Question 1

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither.
-$f(x)=x^{3}+12 x^{2}+48 x+4$

Accepted Answer

A)  Local maximum at -4 ; local minimum at 4 
B)  Local minimum at -4 
C)  No local extrema 
D)  Local maximum at -4 
A)  Local maximum at -4 ; local minimum at 4 
B)  Local minimum at -4 
C)  No local extrema 
D)  Local maximum at -4

Question 2

Solve the problem.
-It is estimated that the total value of a stamp collection is given by the formula $v=44,700+100 t^{2}$, 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. At what value of $t$ is the present value increasing most rapidly?

Accepted Answer

A)  $t=19.66$ 
B)  $t=24.76$ 
C)  $t=19.36$ 
D)  $t=21.66$ 
A)  $t=19.66$ 
B)  $t=24.76$ 
C)  $t=19.36$ 
D)  $t=21.66$

Question 3

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point.
-$f(x)=\left(x^{2}-3 x+2\right)(2 x-6), c=0$

Accepted Answer

A)  $\mathrm{f}^{\prime \prime}(0)=22$ 
B)  $\mathrm{f}^{\prime \prime}(0)=0$ 
C)  $f^{\prime \prime}(0)=-12$ 
D)  $f^{\prime \prime}(0)=-24$ 
A)  $\mathrm{f}^{\prime \prime}(0)=22$ 
B)  $\mathrm{f}^{\prime \prime}(0)=0$ 
C)  $f^{\prime \prime}(0)=-12$ 
D)  $f^{\prime \prime}(0)=-24$

Question 4

Find the largest open intervals where the function is concave upward.
-$f(x)=x^{4}-8 x^{2}$ (exact values)

Accepted Answer

A)  None 
B)  $\left(-\infty,-\frac{2}{\sqrt{3}}\right),\left(\frac{2}{\sqrt{3}}, \infty\right)$ 
C)  $\left(\frac{2}{\sqrt{3}}, \infty\right)$ 
D)  $\left(-\infty,-\frac{2}{\sqrt{3}}\right)$ 
A)  None 
B)  $\left(-\infty,-\frac{2}{\sqrt{3}}\right),\left(\frac{2}{\sqrt{3}}, \infty\right)$ 
C)  $\left(\frac{2}{\sqrt{3}}, \infty\right)$ 
D)  $\left(-\infty,-\frac{2}{\sqrt{3}}\right)$

Question 5

Find the largest open interval where the function is changing as requested.
-Decreasing $f(x)=x^{3}-4 x$

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

A)  $(5,1)$ 
B)  $(5,12)$ 
C)  $(6,1)$ 
D)  $(6,12)$ 
A)  $(5,1)$ 
B)  $(5,12)$ 
C)  $(6,1)$ 
D)  $(6,12)$

Question 8

Use calculus and a graphing calculator to find the approximate location of all relative extrema.
-$f(x)=0.01 x^{5}-x^{4}+x^{3}+8 x^{2}-7 x-10$

Accepted Answer

A)  Approximate local maxima at -1.861 and 2.247 ; approximate local minima at 0.423 and 79.192 
B)  Approximate local maxima at -1.861 and 2.247 ; approximate local minimum at 0.423 
C)  Approximate local maxima at -1.78 and 2.164 ; approximate local minima at 0.464 and 79.109 
D)  Approximate local maxima at -1.774 and 2.263 ; approximate local minima at 0.467 and 79.105 
A)  Approximate local maxima at -1.861 and 2.247 ; approximate local minima at 0.423 and 79.192 
B)  Approximate local maxima at -1.861 and 2.247 ; approximate local minimum at 0.423 
C)  Approximate local maxima at -1.78 and 2.164 ; approximate local minima at 0.464 and 79.109 
D)  Approximate local maxima at -1.774 and 2.263 ; approximate local minima at 0.467 and 79.105

Question 9

Find the equation of the tangent line at the given point on the curve.
-$\mathrm{x}^{2}+\mathrm{y}^{2}=25 ;(-4,3)$

Accepted Answer

A)  $y=-\frac{4}{3} x-\frac{25}{3}$ 
B)  $y=\frac{4}{3} x-\frac{25}{3}$ 
C)  $y=-\frac{4}{3} x+\frac{25}{3}$ 
D)  $y=\frac{4}{3} x+\frac{25}{3}$ 
A)  $y=-\frac{4}{3} x-\frac{25}{3}$ 
B)  $y=\frac{4}{3} x-\frac{25}{3}$ 
C)  $y=-\frac{4}{3} x+\frac{25}{3}$ 
D)  $y=\frac{4}{3} x+\frac{25}{3}$

Question 10

Find $\frac{d y}{d x}$ at the given point.
-$2 x y-x^{2}=1$; (1,1)

Accepted Answer

A)  1 
B)  -1 
C)  $\frac{1}{2}$ 
D)  0 
A)  1 
B)  -1 
C)  $\frac{1}{2}$ 
D)  0

Question 11

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

Accepted Answer

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

Question 12

Solve each problem.
-The velocity of a particle (in $\mathrm{ft} / \mathrm{s}$ ) is given by $\mathrm{v}=\mathrm{t}^{2}-8 \mathrm{t}+3$, where $\mathrm{t}$ is the time (in seconds) for which it has traveled. Find the time at which the velocity is at a minimum.

Accepted Answer

A)  $3 \mathrm{~s}$ 
B)  $1.5 \mathrm{~s}$ 
C)  $8 \mathrm{~s}$ 
D)  $4 \mathrm{~s}$ 
A)  $3 \mathrm{~s}$ 
B)  $1.5 \mathrm{~s}$ 
C)  $8 \mathrm{~s}$ 
D)  $4 \mathrm{~s}$

Question 13

Find $\frac{d y}{d x}$ at the given point.
-$x y+x=2 ;(1,1)$

Accepted Answer

A)  $-\frac{1}{2}$ 
B)  -2 
C)  $\frac{1}{2}$ 
D)  2 
A)  $-\frac{1}{2}$ 
B)  -2 
C)  $\frac{1}{2}$ 
D)  2

Question 14

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

Accepted Answer

A)  $2,4,5$ 
B)  3.67 
C)  4 
D)  $\frac{11-\sqrt{7}}{3}, \frac{11+\sqrt{7}}{3}$ 
A)  $2,4,5$ 
B)  3.67 
C)  4 
D)  $\frac{11-\sqrt{7}}{3}, \frac{11+\sqrt{7}}{3}$

Question 15

Sketch a graph of a single function that has these properties.
-(a) defined for all real numbers
(b) increasing on $(-3,3)$
(c) decreasing on $(-\infty,-3)$ and $(3, \infty)$
(d) concave downward on $(0, \infty)$
(e) concave upward on $(-\infty, 0)$
(f) $\mathrm{f}^{\prime}(-3)=\mathrm{f}^{\prime}(3)=0$
(g) inflection point at $(0,0)$

Accepted Answer

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

Question 16

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$f(x)=0.01 x^{5}-x^{4}+x^{3}+8 x^{2}-7 x+87$

Accepted Answer

A)  Approximate local maxima at -1.861 and 2.247 ; approximate local minimum at 0.423 
B)  Approximate local maxima at -1.85 and 2.165 ; approximate local minima at 0.445 and 79.11 
C)  Approximate local maxima at -1.864 and 2.282 ; approximate local minima at 0.373 and 79.195 
D)  Approximate local maxima at -1.861 and 2.247 ; approximate local minima at 0.423 and 79.192 
A)  Approximate local maxima at -1.861 and 2.247 ; approximate local minimum at 0.423 
B)  Approximate local maxima at -1.85 and 2.165 ; approximate local minima at 0.445 and 79.11 
C)  Approximate local maxima at -1.864 and 2.282 ; approximate local minima at 0.373 and 79.195 
D)  Approximate local maxima at -1.861 and 2.247 ; approximate local minima at 0.423 and 79.192

Question 17

Solve the problem.
-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 $\mathrm{p}(\mathrm{x})=106-\frac{\mathrm{x}}{28}$. How many candy bars must be sold to maximize revenue?

Accepted Answer

A)  2968 thousand candy bars 
B)  1484 thousand candy bars 
C)  1484 candy bars 
D)  2968 candy bars 
A)  2968 thousand candy bars 
B)  1484 thousand candy bars 
C)  1484 candy bars 
D)  2968 candy bars

Question 18

Find the equation of the tangent line at the given point on the curve.
-$x^{2}+3 y^{2}=13 ;(1,2)$

Accepted Answer

A)  $y=-\frac{1}{6} x+\frac{13}{6}$ 
B)  $y=-\frac{1}{3} x+\frac{7}{3}$ 
C)  $y=\frac{1}{3} x+\frac{5}{3}$ 
D)  $y=\frac{1}{6} x+\frac{11}{6}$ 
A)  $y=-\frac{1}{6} x+\frac{13}{6}$ 
B)  $y=-\frac{1}{3} x+\frac{7}{3}$ 
C)  $y=\frac{1}{3} x+\frac{5}{3}$ 
D)  $y=\frac{1}{6} x+\frac{11}{6}$

Question 19

Solve each problem.
-Find the dimensions of the rectangular field of maximum area that can be made from $140 \mathrm{~m}$ of fencing material.

Accepted Answer

A)  $35 \mathrm{~m}$ by $105 \mathrm{~m}$ 
B)  $70 \mathrm{~m}$ by $70 \mathrm{~m}$ 
C)  $14 \mathrm{~m}$ by $126 \mathrm{~m}$ 
D)  $35 \mathrm{~m}$ by $35 \mathrm{~m}$ 
A)  $35 \mathrm{~m}$ by $105 \mathrm{~m}$ 
B)  $70 \mathrm{~m}$ by $70 \mathrm{~m}$ 
C)  $14 \mathrm{~m}$ by $126 \mathrm{~m}$ 
D)  $35 \mathrm{~m}$ by $35 \mathrm{~m}$

Question 20

Solve the problem.
-One airplane is approaching an airport from the north at $135 \mathrm{~km} / \mathrm{hr}$. A second airplane approaches from the east at $201 \mathrm{~km} / \mathrm{hr}$. Find the rate at which the distance between the planes changes when the southbound plane is $33 \mathrm{~km}$ away from the airport and the westbound plane is $21 \mathrm{~km}$ from the airport.

Accepted Answer

A)  $222 \mathrm{~km} / \mathrm{hr}$ 
B)  $108 \mathrm{~km} / \mathrm{hr}$ 
C)  $808 \mathrm{~km} / \mathrm{hr}$ 
D)  $711 \mathrm{~km} / \mathrm{hr}$ 
A)  $222 \mathrm{~km} / \mathrm{hr}$ 
B)  $108 \mathrm{~km} / \mathrm{hr}$ 
C)  $808 \mathrm{~km} / \mathrm{hr}$ 
D)  $711 \mathrm{~km} / \mathrm{hr}$

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - $f(x)=x^{3}+12 x^{2}+48 x+4$

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=\left(x^{2}-3 x+2\right)(2 x-6), c=0$

Find the largest open intervals where the function is concave upward. - $f(x)=x^{4}-8 x^{2}$ (exact values)

Find the largest open interval where the function is changing as requested. -Decreasing $f(x)=x^{3}-4 x$

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

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. - $f(x)=0.01 x^{5}-x^{4}+x^{3}+8 x^{2}-7 x-10$

Find the equation of the tangent line at the given point on the curve. - $\mathrm{x}^{2}+\mathrm{y}^{2}=25 ;(-4,3)$

Find $\frac{d y}{d x}$ at the given point. - $2 x y-x^{2}=1$ ; (1,1)

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

Solve each problem. -The velocity of a particle (in $\mathrm{ft} / \mathrm{s}$ ) is given by $\mathrm{v}=\mathrm{t}^{2}-8 \mathrm{t}+3$ , where $\mathrm{t}$ is the time (in seconds) for which it has traveled. Find the time at which the velocity is at a minimum.

Find $\frac{d y}{d x}$ at the given point. - $x y+x=2 ;(1,1)$

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

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f(x)=0.01 x^{5}-x^{4}+x^{3}+8 x^{2}-7 x+87$

Solve the problem. -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 $\mathrm{p}(\mathrm{x})=106-\frac{\mathrm{x}}{28}$ . How many candy bars must be sold to maximize revenue?

Find the equation of the tangent line at the given point on the curve. - $x^{2}+3 y^{2}=13 ;(1,2)$

Solve each problem. -Find the dimensions of the rectangular field of maximum area that can be made from $140 \mathrm{~m}$ of fencing material.

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Exam 12: Applications of the Derivative

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - f(x)=x3+12x2+48x+4f(x)=x^{3}+12 x^{2}+48 x+4f(x)=x3+12x2+48x+4

Evaluate f′′(c)\mathrm{f}^{\prime \prime}(\mathrm{c})f′′(c) at the point. - f(x)=(x2−3x+2)(2x−6),c=0f(x)=\left(x^{2}-3 x+2\right)(2 x-6), c=0f(x)=(x2−3x+2)(2x−6),c=0

Find the largest open intervals where the function is concave upward. - f(x)=x4−8x2f(x)=x^{4}-8 x^{2}f(x)=x4−8x2 (exact values)

Find the largest open interval where the function is changing as requested. -Decreasing f(x)=x3−4xf(x)=x^{3}-4 xf(x)=x3−4x