 at x = 4 
B)  local minimum: -4 at x = 2; local and absolute maximum: 0 at x = 4 
C)  local minimum: 0 at x = 4; local and absolute maximum:-4 at x = 2 
D)  local and absolute minimum: -4 at x = 2; local maximum: 0 at x = 4

Question 8

The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply.
-Let f have a derivative on an interval I. f' has successive distinct zeros at x = 1 and x = 5. Prove that there can be at most one zero of f on the interval (1, 5).

Accepted Answer

Assume there are 2 (or more) zeros a and

Question 9

Solve the problem.
-The graphs below show the first and second derivatives of a function $y = f ( x )$. Select a possible graph $f$ that passe through the point $P$.

Accepted Answer

A) 
 
[NOTE: Graph vertical scales
 may vary from graph to graph.] 
B) 
 
[NOTE: Graph vertical scales
 may vary from graph to graph.] 
C) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
D)  
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
A) 
 
[NOTE: Graph vertical scales
 may vary from graph to graph.] 
B) 
 
[NOTE: Graph vertical scales
 may vary from graph to graph.] 
C) 
 
[NOTE: Graph vertical scales
may vary from graph to graph.] 
D)  
 
[NOTE: Graph vertical scales
may vary from graph to graph.]

Question 10

Solve the problem.
-Suppose a business can sell x gadgets for p = 250 - 0.01x dollars apiece, and it costs the business c(x) = 1000 + 25x dollars to produce the x gadgets. Determine the production level and cost per gadget required
To maximize profit.

Accepted Answer

A)  111 gadgets at $248.89 each 
B)  13,750 gadgets at $112.50 each 
C)  10,000 gadgets at $150.00 each 
D)  11,250 gadgets at $137.50 each 
A)  111 gadgets at $248.89 each 
B)  13,750 gadgets at $112.50 each 
C)  10,000 gadgets at $150.00 each 
D)  11,250 gadgets at $137.50 each

Question 11

The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply.
-Suppose that f'(x) = 2x for all x. Find f(8) if f(-2) = 2.

Accepted Answer

A)  60 
B)  64 
C)  62 
D)  66 
A)  60 
B)  64 
C)  62 
D)  66

Question 12

Solve the problem.
-Find the table that matches the graph below.

Accepted Answer

A) 
$\begin{array}{l|l}
\mathrm{x} & \mathrm{f}^{\prime}(\mathrm{x}) \
\hline \mathrm{a} & \text { does not exist } \
\hline \mathrm{b} & 0 \
\hline \mathrm{c} & -\frac{1}{2}
\end{array}$ 
B) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & \text { does not exist } \
\hline b & \text { does not exist } \
\hline c & \frac{5}{2}
\end{array}$ 
C) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline \mathrm{a} & 0 \
\hline \mathrm{b} & 0 \
\hline \mathrm{c} & \frac{5}{2}
\end{array}$ 
D) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & 0 \
\hline b & 0 \
\hline c & -\frac{1}{2}
\end{array}$ 
A) 
$\begin{array}{l|l}
\mathrm{x} & \mathrm{f}^{\prime}(\mathrm{x}) \
\hline \mathrm{a} & \text { does not exist } \
\hline \mathrm{b} & 0 \
\hline \mathrm{c} & -\frac{1}{2}
\end{array}$ 
B) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & \text { does not exist } \
\hline b & \text { does not exist } \
\hline c & \frac{5}{2}
\end{array}$ 
C) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline \mathrm{a} & 0 \
\hline \mathrm{b} & 0 \
\hline \mathrm{c} & \frac{5}{2}
\end{array}$ 
D) 
$\begin{array}{l|l}
x & f^{\prime}(x) \
\hline a & 0 \
\hline b & 0 \
\hline c & -\frac{1}{2}
\end{array}$

Question 13

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$$f ( x ) = 0.1 x ^ { 5 } + 5 x ^ { 4 } - 8 x ^ { 3 } - 15 x ^ { 2 } - 6 x - 63$$

Accepted Answer

A)  Approximate local maxima at -41.132 and -0.273; approximate local minima at -0.547 and 1.952 
B)  Approximate local maxima at -41.13 and -0.196; approximate local minima at -0.55 and 2.029 
C)  Approximate local maxima at -41.16 and -0.301; approximate local minima at -0.638 and 2.043 
D)  Approximate local maxima at -41.075 and -0.361; approximate local minima at -0.504 and 1.895 
A)  Approximate local maxima at -41.132 and -0.273; approximate local minima at -0.547 and 1.952 
B)  Approximate local maxima at -41.13 and -0.196; approximate local minima at -0.55 and 2.029 
C)  Approximate local maxima at -41.16 and -0.301; approximate local minima at -0.638 and 2.043 
D)  Approximate local maxima at -41.075 and -0.361; approximate local minima at -0.504 and 1.895

Question 14

Find the largest open interval where the function is changing as requested.
-Decreasing $\quad f ( x ) = x ^ { 3 } - 4 x$

Accepted Answer

A)  $\left( - \infty , - \frac { 2 \sqrt { 3 } } { 3 } \right)$ 
B)  $( - \infty , \infty )$ 
C)  $\left( - \frac { 2 \sqrt { 3 } } { 3 } , \frac { 2 \sqrt { 3 } } { 3 } \right)$ 
D)  $\left( \frac { 2 \sqrt { 3 } } { 3 } , \infty \right)$ 
A)  $\left( - \infty , - \frac { 2 \sqrt { 3 } } { 3 } \right)$ 
B)  $( - \infty , \infty )$ 
C)  $\left( - \frac { 2 \sqrt { 3 } } { 3 } , \frac { 2 \sqrt { 3 } } { 3 } \right)$ 
D)  $\left( \frac { 2 \sqrt { 3 } } { 3 } , \infty \right)$

Question 15

Find the location of the indicated absolute extremum for the function.
-

Accepted Answer

A)  No maximum 
B)  x = -1 
C)  x = 4 
D)  x = 1 
A)  No maximum 
B)  x = -1 
C)  x = 4 
D)  x = 1

Question 16

Find the location of the indicated absolute extremum for the function.
-

Accepted Answer

A)  x = -4 
B)  x = 1 
C)  No maximum 
D)  x = 4 
A)  x = -4 
B)  x = 1 
C)  No maximum 
D)  x = 4

Question 17

Find an antiderivative of the given function.
-$2 \csc 9 x \cot 9 x$

Accepted Answer

A)  $- 2 \csc 9 x$ 
B)  $- \frac { 2 } { 9 } \csc 9 x$ 
C)  $- \frac { 2 } { 9 } \cot 9 x$ 
D)  $\frac { 2 } { 9 } \csc 9 x$ 
A)  $- 2 \csc 9 x$ 
B)  $- \frac { 2 } { 9 } \csc 9 x$ 
C)  $- \frac { 2 } { 9 } \cot 9 x$ 
D)  $\frac { 2 } { 9 } \csc 9 x$

Question 18

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule.
Show each step of your calculation.
-$\lim _ { x \rightarrow - } \frac { 1 - \cos x } { x }$

Accepted Answer

@#IMG-DLM&
Using the graph, students should estima

Question 19

Solve the problem.
-The graphs below show the first and second derivatives of a function $y = f ( x )$. Select a possible graph of $f$ that passes through the point $P$.

Accepted Answer

A) 
 
[NOTE: Graph vertical scales may vary from graph to graph.] 
B)

[NOTE: Graph vertical scales may vary from graph to graph. 
C) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] mav vary from graph to granh.l 
D) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] mav vary from graph to granh.l 
A) 
 
[NOTE: Graph vertical scales may vary from graph to graph.] 
B)

[NOTE: Graph vertical scales may vary from graph to graph. 
C) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] mav vary from graph to granh.l 
D) 
 
[NOTE: Graph vertical scales
[NOTE: Graph vertical scales may vary from graph to graph.] mav vary from graph to granh.l

Question 20

Find the function with the given derivative whose graph passes through the point P.
-$f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,12 )$

Accepted Answer

A)  $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 12$ 
B)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x + 12$ 
C)  $f ( x ) = x ^ { 3 } + 6 x + 12$ 
D)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x$ 
A)  $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 12$ 
B)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x + 12$ 
C)  $f ( x ) = x ^ { 3 } + 6 x + 12$ 
D)  $f ( x ) = \frac { x ^ { 3 } } { 3 } + 6 x$

Find the absolute extreme values of the function on the interval. - $f ( x ) = 2 x - 3 , - 3 \leq x \leq 4$

Find all possible functions with the given derivative. - $y ^ { \prime } = x ^ { 5 }$

Solve the problem. -An object is dropped from $9 \mathrm { ft }$ above the surface of the moon. How long will it take the object to hit the surface of the moon if $\mathrm { d } ^ { 2 } \mathrm {~s} / \mathrm { dt } ^ { 2 } = - 5.2 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ ?

Graph the rational function. - $Graph the rational function. - \begin{array} { l } \frac { 6 x } { x ^ { 2 } + 9 } \\ \end{array}$

Answer each question appropriately. -Suppose the velocity of a body moving along the s-axis is $\frac { \mathrm { ds } } { \mathrm { dt } } = 9.8 \mathrm { t } - 5$ . Is it necessary to know the initial position of the body to find the body's displacement over some time interval? J your answer. ify

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. -

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. - $f ( x ) = x ^ { 2 } - 4 x , - \infty < x \leq 4$

The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply. -Let f have a derivative on an interval I. f' has successive distinct zeros at x = 1 and x = 5. Prove that there can be at most one zero of f on the interval (1, 5).

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph $f$ that passe through the point $P$ .

Solve the problem. -Suppose a business can sell x gadgets for p = 250 - 0.01x dollars apiece, and it costs the business c(x) = 1000 + 25x dollars to produce the x gadgets. Determine the production level and cost per gadget required To maximize profit.

The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply. -Suppose that f'(x) = 2x for all x. Find f(8) if f(-2) = 2.

Solve the problem. -Find the table that matches the graph below.

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 5 } + 5 x ^ { 4 } - 8 x ^ { 3 } - 15 x ^ { 2 } - 6 x - 63$

Find the largest open interval where the function is changing as requested. -Decreasing $\quad f ( x ) = x ^ { 3 } - 4 x$

Find the location of the indicated absolute extremum for the function. -

Find the location of the indicated absolute extremum for the function. -

Find an antiderivative of the given function. - $2 \csc 9 x \cot 9 x$

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. - $\lim _ { x \rightarrow - } \frac { 1 - \cos x } { x }$

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph of $f$ that passes through the point $P$ .

Find the function with the given derivative whose graph passes through the point P. - $f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,12 )$

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 5: Applications of Derivatives

Find the absolute extreme values of the function on the interval. - f(x)=2x−3,−3≤x≤4f ( x ) = 2 x - 3 , - 3 \leq x \leq 4f(x)=2x−3,−3≤x≤4

Find all possible functions with the given derivative. - y′=x5y ^ { \prime } = x ^ { 5 }y′=x5

Graph the rational function. -

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. -

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. - f(x)=x2−4x,−∞<x≤4f ( x ) = x ^ { 2 } - 4 x , - \infty < x \leq 4f(x)=x2−4x,−∞<x≤4

The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply. -Let f have a derivative on an interval I. f' has successive distinct zeros at x = 1 and x = 5. Prove that there can be at most one zero of f on the interval (1, 5).

Solve the problem. -The graphs below show the first and second derivatives of a function y=f(x)y = f ( x )y=f(x) . Select a possible graph fff that passe through the point PPP .

Solve the problem. -Suppose a business can sell x gadgets for p = 250 - 0.01x dollars apiece, and it costs the business c(x) = 1000 + 25x dollars to produce the x gadgets. Determine the production level and cost per gadget required To maximize profit.

The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply. -Suppose that f'(x) = 2x for all x. Find f(8) if f(-2) = 2.

Solve the problem. -Find the table that matches the graph below.

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - f(x)=0.1x5+5x4−8x3−15x2−6x−63f ( x ) = 0.1 x ^ { 5 } + 5 x ^ { 4 } - 8 x ^ { 3 } - 15 x ^ { 2 } - 6 x - 63f(x)=0.1x5+5x4−8x3−15x2−6x−63

Find the largest open interval where the function is changing as requested. -Decreasing f(x)=x3−4x\quad f ( x ) = x ^ { 3 } - 4 xf(x)=x3−4x

Find the location of the indicated absolute extremum for the function. -

Find the location of the indicated absolute extremum for the function. -

Find an antiderivative of the given function. - 2csc⁡9xcot⁡9x2 \csc 9 x \cot 9 x2csc9xcot9x

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. - lim⁡x→−1−cos⁡xx\lim _ { x \rightarrow - } \frac { 1 - \cos x } { x }limx→−​x1−cosx​

Solve the problem. -The graphs below show the first and second derivatives of a function y=f(x)y = f ( x )y=f(x) . Select a possible graph of fff that passes through the point PPP .

Find the function with the given derivative whose graph passes through the point P. - f′(x)=x2+6,P(0,12)f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,12 )f′(x)=x2+6,P(0,12)

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Find the absolute extreme values of the function on the interval. - $f ( x ) = 2 x - 3 , - 3 \leq x \leq 4$

Find all possible functions with the given derivative. - $y ^ { \prime } = x ^ { 5 }$

Graph the rational function. - $Graph the rational function. - \begin{array} { l } \frac { 6 x } { x ^ { 2 } + 9 } \\ \end{array}$

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. - $f ( x ) = x ^ { 2 } - 4 x , - \infty < x \leq 4$

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph $f$ that passe through the point $P$ .

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 5 } + 5 x ^ { 4 } - 8 x ^ { 3 } - 15 x ^ { 2 } - 6 x - 63$

Find the largest open interval where the function is changing as requested. -Decreasing $\quad f ( x ) = x ^ { 3 } - 4 x$

Find an antiderivative of the given function. - $2 \csc 9 x \cot 9 x$

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. - $\lim _ { x \rightarrow - } \frac { 1 - \cos x } { x }$

Solve the problem. -The graphs below show the first and second derivatives of a function $y = f ( x )$ . Select a possible graph of $f$ that passes through the point $P$ .

Find the function with the given derivative whose graph passes through the point P. - $f ^ { \prime } ( x ) = x ^ { 2 } + 6 , P ( 0,12 )$