Question 1

Find all relative extrema of the function $f ( x ) = 4 x ^ { 4 } - 16 x ^ { 3 } + 5$ Use the Second Derivative Test where applicable.

Accepted Answer

A) relative max: $f ( 6 ) = 1733$ ; no relative min 
B) relative max: $f ( 3 ) = 103$ ; no relative min 
C) no relative max or min 
D) relative min: $f ( 6 ) = 1733$ ; no relative max 
E) relative min: $f ( 3 ) = - 103$ ; no relative max 
A) relative max: $f ( 6 ) = 1733$ ; no relative min 
B) relative max: $f ( 3 ) = 103$ ; no relative min 
C) no relative max or min 
D) relative min: $f ( 6 ) = 1733$ ; no relative max 
E) relative min: $f ( 3 ) = - 103$ ; no relative max E

Question 2

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Accepted Answer

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

Question 3

Production. Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function $P ( t ) :$ $P ( t ) = 90 t + 42 t ^ { 2 } - 2 t ^ { 3 }$ . Find the number of hours before the rate of production is maximized. That is, find the point of diminishing returns.

Accepted Answer

A)  $t = 0$ 
B)  $t = 7$ 
C)  $t = 5$ 
D)  $t = 8$ 
E)  $t = 15$ 
A)  $t = 0$ 
B)  $t = 7$ 
C)  $t = 5$ 
D)  $t = 8$ 
E)  $t = 15$ B

Question 4

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. $f ( x ) = x ^ { 3 } - 75 x + 6 , [ - 15,6 ]$

Accepted Answer

A)  $x = - 10$ 
B)  $x = - 5$ 
C)  $x = 0$ 
D)  $x = 5$ 
E)  $x = 6$ 
A)  $x = - 10$ 
B)  $x = - 5$ 
C)  $x = 0$ 
D)  $x = 5$ 
E)  $x = 6$

Question 5

Find the x-values of all relative maxima of the given function. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 12 x + 9$

Accepted Answer

A)  $x = 0$ 
B)  $x = 6$ 
C)  $x = 4$ 
D)  $x = 2$ 
E) no relative maxima 
A)  $x = 0$ 
B)  $x = 6$ 
C)  $x = 4$ 
D)  $x = 2$ 
E) no relative maxima

Question 6

Find all relative extrema of the function $f ( x ) = \sqrt { 36 - x ^ { 2 } }$ . Use the Second-Derivative Test when applicable.

Accepted Answer

A) The relative minimum is $( 0,6 )$ and the relative maximum is $( - 6,0 )$ . 
B) The relative maximum is $( 0,6 )$ . 
C) The relative minimum is $( 0,6 )$ . 
D) The relative maximum is $( 0,6 )$ and the relative minima are $( 6,0 )$ and $( - 6,0 )$ . 
E) The relative minimum is $( 0,6 )$ and the relative maximum is $( 6,0 )$ . 
A) The relative minimum is $( 0,6 )$ and the relative maximum is $( - 6,0 )$ . 
B) The relative maximum is $( 0,6 )$ . 
C) The relative minimum is $( 0,6 )$ . 
D) The relative maximum is $( 0,6 )$ and the relative minima are $( 6,0 )$ and $( - 6,0 )$ . 
E) The relative minimum is $( 0,6 )$ and the relative maximum is $( 6,0 )$ .

Question 7

Assume that x and y are differentiable functions of t. Find dy/dt using the given values. $y = 4 x ^ { 3 } + 7 x ^ { 2 } - x$ for $x = 2 , d x / d t = 2$

Accepted Answer

A) 152 
B) 58 
C) 116 
D) 150 
E) 75 
A) 152 
B) 58 
C) 116 
D) 150 
E) 75

Question 8

Both a function and its derivative are given. Use them to find all critical numbers. $f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }$

Accepted Answer

A)  $x = 0$ 
B)  $x = 216$ 
C)  $x = 0 , x = - 101$ 
D)  $x = 0 , x = 216$ 
E)  $x = - 101 , x = 216$ 
A)  $x = 0$ 
B)  $x = 216$ 
C)  $x = 0 , x = - 101$ 
D)  $x = 0 , x = 216$ 
E)  $x = - 101 , x = 216$

Question 9

Find the slope of the graph at the given point. $( 4 - x ) y ^ { 2 } = x ^ { 3 }$

Accepted Answer

A) 2 
B) 0 
C) 1 
D) 3 
E) 5 
A) 2 
B) 0 
C) 1 
D) 3 
E) 5

Question 10

The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on advertising is $P = - \frac { 1 } { 10 } s ^ { 3 } + 40 s ^ { 2 } + 1000$ The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline. Find the point of diminishing returns. Round your answer to the nearest thousand dollars.

Accepted Answer

A) 67 thousand dollars 
B) 200 thousand dollars 
C) 160 thousand dollars 
D) 267 thousand dollars 
E) 133 thousand dollars 
A) 67 thousand dollars 
B) 200 thousand dollars 
C) 160 thousand dollars 
D) 267 thousand dollars 
E) 133 thousand dollars

Question 11

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }$ $t > 0$ . Find the t-value at which x is maximum. Round your answer to two decimal places.

Accepted Answer

A) 0 hours 
B) 3.61 hours 
C) 693.38 hours 
D) 5.61 hours 
E) 8.80 hours 
A) 0 hours 
B) 3.61 hours 
C) 693.38 hours 
D) 5.61 hours 
E) 8.80 hours

Question 12

A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation $S = 2290 + 90 x + 0.35 x ^ { 2 }$ . The current weekly advertising costs are $1500, and these costs are increasing at a rate of $140 per week. Find the current rate of change of weekly sales.

Accepted Answer

A) 159,600 dollars per week 
B) 161,890 dollars per week 
C) 88,390 dollars per week 
D) 86,100 dollars per week 
E) 802,390 dollars per week 
A) 159,600 dollars per week 
B) 161,890 dollars per week 
C) 88,390 dollars per week 
D) 86,100 dollars per week 
E) 802,390 dollars per week

Question 13

Find the open intervals on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 36 }$ is increasing or decreasing.

Accepted Answer

A) The function is increasing on the interval $- 6 < x < 6$ , and decreasing on the intervals $- \infty < x < - 6$ and $6 < x < \infty$ . 
B) The function is increasing on the interval $- \infty < x < - 6$ , and decreasing on the intervals $- 6 < x < 6$ and $6 < x < \infty$ . 
C) The function is increasing on the interval $6 < x < \infty$ , and decreasing on the intervals $- \infty < x < - 6$ and $- 6 < x < 6$ . 
D) The function is decreasing on the interval $- 6 < x < 6$ , and increasing on the intervals $- \infty < x < - 6$ and $6 < x < \infty$ . 
E) The function is decreasing on the interval $- \infty < x < - 6$ , and increasing on the intervals $- 6 < x < 6$ and $6 < x < \infty$ . 
A) The function is increasing on the interval $- 6 < x < 6$ , and decreasing on the intervals $- \infty < x < - 6$ and $6 < x < \infty$ . 
B) The function is increasing on the interval $- \infty < x < - 6$ , and decreasing on the intervals $- 6 < x < 6$ and $6 < x < \infty$ . 
C) The function is increasing on the interval $6 < x < \infty$ , and decreasing on the intervals $- \infty < x < - 6$ and $- 6 < x < 6$ . 
D) The function is decreasing on the interval $- 6 < x < 6$ , and increasing on the intervals $- \infty < x < - 6$ and $6 < x < \infty$ . 
E) The function is decreasing on the interval $- \infty < x < - 6$ , and increasing on the intervals $- 6 < x < 6$ and $6 < x < \infty$ .

Question 14

Area. The radius, r, of a circle is decreasing at a rate of 2 centimeters per minute. Find the rate of change of area, A, when the radius is $5$ .

Accepted Answer

A)  $\frac { d A } { d t } = - 10 \pi$ 
B)  $\frac { d A } { d t } = - 100 \pi$ 
C)  $\frac { d A } { d t } = 100 \pi$ 
D)  $\frac { d A } { d t } = - 20 \pi$ 
E)  $\frac { d A } { d t } = 20 \pi$ 
A)  $\frac { d A } { d t } = - 10 \pi$ 
B)  $\frac { d A } { d t } = - 100 \pi$ 
C)  $\frac { d A } { d t } = 100 \pi$ 
D)  $\frac { d A } { d t } = - 20 \pi$ 
E)  $\frac { d A } { d t } = 20 \pi$

Question 15

Sketch a graph of a function f having the following characteristics. $$\begin{array} { l } 
f ( - 1 ) = f ( - 3 ) = 0 \
f ^ { \prime }( x )  0 \text { if } x > - 2 \
f ^ { \prime } ( x ) > 0
\end{array}$$

Accepted Answer

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

Question 16

Find the points of inflection and discuss the concavity of the function. $f ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 8$

Accepted Answer

A) inflection point at $x = - \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave upward on $\left( - \frac { 8 } { 21 } , \infty \right)$ 
B) inflection point at $x = - \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave downward on $\left( - \frac { 8 } { 21 } , \infty \right)$ 
C) inflection point at $x = \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave upward on $\left( \frac { 8 } { 21 } , \infty \right)$ 
D) inflection point at $x = \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave downward on $\left( \frac { 8 } { 21 } , \infty \right)$ 
E) none of the above 
A) inflection point at $x = - \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave upward on $\left( - \frac { 8 } { 21 } , \infty \right)$ 
B) inflection point at $x = - \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave downward on $\left( - \frac { 8 } { 21 } , \infty \right)$ 
C) inflection point at $x = \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave upward on $\left( \frac { 8 } { 21 } , \infty \right)$ 
D) inflection point at $x = \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave downward on $\left( \frac { 8 } { 21 } , \infty \right)$ 
E) none of the above

Question 17

Find all relative maxima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 5$

Accepted Answer

A)  $( 0,5 )$ 
B)  $( 2,21 )$ 
C)  $( 4,5 )$ 
D)  $( 0,5 )$ , $( 4,5 )$ 
E) no relative maxima 
A)  $( 0,5 )$ 
B)  $( 2,21 )$ 
C)  $( 4,5 )$ 
D)  $( 0,5 )$ , $( 4,5 )$ 
E) no relative maxima

Question 18

Find all relative extrema of the function $f ( x ) = - 2 x ^ { 2 } - 8 x - 12$ . Use the Second Derivative Test where applicable.

Accepted Answer

A) relative max: $f ( - 2 ) = - 4$ 
B) relative min: $f ( 0 ) = - 12$ 
C) no relative min 
D) no relative max 
E) both A and C
F) both B and D 
A) relative max: $f ( - 2 ) = - 4$ 
B) relative min: $f ( 0 ) = - 12$ 
C) no relative min 
D) no relative max 
E) both A and C
F) both B and D

Question 19

Use the graph of $y = f ( x )$ to identify at which of the indicated points the derivative $f ^ { \prime } ( x )$ changes from positive to negative.

Accepted Answer

A) (5,6) 
B) (-1,2), (2,4) 
C) (-1,2), (5,6) 
D) (-1,2) 
E) (2,4), (5,6) 
A) (5,6) 
B) (-1,2), (2,4) 
C) (-1,2), (5,6) 
D) (-1,2) 
E) (2,4), (5,6)

Question 20

Let x represent the units of labor and y the capital invested in a manufacturing process. When 135,540 units are produced, the relationship between labor and capital can be modeled by $100 x ^ { 0.75 } y ^ { 0.25 } = 135,540$ . Find the rate of change of y with respect to x when $x = 1500 \text { and } y = 135,540$ .

Accepted Answer

A) -2 
B) 0 
C) 3 
D) -7 
E) 5 
A) -2 
B) 0 
C) 3 
D) -7 
E) 5

Find all relative extrema of the function $f ( x ) = 4 x ^ { 4 } - 16 x ^ { 3 } + 5$ Use the Second Derivative Test where applicable.

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. $f ( x ) = x ^ { 3 } - 75 x + 6 , [ - 15,6 ]$

Find the x-values of all relative maxima of the given function. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 12 x + 9$

Find all relative extrema of the function $f ( x ) = \sqrt { 36 - x ^ { 2 } }$ . Use the Second-Derivative Test when applicable.

Assume that x and y are differentiable functions of t. Find dy/dt using the given values. $y = 4 x ^ { 3 } + 7 x ^ { 2 } - x$ for $x = 2 , d x / d t = 2$

Both a function and its derivative are given. Use them to find all critical numbers. $f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }$

Find the slope of the graph at the given point. $( 4 - x ) y ^ { 2 } = x ^ { 3 }$ $Find the slope of the graph at the given point. ( 4 - x ) y ^ { 2 } = x ^ { 3 }$

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }$ $t > 0$ . Find the t-value at which x is maximum. Round your answer to two decimal places.

Find the open intervals on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 36 }$ is increasing or decreasing.

Area. The radius, r, of a circle is decreasing at a rate of 2 centimeters per minute. Find the rate of change of area, A, when the radius is $5$ .

Sketch a graph of a function f having the following characteristics. f(-1)=f(-3)=0 (x)<0 if x<-2 (-2)=0 (x)>0 if x>-2 (x)>0

Find the points of inflection and discuss the concavity of the function. $f ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 8$

Find all relative maxima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 5$

Find all relative extrema of the function $f ( x ) = - 2 x ^ { 2 } - 8 x - 12$ . Use the Second Derivative Test where applicable.

Use the graph of $y = f ( x )$ to identify at which of the indicated points the derivative $f ^ { \prime } ( x )$ changes from positive to negative. $Use the graph of y = f ( x ) to identify at which of the indicated points the derivative f ^ { \prime } ( x ) changes from positive to negative.$

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

Exam 9: Applications of the Derivative

Find all relative extrema of the function f(x)=4x4−16x3+5f ( x ) = 4 x ^ { 4 } - 16 x ^ { 3 } + 5f(x)=4x4−16x3+5 Use the Second Derivative Test where applicable.

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. f(x)=x3−75x+6,[−15,6]f ( x ) = x ^ { 3 } - 75 x + 6 , [ - 15,6 ]f(x)=x3−75x+6,[−15,6]

Find the x-values of all relative maxima of the given function. y=13x3−4x2+12x+9y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 12 x + 9y=31​x3−4x2+12x+9

Find all relative extrema of the function f(x)=36−x2f ( x ) = \sqrt { 36 - x ^ { 2 } }f(x)=36−x2​ . Use the Second-Derivative Test when applicable.

Assume that x and y are differentiable functions of t. Find dy/dt using the given values. y=4x3+7x2−xy = 4 x ^ { 3 } + 7 x ^ { 2 } - xy=4x3+7x2−x for x=2,dx/dt=2x = 2 , d x / d t = 2x=2,dx/dt=2

Both a function and its derivative are given. Use them to find all critical numbers. f(x)=x−9x2/3+7f′(x)=x1/3−6x1/3f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }f(x)=x−9x2/3+7f′(x)=x1/3x1/3−6​

Find the slope of the graph at the given point. (4−x)y2=x3( 4 - x ) y ^ { 2 } = x ^ { 3 }(4−x)y2=x3

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by x(t)=5000tt2+13x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }x(t)=t2+135000t​ t>0t > 0t>0 . Find the t-value at which x is maximum. Round your answer to two decimal places.

Find the open intervals on which the function f(x)=xx2+36f ( x ) = \frac { x } { x ^ { 2 } + 36 }f(x)=x2+36x​ is increasing or decreasing.

Area. The radius, r, of a circle is decreasing at a rate of 2 centimeters per minute. Find the rate of change of area, A, when the radius is 555 .

Sketch a graph of a function f having the following characteristics. f(-1)=f(-3)=0 (x)<0 if x<-2 (-2)=0 (x)>0 if x>-2 (x)>0

Find the points of inflection and discuss the concavity of the function. f(x)=7x3+8x2+5x−8f ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 8f(x)=7x3+8x2+5x−8

Find all relative maxima of the given function. y=x4−8x3+16x2+5y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 5y=x4−8x3+16x2+5

Find all relative extrema of the function f(x)=−2x2−8x−12f ( x ) = - 2 x ^ { 2 } - 8 x - 12f(x)=−2x2−8x−12 . Use the Second Derivative Test where applicable.

Use the graph of y=f(x)y = f ( x )y=f(x) to identify at which of the indicated points the derivative f′(x)f ^ { \prime } ( x )f′(x) changes from positive to negative.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

Find all relative extrema of the function $f ( x ) = 4 x ^ { 4 } - 16 x ^ { 3 } + 5$ Use the Second Derivative Test where applicable.

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. $f ( x ) = x ^ { 3 } - 75 x + 6 , [ - 15,6 ]$

Find the x-values of all relative maxima of the given function. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 12 x + 9$

Find all relative extrema of the function $f ( x ) = \sqrt { 36 - x ^ { 2 } }$ . Use the Second-Derivative Test when applicable.

Assume that x and y are differentiable functions of t. Find dy/dt using the given values. $y = 4 x ^ { 3 } + 7 x ^ { 2 } - x$ for $x = 2 , d x / d t = 2$

Both a function and its derivative are given. Use them to find all critical numbers. $f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }$

Find the slope of the graph at the given point. $( 4 - x ) y ^ { 2 } = x ^ { 3 }$ $Find the slope of the graph at the given point. ( 4 - x ) y ^ { 2 } = x ^ { 3 }$

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }$ $t > 0$ . Find the t-value at which x is maximum. Round your answer to two decimal places.

Find the open intervals on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 36 }$ is increasing or decreasing.

Area. The radius, r, of a circle is decreasing at a rate of 2 centimeters per minute. Find the rate of change of area, A, when the radius is $5$ .

Find the points of inflection and discuss the concavity of the function. $f ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 8$

Find all relative maxima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 5$

Find all relative extrema of the function $f ( x ) = - 2 x ^ { 2 } - 8 x - 12$ . Use the Second Derivative Test where applicable.

Use the graph of $y = f ( x )$ to identify at which of the indicated points the derivative $f ^ { \prime } ( x )$ changes from positive to negative. $Use the graph of y = f ( x ) to identify at which of the indicated points the derivative f ^ { \prime } ( x ) changes from positive to negative.$