Question 1

Find the equation of the tangent line at the given point on the curve.
-$x y^{2}=12 ;(3,-2)$

Accepted Answer

A)  $y=-\frac{1}{3} x-3$ 
B)  $y=\frac{1}{3} x-3$ 
C)  $y=\frac{1}{3} x+3$ 
D)  $y=-\frac{1}{3} x+3$ 
A)  $y=-\frac{1}{3} x-3$ 
B)  $y=\frac{1}{3} x-3$ 
C)  $y=\frac{1}{3} x+3$ 
D)  $y=-\frac{1}{3} x+3$

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

Write the word or phrase that best completes each statement or answers thequestion.
-You are planning to close off a corner of the first quadrant with a line segment 15 units long running from $(x, 0)$ to $(0, y)$. Show that the area of the triangle enclosed by the segment is largest when $\mathrm{x}=\mathrm{y}$.

Accepted Answer

If @#LAT-DLM&, @#LAT-DLM& represent the

Question 5

Assume $x$ and $y$ are functions of $t$. Evaluate $d y / d t$.
-$\mathrm{x}^{3}=14 \mathrm{y}^{5}-6 ; \frac{\mathrm{dx}}{\mathrm{dt}}=7, \mathrm{y}=1$

Accepted Answer

A)  $\frac{5}{6}$ 
B)  $\frac{6}{5}$ 
C)  $\frac{12}{5}$ 
D)  $\frac{3}{5}$ 
A)  $\frac{5}{6}$ 
B)  $\frac{6}{5}$ 
C)  $\frac{12}{5}$ 
D)  $\frac{3}{5}$

Question 6

The rule of the derivative of a function $f$ is given. Find the location of all local extrema.
-$\mathrm{f}^{\prime}(\mathrm{x})=(\mathrm{x}+5)(\mathrm{x}+2)(\mathrm{x}-4)$

Accepted Answer

A)  Local maximum at -2 ; local minima at -5 and 4 
B)  Local maximum at 2 ; local minima at 5 and -4 
C)  Local maxima at 5 and -4 ; local minimum at 2 
D)  Local maxima at -5 and 4 ; local minimum at -2 
A)  Local maximum at -2 ; local minima at -5 and 4 
B)  Local maximum at 2 ; local minima at 5 and -4 
C)  Local maxima at 5 and -4 ; local minimum at 2 
D)  Local maxima at -5 and 4 ; local minimum at -2

Question 7

A trough is to be made with an end of the dimensions shown. The length of the trough is to be 19 feet long. Only the angle $\theta$ can be varied. What value of $\theta$ will maximize the trough's volume?

Accepted Answer

A)  $49^{\circ}$ 
B)  $32^{\circ}$ 
C)  $11^{\circ}$ 
D)  $30^{\circ}$ 
A)  $49^{\circ}$ 
B)  $32^{\circ}$ 
C)  $11^{\circ}$ 
D)  $30^{\circ}$

Question 8

Find the largest open interval where the function is changing as requested.
-Increasing $f(x)=\frac{1}{4} x^{2}-\frac{1}{2} x$

Accepted Answer

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

Question 9

Find the largest open interval where the function is changing as requested.
-Decreasing $f(x)=|x-8|$

Accepted Answer

A)  $(8, \infty)$ 
B)  $(-\infty, 8)$ 
C)  $(-\infty,-8)$ 
D)  $(-8, \infty)$ 
A)  $(8, \infty)$ 
B)  $(-\infty, 8)$ 
C)  $(-\infty,-8)$ 
D)  $(-8, \infty)$

Question 10

Assume $x$ and $y$ are functions of $t$. Evaluate $d y / d t$.
-$\mathrm{xy}+\mathrm{x}=12 ; \frac{\mathrm{dx}}{\mathrm{dt}}=-3, \mathrm{x}=2, \mathrm{y}=5$

Accepted Answer

A)  3 
B)  -3 
C)  -9 
D)  9 
A)  3 
B)  -3 
C)  -9 
D)  9

Question 11

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither.
-$f(x)=x-\frac{10}{x}$

Accepted Answer

A)  Local maximum at $\sqrt{10}$; local minimum at $-\sqrt{10}$ 
B)  Local maximum at -10 ; local minimum at 10 
C)  Local maximum at $-\sqrt{10}$; local minimum at $\sqrt{10}$ 
D)  No critical numbers; no local extrema 
A)  Local maximum at $\sqrt{10}$; local minimum at $-\sqrt{10}$ 
B)  Local maximum at -10 ; local minimum at 10 
C)  Local maximum at $-\sqrt{10}$; local minimum at $\sqrt{10}$ 
D)  No critical numbers; no local extrema

Question 12

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

Accepted Answer

A)  None 
B)  $\left(-4,-\frac{1}{2}\right)$ 
C)  $\left(0, \frac{1}{2}\right)$ 
D)  $\left(1, \frac{1}{3}\right)$ 
A)  None 
B)  $\left(-4,-\frac{1}{2}\right)$ 
C)  $\left(0, \frac{1}{2}\right)$ 
D)  $\left(1, \frac{1}{3}\right)$

Question 13

Solve the problem.
-If the price charged for a bolt is $p$ cents, then $x$ thousand bolts will be sold in a certain hardware store, where $\mathrm{p}=37-\frac{\mathrm{x}}{12}$. How many bolts must be sold to maximize revenue?

Accepted Answer

A)  444 bolts 
B)  222 thousand bolts 
C)  222 bolts 
D)  444 thousand bolts 
A)  444 bolts 
B)  222 thousand bolts 
C)  222 bolts 
D)  444 thousand bolts

Question 14

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point.
-$f(x)=e^{4-x^{2}}, c=2$

Accepted Answer

A)  $f^{\prime \prime}(2)=-4$ 
B)  $f^{\prime \prime}(2)=-20$ 
C)  $\mathrm{f}^{\prime \prime}(2)=1$ 
D)  $\mathrm{f}^{\prime \prime}(2)=14$ 
A)  $f^{\prime \prime}(2)=-4$ 
B)  $f^{\prime \prime}(2)=-20$ 
C)  $\mathrm{f}^{\prime \prime}(2)=1$ 
D)  $\mathrm{f}^{\prime \prime}(2)=14$

Question 15

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

Accepted Answer

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

Question 16

Solve the problem.
-The average daily metabolic rate for a hippopotamus living in the wild can be expressed as a function of weight by $\mathrm{m}=132.9 \mathrm{w} 0.75$, where $\mathrm{w}$ is the weight of the hippopotamus (in $\mathrm{kg}$ ) and $\mathrm{m}$ is the metabolic rate (in $\mathrm{kcal} / \mathrm{day}$ ). Determine $\mathrm{dm} / \mathrm{dt}$ for a $2100-\mathrm{kg}$ hippopotamus that is gaining weight at a rate of $15.75 \mathrm{~kg} /$ day.

Accepted Answer

A)  $309 \mathrm{kcal} / \mathrm{day}^{2}$ 
B)  $232 \mathrm{kcal} / \mathrm{day}^{2}$ 
C)  $15 \mathrm{kcal} / \mathrm{day}^{2}$ 
D)  $10,627 \mathrm{kcal} / \mathrm{day}^{2}$ 
A)  $309 \mathrm{kcal} / \mathrm{day}^{2}$ 
B)  $232 \mathrm{kcal} / \mathrm{day}^{2}$ 
C)  $15 \mathrm{kcal} / \mathrm{day}^{2}$ 
D)  $10,627 \mathrm{kcal} / \mathrm{day}^{2}$

Question 17

Use calculus and a graphing calculator to find the approximate location of all relative extrema.
-$f(x)=0.1 x^{3}-15 x^{2}-14 x+60$

Accepted Answer

A)  Approximate local minimum at -0.465 ; approximate local maximum at 100.465 
B)  Approximate local minimum at -100.465 ; approximate local maximum at 0.465 
C)  Approximate local maximum at -100.465 ; approximate local minimum at 0.465 
D)  Approximate local maximum at -0.465 ; approximate local minimum at 100.465 
A)  Approximate local minimum at -0.465 ; approximate local maximum at 100.465 
B)  Approximate local minimum at -100.465 ; approximate local maximum at 0.465 
C)  Approximate local maximum at -100.465 ; approximate local minimum at 0.465 
D)  Approximate local maximum at -0.465 ; approximate local minimum at 100.465

Question 18

$\mathrm{s}$ 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.
-$s=2 t^{3}+7 t^{2}+7 t+9, t=2$

Accepted Answer

A)  $v=40 \mathrm{ft} / \mathrm{s}, a=26 \mathrm{ft} / \mathrm{s}^{2}$ 
B)  $v=38 \mathrm{ft} / \mathrm{s}, a=59 \mathrm{ft} / \mathrm{s}^{2}$ 
C)  $v=26 \mathrm{ft} / \mathrm{s}, a=40 \mathrm{ft} / \mathrm{s}^{2}$ 
D)  $v=59 \mathrm{ft} / \mathrm{s}, a=38 \mathrm{ft} / \mathrm{s}^{2}$ 
A)  $v=40 \mathrm{ft} / \mathrm{s}, a=26 \mathrm{ft} / \mathrm{s}^{2}$ 
B)  $v=38 \mathrm{ft} / \mathrm{s}, a=59 \mathrm{ft} / \mathrm{s}^{2}$ 
C)  $v=26 \mathrm{ft} / \mathrm{s}, a=40 \mathrm{ft} / \mathrm{s}^{2}$ 
D)  $v=59 \mathrm{ft} / \mathrm{s}, a=38 \mathrm{ft} / \mathrm{s}^{2}$

Question 19

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point.
-$f(x)=\frac{3 x-4}{4 x-3}, c=1$

Accepted Answer

A)  $\mathrm{f}^{\prime \prime}(1)=32$ 
B)  $\left.f^{\prime \prime} 1\right)=7$ 
C)  $\mathrm{f}^{\prime \prime}(1)=44$ 
D)  $f^{\prime \prime}(1)=-56$ 
A)  $\mathrm{f}^{\prime \prime}(1)=32$ 
B)  $\left.f^{\prime \prime} 1\right)=7$ 
C)  $\mathrm{f}^{\prime \prime}(1)=44$ 
D)  $f^{\prime \prime}(1)=-56$

Question 20

The rule of the derivative of a function $f$ is given. Find the location of all points of inflection of the function $f$.
-$f^{\prime}(x)=(x-5)^{2}(x-2)$

Accepted Answer

A)  2 
B)  -4 
C)  $-5,-0.33$ 
D)  $-5,2$ 
A)  2 
B)  -4 
C)  $-5,-0.33$ 
D)  $-5,2$

Find the equation of the tangent line at the given point on the curve. - $x y^{2}=12 ;(3,-2)$

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

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

Assume $x$ and $y$ are functions of $t$ . Evaluate $d y / d t$ . - $\mathrm{x}^{3}=14 \mathrm{y}^{5}-6 ; \frac{\mathrm{dx}}{\mathrm{dt}}=7, \mathrm{y}=1$

The rule of the derivative of a function $f$ is given. Find the location of all local extrema. - $\mathrm{f}^{\prime}(\mathrm{x})=(\mathrm{x}+5)(\mathrm{x}+2)(\mathrm{x}-4)$

Find the largest open interval where the function is changing as requested. -Increasing $f(x)=\frac{1}{4} x^{2}-\frac{1}{2} x$

Find the largest open interval where the function is changing as requested. -Decreasing $f(x)=|x-8|$

Assume $x$ and $y$ are functions of $t$ . Evaluate $d y / d t$ . - $\mathrm{xy}+\mathrm{x}=12 ; \frac{\mathrm{dx}}{\mathrm{dt}}=-3, \mathrm{x}=2, \mathrm{y}=5$

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - $f(x)=x-\frac{10}{x}$

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

Solve the problem. -If the price charged for a bolt is $p$ cents, then $x$ thousand bolts will be sold in a certain hardware store, where $\mathrm{p}=37-\frac{\mathrm{x}}{12}$ . How many bolts must be sold to maximize revenue?

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=e^{4-x^{2}}, c=2$

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

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - $f(x)=0.1 x^{3}-15 x^{2}-14 x+60$

$\mathrm{s}$ 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. - $s=2 t^{3}+7 t^{2}+7 t+9, t=2$

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=\frac{3 x-4}{4 x-3}, c=1$

The rule of the derivative of a function $f$ is given. Find the location of all points of inflection of the function $f$ . - $f^{\prime}(x)=(x-5)^{2}(x-2)$

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Exam 12: Applications of the Derivative

Find the equation of the tangent line at the given point on the curve. - xy2=12;(3,−2)x y^{2}=12 ;(3,-2)xy2=12;(3,−2)

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

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

Assume xxx and yyy are functions of ttt . Evaluate dy/dtd y / d tdy/dt . - x3=14y5−6;dxdt=7,y=1\mathrm{x}^{3}=14 \mathrm{y}^{5}-6 ; \frac{\mathrm{dx}}{\mathrm{dt}}=7, \mathrm{y}=1x3=14y5−6;dtdx​=7,y=1

The rule of the derivative of a function fff is given. Find the location of all local extrema. - f′(x)=(x+5)(x+2)(x−4)\mathrm{f}^{\prime}(\mathrm{x})=(\mathrm{x}+5)(\mathrm{x}+2)(\mathrm{x}-4)f′(x)=(x+5)(x+2)(x−4)

A trough is to be made with an end of the dimensions shown. The length of the trough is to be 19 feet long. Only the angle θ\thetaθ can be varied. What value of θ\thetaθ will maximize the trough's volume?

Find the largest open interval where the function is changing as requested. -Increasing f(x)=14x2−12xf(x)=\frac{1}{4} x^{2}-\frac{1}{2} xf(x)=41​x2−21​x

Find the largest open interval where the function is changing as requested. -Decreasing f(x)=∣x−8∣f(x)=|x-8|f(x)=∣x−8∣

Assume xxx and yyy are functions of ttt . Evaluate dy/dtd y / d tdy/dt . - xy+x=12;dxdt=−3,x=2,y=5\mathrm{xy}+\mathrm{x}=12 ; \frac{\mathrm{dx}}{\mathrm{dt}}=-3, \mathrm{x}=2, \mathrm{y}=5xy+x=12;dtdx​=−3,x=2,y=5

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - f(x)=x−10xf(x)=x-\frac{10}{x}f(x)=x−x10​

Find the absolute extremum within the specified domain. -Minimum of f(x)=1x+2;[−4,1]f(x)=\frac{1}{x+2} ;[-4,1]f(x)=x+21​;[−4,1]

Solve the problem. -If the price charged for a bolt is ppp cents, then xxx thousand bolts will be sold in a certain hardware store, where p=37−x12\mathrm{p}=37-\frac{\mathrm{x}}{12}p=37−12x​ . How many bolts must be sold to maximize revenue?

Evaluate f′′(c)\mathrm{f}^{\prime \prime}(\mathrm{c})f′′(c) at the point. - f(x)=e4−x2,c=2f(x)=e^{4-x^{2}}, c=2f(x)=e4−x2,c=2

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

Use calculus and a graphing calculator to find the approximate location of all relative extrema. - f(x)=0.1x3−15x2−14x+60f(x)=0.1 x^{3}-15 x^{2}-14 x+60f(x)=0.1x3−15x2−14x+60

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=2t3+7t2+7t+9,t=2s=2 t^{3}+7 t^{2}+7 t+9, t=2s=2t3+7t2+7t+9,t=2

Evaluate f′′(c)\mathrm{f}^{\prime \prime}(\mathrm{c})f′′(c) at the point. - f(x)=3x−44x−3,c=1f(x)=\frac{3 x-4}{4 x-3}, c=1f(x)=4x−33x−4​,c=1

The rule of the derivative of a function fff is given. Find the location of all points of inflection of the function fff . - f′(x)=(x−5)2(x−2)f^{\prime}(x)=(x-5)^{2}(x-2)f′(x)=(x−5)2(x−2)