Question 1

Find the largest open intervals where the function is concave upward.
-$f(x)=x^{3}-3 x^{2}-4 x+5$

Accepted Answer

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

Question 2

Find dy/dx by implicit differentiation.
-$2 x y-y^{2}=1$

Accepted Answer

A)  $\frac{x}{x-y}$ 
B)  $\frac{x}{y-x}$ 
C)  $\frac{y}{x-y}$ 
D)  $\frac{y}{y-x}$ 
A)  $\frac{x}{x-y}$ 
B)  $\frac{x}{y-x}$ 
C)  $\frac{y}{x-y}$ 
D)  $\frac{y}{y-x}$

Question 3

The graph of the derivative function $f^{\prime}$ is given. Find the critical numbers of the function $f$.
-

Accepted Answer

A)  - $1,0,1$ 
B)  $-1,1$ 
C)  -1 
D)  0 
A)  - $1,0,1$ 
B)  $-1,1$ 
C)  -1 
D)  0

Question 4

Find the location and value of each local extremum for the function.
-

Accepted Answer

A)  $(3,-2)$ 
B)  $(3,-2),(6,0)$ 
C)  $(0,0),(3,-2),(6,0)$ 
D)  $(0,0),(6,0)$ 
A)  $(3,-2)$ 
B)  $(3,-2),(6,0)$ 
C)  $(0,0),(3,-2),(6,0)$ 
D)  $(0,0),(6,0)$

Question 5

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14$

Accepted Answer

A)  Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 
B)  Approximate local maximum at 1.801; approximate local minima at -6.723 and 12.642 
C)  Approximate local maximum at 1.817; approximate local minima at -6.837 and 12.465 
D)  Approximate local maximum at 1.815 ; approximate local minima at -6.778 and 12.597 
A)  Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 
B)  Approximate local maximum at 1.801; approximate local minima at -6.723 and 12.642 
C)  Approximate local maximum at 1.817; approximate local minima at -6.837 and 12.465 
D)  Approximate local maximum at 1.815 ; approximate local minima at -6.778 and 12.597

Question 6

Find the absolute extremum within the specified domain.
-Minimum of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x-4 ;[-2,5]$

Accepted Answer

A)  $\left(2,-\frac{61}{3}\right)$ 
B)  $\left(2,-\frac{62}{3}\right)$ 
C)  $\left(-2,-\frac{61}{3}\right)$ 
D)  $\left(-2,-\frac{62}{3}\right)$ 
A)  $\left(2,-\frac{61}{3}\right)$ 
B)  $\left(2,-\frac{62}{3}\right)$ 
C)  $\left(-2,-\frac{61}{3}\right)$ 
D)  $\left(-2,-\frac{62}{3}\right)$

Question 7

Sketch a graph of a single function that has these properties.
-(a) defined for all real numbers
(b) decreasing on $(-\infty, 0)$
(c) increasing on $(0, \infty)$
(d) concave downward on $(-\infty, 0) \cup(0, \infty)$
(e) $f(0)=f^{\prime}(0)$ is undefined

Accepted Answer

The answer of Sketch a graph of a single function...

Question 8

Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of $120 \mathrm{ft}$.

Accepted Answer

A)  $30 \mathrm{ft} \times 120 \mathrm{ft}$ 
B)  $10 \mathrm{ft} \times 30 \mathrm{ft}$ 
C)  $60 \mathrm{ft} \times 60 \mathrm{ft}$ 
D)  $30 \mathrm{ft} \times 30 \mathrm{ft}$ 
A)  $30 \mathrm{ft} \times 120 \mathrm{ft}$ 
B)  $10 \mathrm{ft} \times 30 \mathrm{ft}$ 
C)  $60 \mathrm{ft} \times 60 \mathrm{ft}$ 
D)  $30 \mathrm{ft} \times 30 \mathrm{ft}$

Question 9

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

Accepted Answer

A)  None 
B)  $(0,5)$ 
C)  $(-4,2)$ 
D)  $(1,4),(2,4)$ 
A)  None 
B)  $(0,5)$ 
C)  $(-4,2)$ 
D)  $(1,4),(2,4)$

Question 10

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

Accepted Answer

A)  $(\sqrt{11}, 11+11 \ln 11)$ and $(-\sqrt{11}, 11+11 \ln 11)$ 
B)  $(-11-\sqrt{13}, \ln 11)$ 
C)  $(11,-11 \ln 11)$ and $(-11,11 \ln 11)$ 
D)  There are no points of inflection. 
A)  $(\sqrt{11}, 11+11 \ln 11)$ and $(-\sqrt{11}, 11+11 \ln 11)$ 
B)  $(-11-\sqrt{13}, \ln 11)$ 
C)  $(11,-11 \ln 11)$ and $(-11,11 \ln 11)$ 
D)  There are no points of inflection.

Question 11

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

Accepted Answer

A)  None 
B)  $(-5,0)$ 
C)  $(-3,3)$ 
D)  $(3,-4)$ 
A)  None 
B)  $(-5,0)$ 
C)  $(-3,3)$ 
D)  $(3,-4)$

Question 12

Find the absolute extremum within the specified domain.
-Maximum of $f(x)=(x+1)^{2}(x-2)$; $[-2,1]$

Accepted Answer

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

Question 13

If the price charged for a candy bar is $p(x)$ cents, then $x$ thousand candy bars will be sold in a certain city, where $\mathrm{p}(\mathrm{x})=49-\frac{\mathrm{x}}{20}$. How many candy bars must be sold to maximize revenue?

Accepted Answer

A)  490 thousand candy bars 
B)  980 thousand candy bars 
C)  980 candy bars 
D)  490 candy bars 
A)  490 thousand candy bars 
B)  980 thousand candy bars 
C)  980 candy bars 
D)  490 candy bars

Question 14

Solve the problem.
-Given the revenue and cost functions $R=36 x-0.7 x^{2}$ and $C=6 x+13$, where $x$ is the daily production, find the rate of change of profit with respect to time when 15 units are produced and the rate of change of production is 4 units per day.

Accepted Answer

A)  $\$ 114.40$ per day 
B)  $\$ 99.00$ per day 
C)  $\$ 36.00$ per day 
D)  $\$ 60.00$ per day 
A)  $\$ 114.40$ per day 
B)  $\$ 99.00$ per day 
C)  $\$ 36.00$ per day 
D)  $\$ 60.00$ per day

Question 15

Solve the problem.
-A truck burns fuel at the rate (gallons per hr) of $G(x)=\frac{1}{35}\left(\frac{64}{x}+\frac{x}{49}\right)$ while traveling at $x \mathrm{mph}$. If fuel costs $\$ 1.27$ per gallon, find the speed that minimizes total cost for a 200 -mile trip. Round to the nearest tenth.

Accepted Answer

A)  $49.5 \mathrm{mph}$ 
B)  $56.5 \mathrm{mph}$ 
C)  $56.0 \mathrm{mph}$ 
D)  $62.2 \mathrm{mph}$ 
A)  $49.5 \mathrm{mph}$ 
B)  $56.5 \mathrm{mph}$ 
C)  $56.0 \mathrm{mph}$ 
D)  $62.2 \mathrm{mph}$

Question 16

Find the largest open intervals where the function is concave upward.
-$f(x)=-3 x^{2}+18 x+16$

Accepted Answer

A)  $(3, \infty)$ 
B)  None 
C)  $(-\infty, \infty)$ 
D)  $(-\infty, 3)$ 
A)  $(3, \infty)$ 
B)  None 
C)  $(-\infty, \infty)$ 
D)  $(-\infty, 3)$

Question 17

Solve each problem.
-An architect needs to design a rectangular room with an area of $91 \mathrm{ft}^{2}$. What dimensions should she use in order to minimize the perimeter?

Accepted Answer

A)  $9.54 \mathrm{ft} \times 9.54 \mathrm{ft}$ 
B)  $9.54 \mathrm{ft} \times 22.75 \mathrm{ft}$ 
C)  $18.2 \mathrm{ft} \times 91 \mathrm{ft}$ 
D)  $22.75 \mathrm{ft} \times 22.75 \mathrm{ft}$ 
A)  $9.54 \mathrm{ft} \times 9.54 \mathrm{ft}$ 
B)  $9.54 \mathrm{ft} \times 22.75 \mathrm{ft}$ 
C)  $18.2 \mathrm{ft} \times 91 \mathrm{ft}$ 
D)  $22.75 \mathrm{ft} \times 22.75 \mathrm{ft}$

Question 18

Find the location and value of each local extremum for the function.
-

Accepted Answer

A)  None 
B)  $(-2,5),(0,0),(2,5)$ 
C)  $(-2,5),(2,5)$ 
D)  $(0,0)$ 
A)  None 
B)  $(-2,5),(0,0),(2,5)$ 
C)  $(-2,5),(2,5)$ 
D)  $(0,0)$

Question 19

Solve the problem.
-A company knows that unit cost $C$ and unit revenue $R$ from the production and sale of $x$ units are related by $C=\frac{R^{2}}{138,000}+2676$. Find the rate of change of revenue per unit when the cost per unit is changing by $\$ 11$ and the revenue is $\$ 1000$.

Accepted Answer

A)  $\$ 55.00$ 
B)  $\$ 267.60$ 
C)  $\$ 759.00$ 
D)  $\$ 513.30$ 
A)  $\$ 55.00$ 
B)  $\$ 267.60$ 
C)  $\$ 759.00$ 
D)  $\$ 513.30$

Question 20

Find the largest open intervals where the function is concave upward.
-$f(x)=\frac{x}{x^{2}+1}$ (exact values)

Accepted Answer

A)  $(-\infty,-1),(-1, \infty)$ 
B)  $(\sqrt{3}, \infty)$ 
C)  $(-\infty,-1)$ 
D)  None 
A)  $(-\infty,-1),(-1, \infty)$ 
B)  $(\sqrt{3}, \infty)$ 
C)  $(-\infty,-1)$ 
D)  None

Find the largest open intervals where the function is concave upward. - $f(x)=x^{3}-3 x^{2}-4 x+5$

Find dy/dx by implicit differentiation. - $2 x y-y^{2}=1$

The graph of the derivative function $f^{\prime}$ is given. Find the critical numbers of the function $f$ . - $The graph of the derivative function f^{\prime} is given. Find the critical numbers of the function f . -$

Find the location and value of each local extremum for the function. -

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14$

Find the absolute extremum within the specified domain. -Minimum of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x-4 ;[-2,5]$

Sketch a graph of a single function that has these properties. -(a) defined for all real numbers (b) decreasing on $(-\infty, 0)$ (c) increasing on $(0, \infty)$ (d) concave downward on $(-\infty, 0) \cup(0, \infty)$ (e) $f(0)=f^{\prime}(0)$ is undefined

Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of $120 \mathrm{ft}$ .

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

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

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

Find the absolute extremum within the specified domain. -Maximum of $f(x)=(x+1)^{2}(x-2)$ ; $[-2,1]$

If the price charged for a candy bar is $p(x)$ cents, then $x$ thousand candy bars will be sold in a certain city, where $\mathrm{p}(\mathrm{x})=49-\frac{\mathrm{x}}{20}$ . How many candy bars must be sold to maximize revenue?

Solve the problem. -Given the revenue and cost functions $R=36 x-0.7 x^{2}$ and $C=6 x+13$ , where $x$ is the daily production, find the rate of change of profit with respect to time when 15 units are produced and the rate of change of production is 4 units per day.

Solve the problem. -A truck burns fuel at the rate (gallons per hr) of $G(x)=\frac{1}{35}\left(\frac{64}{x}+\frac{x}{49}\right)$ while traveling at $x \mathrm{mph}$ . If fuel costs $\$ 1.27$ per gallon, find the speed that minimizes total cost for a 200 -mile trip. Round to the nearest tenth.

Find the largest open intervals where the function is concave upward. - $f(x)=-3 x^{2}+18 x+16$

Solve each problem. -An architect needs to design a rectangular room with an area of $91 \mathrm{ft}^{2}$ . What dimensions should she use in order to minimize the perimeter?

Find the location and value of each local extremum for the function. -

Solve the problem. -A company knows that unit cost $C$ and unit revenue $R$ from the production and sale of $x$ units are related by $C=\frac{R^{2}}{138,000}+2676$ . Find the rate of change of revenue per unit when the cost per unit is changing by $\$ 11$ and the revenue is $\$ 1000$ .

Find the largest open intervals where the function is concave upward. - $f(x)=\frac{x}{x^{2}+1}$ (exact values)

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

Exam 12: Applications of the Derivative

Find the largest open intervals where the function is concave upward. - f(x)=x3−3x2−4x+5f(x)=x^{3}-3 x^{2}-4 x+5f(x)=x3−3x2−4x+5

Find dy/dx by implicit differentiation. - 2xy−y2=12 x y-y^{2}=12xy−y2=1

The graph of the derivative function f′f^{\prime}f′ is given. Find the critical numbers of the function fff . -

Find the location and value of each local extremum for the function. -

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+14f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14f(x)=0.1x4−x3−15x2+59x+14

Find the absolute extremum within the specified domain. -Minimum of f(x)=13x3−2x2+3x−4;[−2,5]f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x-4 ;[-2,5]f(x)=31​x3−2x2+3x−4;[−2,5]

Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 120ft120 \mathrm{ft}120ft .

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

Find the coordinates of the points of inflection for the function. - f(x)=x2+11ln⁡x2f(x)=x^{2}+11 \ln x^{2}f(x)=x2+11lnx2

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

Find the absolute extremum within the specified domain. -Maximum of f(x)=(x+1)2(x−2)f(x)=(x+1)^{2}(x-2)f(x)=(x+1)2(x−2) ; [−2,1][-2,1][−2,1]

If the price charged for a candy bar is p(x)p(x)p(x) cents, then xxx thousand candy bars will be sold in a certain city, where p(x)=49−x20\mathrm{p}(\mathrm{x})=49-\frac{\mathrm{x}}{20}p(x)=49−20x​ . How many candy bars must be sold to maximize revenue?

Find the largest open intervals where the function is concave upward. - f(x)=−3x2+18x+16f(x)=-3 x^{2}+18 x+16f(x)=−3x2+18x+16

Solve each problem. -An architect needs to design a rectangular room with an area of 91ft291 \mathrm{ft}^{2}91ft2 . What dimensions should she use in order to minimize the perimeter?

Find the location and value of each local extremum for the function. -

Find the largest open intervals where the function is concave upward. - f(x)=xx2+1f(x)=\frac{x}{x^{2}+1}f(x)=x2+1x​ (exact values)

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

Find the largest open intervals where the function is concave upward. - $f(x)=x^{3}-3 x^{2}-4 x+5$

Find dy/dx by implicit differentiation. - $2 x y-y^{2}=1$

The graph of the derivative function $f^{\prime}$ is given. Find the critical numbers of the function $f$ . - $The graph of the derivative function f^{\prime} is given. Find the critical numbers of the function f . -$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f(x)=0.1 x^{4}-x^{3}-15 x^{2}+59 x+14$

Find the absolute extremum within the specified domain. -Minimum of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x-4 ;[-2,5]$

Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of $120 \mathrm{ft}$ .

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

Find the absolute extremum within the specified domain. -Maximum of $f(x)=(x+1)^{2}(x-2)$ ; $[-2,1]$

If the price charged for a candy bar is $p(x)$ cents, then $x$ thousand candy bars will be sold in a certain city, where $\mathrm{p}(\mathrm{x})=49-\frac{\mathrm{x}}{20}$ . How many candy bars must be sold to maximize revenue?

Find the largest open intervals where the function is concave upward. - $f(x)=-3 x^{2}+18 x+16$

Solve each problem. -An architect needs to design a rectangular room with an area of $91 \mathrm{ft}^{2}$ . What dimensions should she use in order to minimize the perimeter?

Find the largest open intervals where the function is concave upward. - $f(x)=\frac{x}{x^{2}+1}$ (exact values)