Question 1

Find the absolute extremum within the specified domain.
-Minimum of $f(x)=\frac{5}{\sqrt{x^{2}+9}} ;[-3,2]$

Accepted Answer

A)  $\left(2, \frac{5 \sqrt{13}}{13}\right)$ 
B)  None 
C)  $\left(-3, \frac{5 \sqrt{2}}{6}\right)$ 
D)  $\left(0, \frac{5}{3}\right)$ 
A)  $\left(2, \frac{5 \sqrt{13}}{13}\right)$ 
B)  None 
C)  $\left(-3, \frac{5 \sqrt{2}}{6}\right)$ 
D)  $\left(0, \frac{5}{3}\right)$

Question 2

A rectangular sheet of perimeter $33 \mathrm{~cm}$ and dimensions $x \mathrm{~cm}$ by $y \mathrm{~cm}$ is to be rolled into a cylinder as shown in part (a) of the figure. What values of $x$ and $y$ give the largest volume?

Accepted Answer

A)  $x=12 \mathrm{~cm} ; y=\frac{9}{2} \mathrm{~cm}$ 
B)  $x=10 \mathrm{~cm} ; y=\frac{13}{2} \mathrm{~cm}$ 
C)  $x=13 \mathrm{~cm} ; y=\frac{7}{2} \mathrm{~cm}$ 
D)  $x=11 \mathrm{~cm} ; y=\frac{11}{2} \mathrm{~cm}$ 
A)  $x=12 \mathrm{~cm} ; y=\frac{9}{2} \mathrm{~cm}$ 
B)  $x=10 \mathrm{~cm} ; y=\frac{13}{2} \mathrm{~cm}$ 
C)  $x=13 \mathrm{~cm} ; y=\frac{7}{2} \mathrm{~cm}$ 
D)  $x=11 \mathrm{~cm} ; y=\frac{11}{2} \mathrm{~cm}$

Question 3

Solve the problem.
-Electrical systems are governed by Ohm's law, which states that $V=I R$, where $V=$ voltage, $I$ = current, and $R=$ resistance. If the current in an electrical system is decreasing at a rate of $7 \mathrm{~A} / \mathrm{s}$ while the voltage remains constant at $20 \mathrm{~V}$, at what rate is the resistance increasing when the current is $60 \mathrm{~A}$ ?

Accepted Answer

A)  $\frac{3}{7} \mathrm{ohms} / \mathrm{s}$ 
B)  $\frac{7}{180} \mathrm{ohms} / \mathrm{s}$ 
C)  $\frac{49}{3} \mathrm{ohms} / \mathrm{s}$ 
D)  $\frac{7}{3} \mathrm{ohms} / \mathrm{s}$ 
A)  $\frac{3}{7} \mathrm{ohms} / \mathrm{s}$ 
B)  $\frac{7}{180} \mathrm{ohms} / \mathrm{s}$ 
C)  $\frac{49}{3} \mathrm{ohms} / \mathrm{s}$ 
D)  $\frac{7}{3} \mathrm{ohms} / \mathrm{s}$

Question 4

Find the absolute extremum within the specified domain.
-Maximum of $f(x)=3 x^{4}+16 x^{3}+24 x^{2}+32 ;[-3,1]$

Accepted Answer

A)  $(1,75)$ 
B)  $(-3,32)$ 
C)  $(-2,85)$ 
D)  $(0,32)$ 
A)  $(1,75)$ 
B)  $(-3,32)$ 
C)  $(-2,85)$ 
D)  $(0,32)$

Question 5

Solve the problem.
-The demand for tickets at a concert hall can be approximated by $p=250-\frac{q}{4}$, where $p$ is the price (in dollars) and q is the quantity demanded. Use implicit differentiation to find and interpret $\frac{\mathrm{dq}}{\mathrm{dp}}$ when $\mathrm{q}=600$.

Accepted Answer

A)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-\frac{1}{4}$ means that if the price is increased by $\$ 1$, the quantity demanded decreases by approximately $\frac{1}{4}$ tickets. 
B)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-4$ means that if the price is decreased by $\$ 4$, the quantity demanded increases by 1 ticket. 
C)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-\frac{1}{4}$ means that if the price is decreased by $\$ 0.25$, the quantity demanded increases by 1 ticket. 
D)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-4$ means that if the price is increased by $\$ 1$, the quantity demanded decreases by approximately 4 tickets. 
A)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-\frac{1}{4}$ means that if the price is increased by $\$ 1$, the quantity demanded decreases by approximately $\frac{1}{4}$ tickets. 
B)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-4$ means that if the price is decreased by $\$ 4$, the quantity demanded increases by 1 ticket. 
C)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-\frac{1}{4}$ means that if the price is decreased by $\$ 0.25$, the quantity demanded increases by 1 ticket. 
D)  $\frac{\mathrm{dq}}{\mathrm{dp}}=-4$ means that if the price is increased by $\$ 1$, the quantity demanded decreases by approximately 4 tickets.

Question 6

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

Accepted Answer

A)  $(-.57,3.64)$ and $(.57,3.64)$ 
B)  $(-.71,3.64)$ and $(.71,3.64)$ 
C)  $(-.71,3.16)$ and $(.71,3.16)$ 
D)  $(-.57,3.16)$ and $(.57,3.16)$ 
A)  $(-.57,3.64)$ and $(.57,3.64)$ 
B)  $(-.71,3.64)$ and $(.71,3.64)$ 
C)  $(-.71,3.16)$ and $(.71,3.16)$ 
D)  $(-.57,3.16)$ and $(.57,3.16)$

Question 7

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

Accepted Answer

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

Question 8

Find the largest open intervals where the function is concave upward.
-$f(x)=4 x^{3}-45 x^{2}+150 x$

Accepted Answer

A)  $\left(-\frac{15}{4}, \infty\right)$ 
B)  $\left(\frac{15}{4}, \infty\right)$ 
C)  $\left(-\infty,-\frac{15}{4}\right)$ 
D)  $\left(-\infty, \frac{15}{4}\right)$ 
A)  $\left(-\frac{15}{4}, \infty\right)$ 
B)  $\left(\frac{15}{4}, \infty\right)$ 
C)  $\left(-\infty,-\frac{15}{4}\right)$ 
D)  $\left(-\infty, \frac{15}{4}\right)$

Question 9

Sketch the graph and show all local extrema and inflection points.
-$f(x)=4 x^{2}+24 x$

Accepted Answer

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

Question 10

Assume $x$ and $y$ are functions of $t$. Evaluate $d y / d t$.
-$\mathrm{x}^{3}+\mathrm{y}^{3}=9 ; \frac{\mathrm{dx}}{\mathrm{dt}}=-5, \mathrm{x}=2$

Accepted Answer

A)  $\frac{5}{4}$ 
B)  $\frac{4}{5}$ 
C)  -20 
D)  20 
A)  $\frac{5}{4}$ 
B)  $\frac{4}{5}$ 
C)  -20 
D)  20

Question 11

Solve the problem.
-It is estimated that the total value of a stamp collection is given by the formula $v=45,600+100 t^{2}$, where $t$ is the number of years from now. If the inflation rate is running continuously at $4 \%$ per year so that the (discounted) present value of an item that will be worth $\$ v$ in $t$ years' time is given by $\mathrm{p}=\mathrm{ve}^{-.04 \mathrm{t}}$. Sketch the graph of the discounted value as a function of time at which the stamp collection is sold. The graph has an absolute maximum. What is the value of $t$ at the absolute maximum? What is the discounted value of the collection at that time?

Accepted Answer

A)  $\mathrm{t}=0$; at which time the discounted value is $\$ 45,600$. 
B)  $\mathrm{t}=12.63$; at which time the discounted value is $\$ 47,592.54$. 
C)  $t=36.23$; at which time the discounted value is $\$ 49,612.90$. 
D)  $t=38.45$; at which time the discounted value is $\$ 46,734.12$. 
A)  $\mathrm{t}=0$; at which time the discounted value is $\$ 45,600$. 
B)  $\mathrm{t}=12.63$; at which time the discounted value is $\$ 47,592.54$. 
C)  $t=36.23$; at which time the discounted value is $\$ 49,612.90$. 
D)  $t=38.45$; at which time the discounted value is $\$ 46,734.12$.

Question 12

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

Accepted Answer

A)  0,2 
B)  $\frac{2}{3}, 2$ 
C)  $0, \frac{2}{3}, 2$ 
D)  none 
A)  0,2 
B)  $\frac{2}{3}, 2$ 
C)  $0, \frac{2}{3}, 2$ 
D)  none

Question 13

Solve the problem.
-A rectangular field is to be enclosed on four sides with a fence. Fencing costs $\$ 5$ per foot for two opposite sides, and $\$ 8$ per foot for the other two sides. Find the dimensions of the field of area $830 \mathrm{ft}^{2}$ that would be the cheapest to enclose. Round to the nearest tenth.

Accepted Answer

A)  $36.4 \mathrm{ft} @ \$ 5$ by $22.8 \mathrm{ft} @ \$ 8$ 
B)  $22.8 \mathrm{ft} @ \$ 5$ by $36.4 \mathrm{ft} @ \$ 8$ 
C)  $46.1 \mathrm{ft} @ \$ 5$ by $18 \mathrm{ft} @ \$ 8$ 
D)  $18 \mathrm{ft} @ \$ 5$ by $46.1 \mathrm{ft} @ \$ 8$ 
A)  $36.4 \mathrm{ft} @ \$ 5$ by $22.8 \mathrm{ft} @ \$ 8$ 
B)  $22.8 \mathrm{ft} @ \$ 5$ by $36.4 \mathrm{ft} @ \$ 8$ 
C)  $46.1 \mathrm{ft} @ \$ 5$ by $18 \mathrm{ft} @ \$ 8$ 
D)  $18 \mathrm{ft} @ \$ 5$ by $46.1 \mathrm{ft} @ \$ 8$

Question 14

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

Accepted Answer

A)  $(2,4)$ 
B)  $(0, \sqrt[3]{16})$ 
C)  $(2, \sqrt[3]{16})$ 
D)  $(0,4)$ 
A)  $(2,4)$ 
B)  $(0, \sqrt[3]{16})$ 
C)  $(2, \sqrt[3]{16})$ 
D)  $(0,4)$

Question 15

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

Accepted Answer

A)  Local maximum at $-\sqrt{2}$; local minimum at $\sqrt{2}$ 
B)  Local maximum at -2 ; local minimum at 2 
C)  Local maximum at $\sqrt{2}$; local minimum at $-\sqrt{2}$ 
D)  No critical numbers; no local extrema 
A)  Local maximum at $-\sqrt{2}$; local minimum at $\sqrt{2}$ 
B)  Local maximum at -2 ; local minimum at 2 
C)  Local maximum at $\sqrt{2}$; local minimum at $-\sqrt{2}$ 
D)  No critical numbers; no local extrema

Question 16

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 $114 \mathrm{in}$. What dimensions will give a box with a square end the largest possible volume?

Accepted Answer

A)  19 in. $x 19$ in. $x 38$ in. 
B)  19 in. $x 38$ in. $x 38$ in. 
C)  19 in. x 19 in. x 95 in. 
D)  38 in. $x 38$ in. $x 38$ in. 
A)  19 in. $x 19$ in. $x 38$ in. 
B)  19 in. $x 38$ in. $x 38$ in. 
C)  19 in. x 19 in. x 95 in. 
D)  38 in. $x 38$ in. $x 38$ in.

Question 17

Solve the problem.
-Water is discharged from a pipeline at a velocity $v$ given by $v=1428 p(1 / 2)$, where $p$ is the pressure (in $\mathrm{psi}$ ). If the water pressure is changing at a rate of $0.401 \mathrm{psi} / \mathrm{second}$, find the acceleration $(\mathrm{dv} / \mathrm{dt})$ of the water when $\mathrm{p}=49 \mathrm{psi}$.

Accepted Answer

A)  $2004.2 \mathrm{ft} / \mathrm{s}^{2}$ 
B)  $102 \mathrm{ft} / \mathrm{s}^{2}$ 
C)  $49.98 \mathrm{ft} / \mathrm{s}^{2}$ 
D)  $40.9 \mathrm{ft} / \mathrm{s}^{2}$ 
A)  $2004.2 \mathrm{ft} / \mathrm{s}^{2}$ 
B)  $102 \mathrm{ft} / \mathrm{s}^{2}$ 
C)  $49.98 \mathrm{ft} / \mathrm{s}^{2}$ 
D)  $40.9 \mathrm{ft} / \mathrm{s}^{2}$

Question 18

Solve the problem.
-The volume of a sphere is increasing at a rate of $2 \mathrm{~cm}^{3} / \mathrm{s}$. Find the rate of change of its surface area when its volume is $\frac{256 \pi}{3} \mathrm{~cm}^{3}$

Accepted Answer

A)  $1 \mathrm{~cm}^{2} / \mathrm{s}$ 
B)  $\frac{64}{3} \mathrm{~cm}^{2} / \mathrm{s}$ 
C)  $\frac{1}{3} \mathrm{~cm}^{2} / \mathrm{s}$ 
D)  $\frac{2}{3} \mathrm{~cm}^{2} / \mathrm{s}$ 
A)  $1 \mathrm{~cm}^{2} / \mathrm{s}$ 
B)  $\frac{64}{3} \mathrm{~cm}^{2} / \mathrm{s}$ 
C)  $\frac{1}{3} \mathrm{~cm}^{2} / \mathrm{s}$ 
D)  $\frac{2}{3} \mathrm{~cm}^{2} / \mathrm{s}$

Question 19

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither.
-$f(x)=\frac{x^{4}}{4}+1.67 x^{3}-1 x^{2}-24 x+4$

Accepted Answer

A)  Local maximum at -3 ; local minima at -4 and 2 
B)  Local maxima at 4 and -2 ; local minimum at -3 
C)  Local maxima at -4 and 2 ; local minimum at -3 
D)  Local maximum at -4 ; local minimum at 2 
A)  Local maximum at -3 ; local minima at -4 and 2 
B)  Local maxima at 4 and -2 ; local minimum at -3 
C)  Local maxima at -4 and 2 ; local minimum at -3 
D)  Local maximum at -4 ; local minimum at 2

Question 20

Solve the problem.
-It is estimated that the total value of a stamp collection is given by the formula $v=45,600+100 t^{2}$, where $t$ is the number of years from now. If the inflation rate is running continuously at $4 \%$ per year so that the (discounted) present value of an item that will be worth $\$ v$ in $t$ years' time is given by $\mathrm{p}=\mathrm{ve}^{-.04 \mathrm{t}}$. Sketch the graph of the discounted value as a function of time at which the stamp collection is sold. What are the values of $t$ at the points of inflection?

Accepted Answer

A)  $t=19.52$ and $t=81.28$. 
B)  $\mathrm{t}=24.92$ and $\mathrm{t}=77.38$. 
C)  $t=19.82$ and $t=86.18$. 
D)  $\mathrm{t}=21.82$ and $\mathrm{t}=78.18$. 
A)  $t=19.52$ and $t=81.28$. 
B)  $\mathrm{t}=24.92$ and $\mathrm{t}=77.38$. 
C)  $t=19.82$ and $t=86.18$. 
D)  $\mathrm{t}=21.82$ and $\mathrm{t}=78.18$.

Find the absolute extremum within the specified domain. -Minimum of $f(x)=\frac{5}{\sqrt{x^{2}+9}} ;[-3,2]$

Find the absolute extremum within the specified domain. -Maximum of $f(x)=3 x^{4}+16 x^{3}+24 x^{2}+32 ;[-3,1]$

Solve the problem. -The demand for tickets at a concert hall can be approximated by $p=250-\frac{q}{4}$ , where $p$ is the price (in dollars) and q is the quantity demanded. Use implicit differentiation to find and interpret $\frac{\mathrm{dq}}{\mathrm{dp}}$ when $\mathrm{q}=600$ .

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

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

Find the largest open intervals where the function is concave upward. - $f(x)=4 x^{3}-45 x^{2}+150 x$

Sketch the graph and show all local extrema and inflection points. - $f(x)=4 x^{2}+24 x$ $Sketch the graph and show all local extrema and inflection points. - f(x)=4 x^{2}+24 x$

Assume $x$ and $y$ are functions of $t$ . Evaluate $d y / d t$ . - $\mathrm{x}^{3}+\mathrm{y}^{3}=9 ; \frac{\mathrm{dx}}{\mathrm{dt}}=-5, \mathrm{x}=2$

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

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

Solve the problem. -The volume of a sphere is increasing at a rate of $2 \mathrm{~cm}^{3} / \mathrm{s}$ . Find the rate of change of its surface area when its volume is $\frac{256 \pi}{3} \mathrm{~cm}^{3}$

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - $f(x)=\frac{x^{4}}{4}+1.67 x^{3}-1 x^{2}-24 x+4$

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Exam 12: Applications of the Derivative

Find the absolute extremum within the specified domain. -Minimum of f(x)=5x2+9;[−3,2]f(x)=\frac{5}{\sqrt{x^{2}+9}} ;[-3,2]f(x)=x2+9​5​;[−3,2]

A rectangular sheet of perimeter 33 cm33 \mathrm{~cm}33 cm and dimensions x cmx \mathrm{~cm}x cm by y cmy \mathrm{~cm}y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of xxx and yyy give the largest volume?

Find the absolute extremum within the specified domain. -Maximum of f(x)=3x4+16x3+24x2+32;[−3,1]f(x)=3 x^{4}+16 x^{3}+24 x^{2}+32 ;[-3,1]f(x)=3x4+16x3+24x2+32;[−3,1]

Find the coordinates of the points of inflection for the function. - f(x)=6e−x2f(x)=6 e^{-x^{2}}f(x)=6e−x2