Question 1

Identify the intervals where the function is changing as requested.
-Decreasing

Accepted Answer

A)  $(0,3)$ 
B)  $(-\infty,-2)$ 
C)  $(0,-2)$ 
D)  $(-\infty, 3)$ 
A)  $(0,3)$ 
B)  $(-\infty,-2)$ 
C)  $(0,-2)$ 
D)  $(-\infty, 3)$

Question 2

Use calculus and a graphing calculator to find the approximate location of all relative extrema.
-$f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14$

Accepted Answer

A)  Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 
B)  Approximate local maximum at 1.755 ; approximate local minima at -6.703 and 12.552 
C)  Approximate local maximum at 1.661 ; approximate local minima at -6.768 and 12.52 
D)  Approximate local maximum at 1.791; approximate local minima at -6.785 and 12.601 
A)  Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 
B)  Approximate local maximum at 1.755 ; approximate local minima at -6.703 and 12.552 
C)  Approximate local maximum at 1.661 ; approximate local minima at -6.768 and 12.52 
D)  Approximate local maximum at 1.791; approximate local minima at -6.785 and 12.601

Question 3

Solve each problem.
-Find two numbers whose sum is 440 and whose product is as large as possible.

Accepted Answer

A)  219 and 221 
B)  1 and 439 
C)  220 and 220 
D)  10 and 430 
A)  219 and 221 
B)  1 and 439 
C)  220 and 220 
D)  10 and 430

Question 4

Find the location of the indicated absolute extrema for the function.
-Maximum

Accepted Answer

A)  None 
B)  $(-3,2),(3,2)$ 
C)  $(-3,2)$ 
D)  $(3,2),(0,-1)$ 
A)  None 
B)  $(-3,2),(3,2)$ 
C)  $(-3,2)$ 
D)  $(3,2),(0,-1)$

Question 5

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point.
-$f(x)=\frac{x^{2}+2}{3 x^{2}-1}, c=0$

Accepted Answer

A)  $f^{\prime \prime}(0)=-14$ 
B)  $f^{\prime \prime}(0)=-10$ 
C)  $\mathrm{f}^{\prime \prime}(0)=1$ 
D)  $\mathrm{f}^{\prime \prime}(0)=0$ 
A)  $f^{\prime \prime}(0)=-14$ 
B)  $f^{\prime \prime}(0)=-10$ 
C)  $\mathrm{f}^{\prime \prime}(0)=1$ 
D)  $\mathrm{f}^{\prime \prime}(0)=0$

Question 6

Find the absolute extremum within the specified domain.
-Minimum of $f(x)=x^{3}-3 x^{2} ;[-0.5,4]$

Accepted Answer

A)  $(4,-5)$ 
B)  $(4,-4)$ 
C)  $(2,-5)$ 
D)  $(2,-4)$ 
A)  $(4,-5)$ 
B)  $(4,-4)$ 
C)  $(2,-5)$ 
D)  $(2,-4)$

Question 7

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither.
-$f(x)=-x^{3}-3 x^{2}+9 x+2$

Accepted Answer

A)  Local maximum at 1 ; local minimum at -3 
B)  Local maximum at -3 ; local minimum at 1 
C)  Local maximum at -1 ; local minimum at 3 
D)  Local maximum at 3 ; local minimum at -1 
A)  Local maximum at 1 ; local minimum at -3 
B)  Local maximum at -3 ; local minimum at 1 
C)  Local maximum at -1 ; local minimum at 3 
D)  Local maximum at 3 ; local minimum at -1

Question 8

Sketch the graph and show all local extrema and inflection points.
-$f(x)=\frac{1}{x^{2}-2 x-3}$

Accepted Answer

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

Question 9

A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed $90 \mathrm{in}$. Suppose you want to mail a box with square sides so that its dimensions are $\mathrm{h}$ by $\mathrm{h}$ by $\mathrm{w}$ and it's girth is $2 \mathrm{~h}+2 \mathrm{w}$. What dimensions will give the box its largest volume?

Accepted Answer

A)  20 in. $x 20$ in. $x 15$ in. 
B)  15 in. $x 15$ in. $x 30$ in. 
C)  30 in. $x 15$ in. $x 30$ in. 
D)  15 in. $x 15$ in. $x 75$ in. 
A)  20 in. $x 20$ in. $x 15$ in. 
B)  15 in. $x 15$ in. $x 30$ in. 
C)  30 in. $x 15$ in. $x 30$ in. 
D)  15 in. $x 15$ in. $x 75$ in.

Question 10

Identify the intervals where the function is changing as requested.
-Increasing

Accepted Answer

A)  $(-1,0)$ 
B)  $(0,3)$ 
C)  $(-\infty,-1)$ 
D)  $(-\infty, 0)$ 
A)  $(-1,0)$ 
B)  $(0,3)$ 
C)  $(-\infty,-1)$ 
D)  $(-\infty, 0)$

Question 11

Find the coordinates of the points of inflection for the function.
-$f(x)=x^{2}+6 x+11$

Accepted Answer

A)  $(-4,1)$ 
B)  $(-3,2)$ 
C)  $(-2,3)$ 
D)  There are no points of inflection. 
A)  $(-4,1)$ 
B)  $(-3,2)$ 
C)  $(-2,3)$ 
D)  There are no points of inflection.

Question 12

Find the coordinates of the points of inflection for the function.
-$f(x)=x^{1 / 3}\left(x^{2}-28\right)$

Accepted Answer

A)  $(0,1)$ 
B)  $\left( \pm 2,-\frac{10}{3}\right)$ 
C)  $(0,0)$ 
D)  There are no points of inflection. 
A)  $(0,1)$ 
B)  $\left( \pm 2,-\frac{10}{3}\right)$ 
C)  $(0,0)$ 
D)  There are no points of inflection.

Question 13

Identify the intervals where the function is changing as requested.
-Increasing

Accepted Answer

A)   
B)  $(0,5)$ 
C)  $(1,6)$ 
D)  $(0,6)$ 
A)   
B)  $(0,5)$ 
C)  $(1,6)$ 
D)  $(0,6)$

Question 14

Solve the problem.
-Because of material shortages, it is increasingly expensive to produce 6.0L diesel engines. In fact, the profit in millions of dollars from producing $x$ hundred thousand engines is approximated by $P(x)=-x^{3}+27 x^{2}+15 x-52$, where $0 \leq x \leq 20$. Find the point of diminishing returns.

Accepted Answer

A)  $(9.00,1415.00)$ 
B)  $(9.00,1541.00)$ 
C)  $(6.75,1541.00)$ 
D)  $(9.00,137.00)$ 
A)  $(9.00,1415.00)$ 
B)  $(9.00,1541.00)$ 
C)  $(6.75,1541.00)$ 
D)  $(9.00,137.00)$

Question 15

Identify the intervals where the function is changing as requested.
-Decreasing

Accepted Answer

A)  $(-1,2)$ 
B)  $(2,1)$ 
C)  $(2,-1)$ 
D)  $(1,2)$ 
A)  $(-1,2)$ 
B)  $(2,1)$ 
C)  $(2,-1)$ 
D)  $(1,2)$

Question 16

Solve the problem.
-The cost function for the manufacture of graphing calculators is given by $C(x)=100,000+21 x+\frac{x^{2}}{10,000}$, where $x$ is the number of graphing calculators manufactured.
Using the appropriate domain, sketch the graph of the average cost $\bar{C}$ to manufacture $x$ graphing calculators. Find the absolute minimum on the graph of $\overline{\mathrm{C}}$. What do the coordinates of the absolute minimum tell us?

Accepted Answer

A)  The absolute minimum is at $(31,622.78,29.11)$. This tells us that the average cost of a graphing calculator is minimized at $\$ 29.11$ per calculator when approximately 31,623 are produced. 
B)  The absolute minimum is at $(31,622.78,27.32)$. This tells us that the average cost of a graphing calculator is minimized at $\$ 27.32$ per calculator when approximately 31,623 are produced. 
C)  The absolute minimum is at $(34,825.23,27.32)$. This tells us that the average cost of a graphing calculator is minimized at $\$ 27.32$ per calculator when approximately $34,825.23$ are produced. 
D)  There is no absolute maximum. 
A)  The absolute minimum is at $(31,622.78,29.11)$. This tells us that the average cost of a graphing calculator is minimized at $\$ 29.11$ per calculator when approximately 31,623 are produced. 
B)  The absolute minimum is at $(31,622.78,27.32)$. This tells us that the average cost of a graphing calculator is minimized at $\$ 27.32$ per calculator when approximately 31,623 are produced. 
C)  The absolute minimum is at $(34,825.23,27.32)$. This tells us that the average cost of a graphing calculator is minimized at $\$ 27.32$ per calculator when approximately $34,825.23$ are produced. 
D)  There is no absolute maximum.

Question 17

$\mathrm{s}$ is the distance (in $\mathrm{ft}$ ) traveled in time $\mathrm{t}$ (in s) by a particle. Find the velocity and acceleration at the given time.
-$s=5 t^{3}+6 t^{2}+9 t+2, t=1$

Accepted Answer

A)  $\mathrm{v}=36 \mathrm{ft} / \mathrm{s}, \mathrm{a}=42 \mathrm{ft} / \mathrm{s}^{2}$ 
B)  $v=27 \mathrm{ft} / \mathrm{s}, a=3 \mathrm{ft} / \mathrm{s}^{2}$ 
C)  $v=42 \mathrm{ft} / \mathrm{s}, a=36 \mathrm{ft} / \mathrm{s}^{2}$ 
D)  $v=3 \mathrm{ft} / \mathrm{s}, a=27 \mathrm{ft} / \mathrm{s}^{2}$ 
A)  $\mathrm{v}=36 \mathrm{ft} / \mathrm{s}, \mathrm{a}=42 \mathrm{ft} / \mathrm{s}^{2}$ 
B)  $v=27 \mathrm{ft} / \mathrm{s}, a=3 \mathrm{ft} / \mathrm{s}^{2}$ 
C)  $v=42 \mathrm{ft} / \mathrm{s}, a=36 \mathrm{ft} / \mathrm{s}^{2}$ 
D)  $v=3 \mathrm{ft} / \mathrm{s}, a=27 \mathrm{ft} / \mathrm{s}^{2}$

Question 18

Find the largest open intervals where the function is concave upward.
-$f(x)=x^{3}+3 x^{2}-9 x+5$

Accepted Answer

A)  None 
B)  $(-1, \infty),(-\infty,-1)$ 
C)  $(-1, \infty)$ 
D)  $(-\infty,-1)$ 
A)  None 
B)  $(-1, \infty),(-\infty,-1)$ 
C)  $(-1, \infty)$ 
D)  $(-\infty,-1)$

Question 19

Solve the problem.
-A zoom lens in a camera makes a rectangular image on the film that is $7 \mathrm{~cm} \times 6 \mathrm{~cm}$. As the lens zooms in and out, the size of the image changes. Find the rate at which the area of the image begins to change $(\mathrm{dA} / \mathrm{df})$ if the length of the frame changes at $0.8 \mathrm{~cm} / \mathrm{s}$ and the width of the frame changes at $0.2 \mathrm{~cm} / \mathrm{s}$.

Accepted Answer

A)  $6.8 \mathrm{~m}^{2} / \mathrm{s}$ 
B)  $6.2 \mathrm{~m}^{2} / \mathrm{s}$ 
C)  $\frac{7}{6} \mathrm{~m}^{2} / 2$ 
D)  $1.0333 \mathrm{~m}^{2} / \mathrm{s}$ 
A)  $6.8 \mathrm{~m}^{2} / \mathrm{s}$ 
B)  $6.2 \mathrm{~m}^{2} / \mathrm{s}$ 
C)  $\frac{7}{6} \mathrm{~m}^{2} / 2$ 
D)  $1.0333 \mathrm{~m}^{2} / \mathrm{s}$

Question 20

Find the coordinates of the points of inflection for the function.
-$f(x)=\sqrt{x+10}$

Accepted Answer

A)  $(-6,2)$ 
B)  $(-9,1)$ 
C)  $(-10,0)$ 
D)  There are no points of inflection. 
A)  $(-6,2)$ 
B)  $(-9,1)$ 
C)  $(-10,0)$ 
D)  There are no points of inflection.

Identify the intervals where the function is changing as requested. -Decreasing

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - $f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14$

Solve each problem. -Find two numbers whose sum is 440 and whose product is as large as possible.

Find the location of the indicated absolute extrema for the function. -Maximum

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=\frac{x^{2}+2}{3 x^{2}-1}, c=0$

Find the absolute extremum within the specified domain. -Minimum of $f(x)=x^{3}-3 x^{2} ;[-0.5,4]$

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - $f(x)=-x^{3}-3 x^{2}+9 x+2$

Sketch the graph and show all local extrema and inflection points. - $f(x)=\frac{1}{x^{2}-2 x-3}$ $Sketch the graph and show all local extrema and inflection points. - f(x)=\frac{1}{x^{2}-2 x-3}$

Identify the intervals where the function is changing as requested. -Increasing

Find the coordinates of the points of inflection for the function. - $f(x)=x^{2}+6 x+11$

Find the coordinates of the points of inflection for the function. - $f(x)=x^{1 / 3}\left(x^{2}-28\right)$

Identify the intervals where the function is changing as requested. -Increasing

Identify the intervals where the function is changing as requested. -Decreasing

$\mathrm{s}$ is the distance (in $\mathrm{ft}$ ) traveled in time $\mathrm{t}$ (in s) by a particle. Find the velocity and acceleration at the given time. - $s=5 t^{3}+6 t^{2}+9 t+2, t=1$

Find the largest open intervals where the function is concave upward. - $f(x)=x^{3}+3 x^{2}-9 x+5$

Find the coordinates of the points of inflection for the function. - $f(x)=\sqrt{x+10}$

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Systems of Linear Equations and Matrices

Linear Programming

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Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

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Exam 12: Applications of the Derivative

Identify the intervals where the function is changing as requested. -Decreasing

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - f(x)=0.1x4−x3−15x2+59x+14f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14f(x)=0.1x4−x3−15x2+59x+14

Solve each problem. -Find two numbers whose sum is 440 and whose product is as large as possible.

Find the location of the indicated absolute extrema for the function. -Maximum

Evaluate f′′(c)\mathrm{f}^{\prime \prime}(\mathrm{c})f′′(c) at the point. - f(x)=x2+23x2−1,c=0f(x)=\frac{x^{2}+2}{3 x^{2}-1}, c=0f(x)=3x2−1x2+2​,c=0

Find the absolute extremum within the specified domain. -Minimum of f(x)=x3−3x2;[−0.5,4]f(x)=x^{3}-3 x^{2} ;[-0.5,4]f(x)=x3−3x2;[−0.5,4]

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - f(x)=−x3−3x2+9x+2f(x)=-x^{3}-3 x^{2}+9 x+2f(x)=−x3−3x2+9x+2

Sketch the graph and show all local extrema and inflection points. - f(x)=1x2−2x−3f(x)=\frac{1}{x^{2}-2 x-3}f(x)=x2−2x−31​

Identify the intervals where the function is changing as requested. -Increasing

Find the coordinates of the points of inflection for the function. - f(x)=x2+6x+11f(x)=x^{2}+6 x+11f(x)=x2+6x+11

Find the coordinates of the points of inflection for the function. - f(x)=x1/3(x2−28)f(x)=x^{1 / 3}\left(x^{2}-28\right)f(x)=x1/3(x2−28)

Identify the intervals where the function is changing as requested. -Increasing

Identify the intervals where the function is changing as requested. -Decreasing

s\mathrm{s}s is the distance (in ft\mathrm{ft}ft ) traveled in time t\mathrm{t}t (in s) by a particle. Find the velocity and acceleration at the given time. - s=5t3+6t2+9t+2,t=1s=5 t^{3}+6 t^{2}+9 t+2, t=1s=5t3+6t2+9t+2,t=1

Find the largest open intervals where the function is concave upward. - f(x)=x3+3x2−9x+5f(x)=x^{3}+3 x^{2}-9 x+5f(x)=x3+3x2−9x+5

Find the coordinates of the points of inflection for the function. - f(x)=x+10f(x)=\sqrt{x+10}f(x)=x+10​

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Systems of Linear Equations and Matrices

Linear Programming

Sets and Probability

Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

Differential Calculus

Integral Calculus

Multivariate Calculus

Filters

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - $f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14$

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=\frac{x^{2}+2}{3 x^{2}-1}, c=0$

Find the absolute extremum within the specified domain. -Minimum of $f(x)=x^{3}-3 x^{2} ;[-0.5,4]$

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - $f(x)=-x^{3}-3 x^{2}+9 x+2$

Sketch the graph and show all local extrema and inflection points. - $f(x)=\frac{1}{x^{2}-2 x-3}$ $Sketch the graph and show all local extrema and inflection points. - f(x)=\frac{1}{x^{2}-2 x-3}$

Find the coordinates of the points of inflection for the function. - $f(x)=x^{2}+6 x+11$

Find the coordinates of the points of inflection for the function. - $f(x)=x^{1 / 3}\left(x^{2}-28\right)$

$\mathrm{s}$ is the distance (in $\mathrm{ft}$ ) traveled in time $\mathrm{t}$ (in s) by a particle. Find the velocity and acceleration at the given time. - $s=5 t^{3}+6 t^{2}+9 t+2, t=1$

Find the largest open intervals where the function is concave upward. - $f(x)=x^{3}+3 x^{2}-9 x+5$

Find the coordinates of the points of inflection for the function. - $f(x)=\sqrt{x+10}$