Question 1

Find the absolute extreme values of the function on the interval.
-$$h ( x ) = \frac { 1 } { 2 } x + 4 , - 2 \leq x \leq 4$$

Accepted Answer

A)  absolute maximum is $- 2$ at $x = - 4$; absolute minimum is $- 4$ at $x = 2$ 
B)  absolute maximum is $- 2$ at $x = - 2$; absolute minimum is 3 at $x = 4$ 
C)  absolute maximum is $- 2$ at $x = 4$; absolute minimum is 3 at $x = - 2$ 
D)  absolute maximum is 6 at $x = 4$; absolute minimum is 3 at $x = - 2$ 
A)  absolute maximum is $- 2$ at $x = - 4$; absolute minimum is $- 4$ at $x = 2$ 
B)  absolute maximum is $- 2$ at $x = - 2$; absolute minimum is 3 at $x = 4$ 
C)  absolute maximum is $- 2$ at $x = 4$; absolute minimum is 3 at $x = - 2$ 
D)  absolute maximum is 6 at $x = 4$; absolute minimum is 3 at $x = - 2$

Question 2

Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval.
-An approximation to the total profit (in thousands of dollars) from the sale of $x$ hundred thousand tires is given by $p = - x ^ { 3 } + 15 x ^ { 2 } - 48 x + 450 , x \geq 3$. Find the number of hundred thousands of tires that must be sold to maximize profit.

Accepted Answer

A)  5 hundred thousand 
B)  3 hundred thousand 
C)  10 hundred thousand 
D)  8 hundred thousand 
A)  5 hundred thousand 
B)  3 hundred thousand 
C)  10 hundred thousand 
D)  8 hundred thousand

Question 3

Find the largest open interval where the function is changing as requested.
-Decreasing $y = \frac { 1 } { x ^ { 2 } } + 7$

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 4

Find the largest open interval where the function is changing as requested.
-Increasing $\quad f ( x ) = \frac { 1 } { x ^ { 2 } + 1 }$

Accepted Answer

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

Question 5

Find the extreme values of the function and where they occur.
-$$y = \frac { 1 } { x ^ { 2 } + 1 }$$

Accepted Answer

A)  The minimum value is $- 1$ at $x = 0.5$. 
B)  The maximum value is 1 at $x = 0.5$, the minimum value is $- 1$ at $x = 0.5$. 
C)  The maximum value is 1 at $x = 0.5$. 
D)  The maximum value is 1 at $x = 0$. 
A)  The minimum value is $- 1$ at $x = 0.5$. 
B)  The maximum value is 1 at $x = 0.5$, the minimum value is $- 1$ at $x = 0.5$. 
C)  The maximum value is 1 at $x = 0.5$. 
D)  The maximum value is 1 at $x = 0$.

Question 6

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

Accepted Answer

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

Question 7

Find the derivative at each critical point and determine the local extreme values.
-$$y = x ^ { 2 } \sqrt { 1 - x }$$

Accepted Answer

A) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=\frac{4}{5} & 0 & \text { local max } & \frac{16}{125} \sqrt{5}
\end{array}$ 
B) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=0 & 0 & \text { min } & 0 \
x=1 & 0 & \text { min } & 0 \
x=\frac{4}{5} & 0 & \text { local } \max & \frac{16}{125} \sqrt{5}
\end{array}$ 
C) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=0 & 0 & \min & 0 \
x=1 & \text { undefined } & \text { min } & 0 \
x=\frac{4}{5} & 0 & \text { local max } & \frac{16}{125} \sqrt{5}
\end{array}$ 
D) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=1 & \text { undefined } & \text { min } & 0 \
x=\frac{4}{5} & 0 & \text { local } \max & \frac{16}{125} \sqrt{5}
\end{array}$ 
A) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=\frac{4}{5} & 0 & \text { local max } & \frac{16}{125} \sqrt{5}
\end{array}$ 
B) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=0 & 0 & \text { min } & 0 \
x=1 & 0 & \text { min } & 0 \
x=\frac{4}{5} & 0 & \text { local } \max & \frac{16}{125} \sqrt{5}
\end{array}$ 
C) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=0 & 0 & \min & 0 \
x=1 & \text { undefined } & \text { min } & 0 \
x=\frac{4}{5} & 0 & \text { local max } & \frac{16}{125} \sqrt{5}
\end{array}$ 
D) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=1 & \text { undefined } & \text { min } & 0 \
x=\frac{4}{5} & 0 & \text { local } \max & \frac{16}{125} \sqrt{5}
\end{array}$

Question 8

Find the derivative at each critical point and determine the local extreme values.
-$$y = \left\{ \begin{array} { l l } 
- \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } x + \frac { 15 } { 4 } , & x \leq 1 \
x ^ { 3 } - 6 x ^ { 2 } + 8 x , & x > 1
\end{array} \right.$$

Accepted Answer

A) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-1 & 0 & \text { local max } & 4 \
x=3.15 & 0 & \text { local min } & -3.08
\end{array}$ 
B) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-1 & \text { undefined } & \text { local min } & 4 \
x=3.15 & 0 & \text { local max } & -3.08
\end{array}$ 
C) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-1 & 0 & \text { local min } & 4 \
x=3.15 & 0 & \text { local max } & -3.08
\end{array}$ 
D) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=1 & 0 & \text { local max } & 4 \
x=3.15 & \text { undefined } & \text { local max } & -3.08
\end{array}$ 
A) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-1 & 0 & \text { local max } & 4 \
x=3.15 & 0 & \text { local min } & -3.08
\end{array}$ 
B) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-1 & \text { undefined } & \text { local min } & 4 \
x=3.15 & 0 & \text { local max } & -3.08
\end{array}$ 
C) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=-1 & 0 & \text { local min } & 4 \
x=3.15 & 0 & \text { local max } & -3.08
\end{array}$ 
D) 
$\begin{array}{l|l|l|l}
\text { Critical Pt. } & \text { derivative } & \text { Extremum } & \text { Value } \
\hline x=1 & 0 & \text { local max } & 4 \
x=3.15 & \text { undefined } & \text { local max } & -3.08
\end{array}$

Question 9

Identify the function's local and absolute extreme values, if any, saying where they occur.
-$$f ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x - 3$$

Accepted Answer

A)  local maximum at $x = - 3$ 
B)  local minimum at $x = - 3$ 
C)  local maximum at $x = - 3$; local minimum at $x = 3$ 
D)  no local extrema 
A)  local maximum at $x = - 3$ 
B)  local minimum at $x = - 3$ 
C)  local maximum at $x = - 3$; local minimum at $x = 3$ 
D)  no local extrema

Question 10

Show that the function has exactly one zero in the given interval.
-$$r ( \theta ) = 5 \cot \theta + \frac { 1 } { \theta ^ { 2 } } + 6 , ( 0 , \pi )$$

Accepted Answer

The function @#LAT-DLM& is continuous on the open

Question 11

Using the derivative of f(x) given below, determine the critical points of f(x).
-f'(x) = (x - 5) e-x

Accepted Answer

A)  5 
B)  6 
C)  -6 
D)  -5 
A)  5 
B)  6 
C)  -6 
D)  -5

Question 12

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval.
-$$f ( \theta ) = \left\{ \begin{array} { c c } 
\frac { \cos \theta } { \theta } , & - \pi \leq \theta < 0 \
0 , & \theta = 0
\end{array} \right.$$

Accepted Answer

A) True 
 B)False

Question 13

Find all possible functions with the given derivative.
-$$y ^ { \prime } = 3 t - \frac { 2 } { \sqrt { t } }$$

Accepted Answer

A)  $3 t ^ { 2 } - 4 \sqrt { t } + C$ 
B)  $t ^ { 2 } - 2 \sqrt { t } + C$ 
C)  $3 t ^ { 2 } - \frac { 2 } { \sqrt { t } } + C$ 
D)  $\frac { 3 } { 2 } t ^ { 2 } - 4 \sqrt { t } + C$ 
A)  $3 t ^ { 2 } - 4 \sqrt { t } + C$ 
B)  $t ^ { 2 } - 2 \sqrt { t } + C$ 
C)  $3 t ^ { 2 } - \frac { 2 } { \sqrt { t } } + C$ 
D)  $\frac { 3 } { 2 } t ^ { 2 } - 4 \sqrt { t } + C$

Question 14

Determine all critical points for the function.
-$$f ( x ) = x ^ { 3 } - 12 x + 5$$

Accepted Answer

A)  $x = - 2$ and $x = 2$ 
B)  $x = 2$ 
C)  $x = - 2$ 
D)  $x = - 2 , x = 0$, and $x = 2$ 
A)  $x = - 2$ and $x = 2$ 
B)  $x = 2$ 
C)  $x = - 2$ 
D)  $x = - 2 , x = 0$, and $x = 2$

Question 15

Find the absolute extreme values of the function on the interval.
-$$g ( x ) = 6 - 7 x ^ { 2 } , - 4 \leq x \leq 5$$

Accepted Answer

A)  absolute maximum is 12 at $x = 0$; absolute minimum is $- 106$ at $x = 5$ 
B)  absolute maximum is 6 at $x = 0$; absolute minimum is $- 169$ at $x = 5$ 
C)  absolute maximum is 7 at $x = 0$; absolute minimum is $- 181$ at $x = 5$ 
D)  absolute maximum is 42 at $x = 0$; absolute minimum is $- 106$ at $x = - 4$ 
A)  absolute maximum is 12 at $x = 0$; absolute minimum is $- 106$ at $x = 5$ 
B)  absolute maximum is 6 at $x = 0$; absolute minimum is $- 169$ at $x = 5$ 
C)  absolute maximum is 7 at $x = 0$; absolute minimum is $- 181$ at $x = 5$ 
D)  absolute maximum is 42 at $x = 0$; absolute minimum is $- 106$ at $x = - 4$

Question 16

Find the largest open interval where the function is changing as requested.
-Increasing $\quad y = \left( x ^ { 2 } - 9 \right) ^ { 2 }$

Accepted Answer

A)  $( - \infty , 0 )$ 
B)  $( - 3,0 )$ 
C)  $( 3 , \infty )$ 
D)  $( - 3,3 )$ 
A)  $( - \infty , 0 )$ 
B)  $( - 3,0 )$ 
C)  $( 3 , \infty )$ 
D)  $( - 3,3 )$

Question 17

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$$f ( x ) = 0.1 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 59 x + 8$$

Accepted Answer

A)  Approximate local maximum at 1.805; approximate local minima at -6.72 and 12.449 
B)  Approximate local maximum at 1.703; approximate local minima at -6.754 and 12.588 
C)  Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 
D)  Approximate local maximum at 1.77; approximate local minima at -6.771 and 12.53 
A)  Approximate local maximum at 1.805; approximate local minima at -6.72 and 12.449 
B)  Approximate local maximum at 1.703; approximate local minima at -6.754 and 12.588 
C)  Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 
D)  Approximate local maximum at 1.77; approximate local minima at -6.771 and 12.53

Question 18

Provide an appropriate response.
-The function $f ( x ) = \left\{ \begin{array} { l l } 7 x & 0 \leq x < 1 \ 0 & x = 1 \end{array} \right.$ is zero at $x = 0$ and $x = 1$ and differentiable on $( 0,1 )$, but its derivative on $( 0,1 )$ is never zero. Does this example contradict Rolle's Theorem?

Accepted Answer

This example does not contradi

Question 19

Find all possible functions with the given derivative.
-$$y ^ { \prime } = x ^ { 3 } - 6 x$$

Accepted Answer

A)  $\frac { x ^ { 4 } } { 4 } - \frac { 6 x ^ { 2 } } { 2 } + C$ 
B)  $\frac { x ^ { 3 } } { 3 } - \frac { 6 x ^ { 2 } } { 2 } + C$ 
C)  $3 x ^ { 2 } - 6 + C$ 
D)  $\frac { x ^ { 4 } } { 4 } + 6 x ^ { 2 } + C$ 
A)  $\frac { x ^ { 4 } } { 4 } - \frac { 6 x ^ { 2 } } { 2 } + C$ 
B)  $\frac { x ^ { 3 } } { 3 } - \frac { 6 x ^ { 2 } } { 2 } + C$ 
C)  $3 x ^ { 2 } - 6 + C$ 
D)  $\frac { x ^ { 4 } } { 4 } + 6 x ^ { 2 } + C$

Question 20

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

Accepted Answer

A)  x = -3 
B)  x = -4 
C)  x = 2 
D)  x = 0 
A)  x = -3 
B)  x = -4 
C)  x = 2 
D)  x = 0

Find the absolute extreme values of the function on the interval. - $h ( x ) = \frac { 1 } { 2 } x + 4 , - 2 \leq x \leq 4$

Find the largest open interval where the function is changing as requested. -Decreasing $y = \frac { 1 } { x ^ { 2 } } + 7$

Find the largest open interval where the function is changing as requested. -Increasing $\quad f ( x ) = \frac { 1 } { x ^ { 2 } + 1 }$

Find the extreme values of the function and where they occur. - $y = \frac { 1 } { x ^ { 2 } + 1 }$

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

Find the derivative at each critical point and determine the local extreme values. - $y = x ^ { 2 } \sqrt { 1 - x }$

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l l } - \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } x + \frac { 15 } { 4 } , & x \leq 1 \\x ^ { 3 } - 6 x ^ { 2 } + 8 x , & x > 1\end{array} \right.$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x - 3$

Show that the function has exactly one zero in the given interval. - $r ( \theta ) = 5 \cot \theta + \frac { 1 } { \theta ^ { 2 } } + 6 , ( 0 , \pi )$

Using the derivative of f(x) given below, determine the critical points of f(x). -f'(x) = (x - 5) e^-x

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. - $f ( \theta ) = \left\{ \begin{array} { c c } \frac { \cos \theta } { \theta } , & - \pi \leq \theta < 0 \\0 , & \theta = 0\end{array} \right.$

Find all possible functions with the given derivative. - $y ^ { \prime } = 3 t - \frac { 2 } { \sqrt { t } }$

Determine all critical points for the function. - $f ( x ) = x ^ { 3 } - 12 x + 5$

Find the absolute extreme values of the function on the interval. - $g ( x ) = 6 - 7 x ^ { 2 } , - 4 \leq x \leq 5$

Find the largest open interval where the function is changing as requested. -Increasing $\quad y = \left( x ^ { 2 } - 9 \right) ^ { 2 }$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 59 x + 8$

Provide an appropriate response. -The function $f ( x ) = \left\{ \begin{array} { l l } 7 x & 0 \leq x < 1 \\ 0 & x = 1 \end{array} \right.$ is zero at $x = 0$ and $x = 1$ and differentiable on $( 0,1 )$ , but its derivative on $( 0,1 )$ is never zero. Does this example contradict Rolle's Theorem?

Find all possible functions with the given derivative. - $y ^ { \prime } = x ^ { 3 } - 6 x$

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

Functions

Limits and Derivatives

Differentiation

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Filters

Exam 4: Applications of Derivatives

Find the absolute extreme values of the function on the interval. - h(x)=12x+4,−2≤x≤4h ( x ) = \frac { 1 } { 2 } x + 4 , - 2 \leq x \leq 4h(x)=21​x+4,−2≤x≤4

Find the largest open interval where the function is changing as requested. -Decreasing y=1x2+7y = \frac { 1 } { x ^ { 2 } } + 7y=x21​+7

Find the largest open interval where the function is changing as requested. -Increasing f(x)=1x2+1\quad f ( x ) = \frac { 1 } { x ^ { 2 } + 1 }f(x)=x2+11​

Find the extreme values of the function and where they occur. - y=1x2+1y = \frac { 1 } { x ^ { 2 } + 1 }y=x2+11​

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

Find the derivative at each critical point and determine the local extreme values. - y=x21−xy = x ^ { 2 } \sqrt { 1 - x }y=x21−x​

Identify the function's local and absolute extreme values, if any, saying where they occur. - f(x)=x3+9x2+27x−3f ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x - 3f(x)=x3+9x2+27x−3

Show that the function has exactly one zero in the given interval. - r(θ)=5cot⁡θ+1θ2+6,(0,π)r ( \theta ) = 5 \cot \theta + \frac { 1 } { \theta ^ { 2 } } + 6 , ( 0 , \pi )r(θ)=5cotθ+θ21​+6,(0,π)

Using the derivative of f(x) given below, determine the critical points of f(x). -f'(x) = (x - 5) e-x

Find all possible functions with the given derivative. - y′=3t−2ty ^ { \prime } = 3 t - \frac { 2 } { \sqrt { t } }y′=3t−t​2​

Determine all critical points for the function. - f(x)=x3−12x+5f ( x ) = x ^ { 3 } - 12 x + 5f(x)=x3−12x+5

Find the absolute extreme values of the function on the interval. - g(x)=6−7x2,−4≤x≤5g ( x ) = 6 - 7 x ^ { 2 } , - 4 \leq x \leq 5g(x)=6−7x2,−4≤x≤5

Find the largest open interval where the function is changing as requested. -Increasing y=(x2−9)2\quad y = \left( x ^ { 2 } - 9 \right) ^ { 2 }y=(x2−9)2

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - f(x)=0.1x4−x3−15x2+59x+8f ( x ) = 0.1 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 59 x + 8f(x)=0.1x4−x3−15x2+59x+8

Find all possible functions with the given derivative. - y′=x3−6xy ^ { \prime } = x ^ { 3 } - 6 xy′=x3−6x