Question 1

​Analyze and sketch a graph of the function   ​

Accepted Answer

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

Question 2

​Average costs.Suppose the average costs of a mining operation depend on the number of machines used,and average costs,in dollars,are given by   ,   ,where x is the number of machines used.How many machines give minimum average costs? ​

Accepted Answer

A)  ​Using 14 machines gives the minimum average costs. 
B)  ​Using zero machines gives the minimum average costs. 
C)  ​Using 24 machines gives the minimum average costs. 
D)  ​Using 28 machines gives the minimum average costs. 
E)  ​Using 33 machines gives the minimum average costs. 
A)  ​Using 14 machines gives the minimum average costs. 
B)  ​Using zero machines gives the minimum average costs. 
C)  ​Using 24 machines gives the minimum average costs. 
D)  ​Using 28 machines gives the minimum average costs. 
E)  ​Using 33 machines gives the minimum average costs.

Question 3

​This problem contains a function and its graph,where   Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically. ​
​   ​
​   ​

Accepted Answer

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

Question 4

​For the given function,find all critical numbers. ​
​   ​

Accepted Answer

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

Question 5

​Locate the absolute extrema of the function   on the closed interval   ​

Accepted Answer

A)  ​no absolute max; absolute min:   
B)  ​absolute max:   ; absolute min:   
C)  ​absolute max:   ; no absolute min 
D)  ​absolute max:   ;   
E)  ​no absolute max or min 
A)  ​no absolute max; absolute min:   
B)  ​absolute max:   ; absolute min:   
C)  ​absolute max:   ; no absolute min 
D)  ​absolute max:   ;   
E)  ​no absolute max or min

Question 6

Find any horizontal asymptotes for the given function. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  no horizontal asymptotes 
A)    
B)    
C)    
D)    
E)  no horizontal asymptotes

Question 7

​The revenue R for a company selling x units is   Use differentials to approximate the change in revenue if sales increase from   to   units. ​

Accepted Answer

A)  ​48,000 dollars 
B)  ​50,000 dollars 
C)  ​45,000 dollars 
D)  ​53,000 dollars 
E)  ​60000 dollars 
A)  ​48,000 dollars 
B)  ​50,000 dollars 
C)  ​45,000 dollars 
D)  ​53,000 dollars 
E)  ​60000 dollars

Question 8

​Determine the open intervals on which the graph of   is concave downward or concave upward.

Accepted Answer

A)  ​concave upward on   ; concave downward on   
B)  ​concave downward on   
C)  ​concave upward on   
D)  ​concave downward on   ; ​concave upward   
E)  ​​concave upward on   ; concave downward on   
A)  ​concave upward on   ; concave downward on   
B)  ​concave downward on   
C)  ​concave upward on   
D)  ​concave downward on   ; ​concave upward   
E)  ​​concave upward on   ; concave downward on

Question 9

For the function   ​ (a)Find the critical numbers of f (if any);
(b)Find the open intervals where the function is increasing or decreasing; and
(c)Apply the First Derivative Test to identify all relative extrema.
​

Accepted Answer

A)  (a)  
(b) increasing:   ; decreasing:  
(c) relative max:   ; relative min:  
​ 
B)  (a)  
(b) decreasing:   ; increasing:  
(c) relative min:   ; relative max:  
​ 
C)  (a)  
(b) increasing:   ; decreasing:  
(c) relative max:   ; relative min:  
​ 
D)  (a)  
(b) decreasing:   ; increasing:  
(c) relative min:   ; relative max:  
​ 
E)  (a)  
(b)increasing:   ; decreasing:  
(c)relative max:   ; relative min: no relative min.
​ 
A)  (a)  
(b) increasing:   ; decreasing:  
(c) relative max:   ; relative min:  
​ 
B)  (a)  
(b) decreasing:   ; increasing:  
(c) relative min:   ; relative max:  
​ 
C)  (a)  
(b) increasing:   ; decreasing:  
(c) relative max:   ; relative min:  
​ 
D)  (a)  
(b) decreasing:   ; increasing:  
(c) relative min:   ; relative max:  
​ 
E)  (a)  
(b)increasing:   ; decreasing:  
(c)relative max:   ; relative min: no relative min.
​

Question 10

​Find all relative minima of the given function. ​
​   ​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​no relative maxima 
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​no relative maxima

Question 11

Graph a function on the interval   having the following characteristics. Absolute maximum at   Absolute minimum at   Relative minimum at   Relative maximum at   ​

Accepted Answer

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

Question 12

​The measurement of the edge of a cube is found to be 18 inches,with a possible error of 0.7 inch.Use differentials to estimate the propagated error in computing (a)the volume of the cube and (b)the surface area of the cube.Give your answers to two decimal places. ​

Accepted Answer

A)  ​816.48, 151.20 
B)  ​544.32, 136.08 
C)  ​612.36, 136.08 
D)  ​544.32, 120.96 
E)  ​680.40,151.20 
A)  ​816.48, 151.20 
B)  ​544.32, 136.08 
C)  ​612.36, 136.08 
D)  ​544.32, 120.96 
E)  ​680.40,151.20

Question 13

​Sketch the graph of the equation given below.Use intercepts,extrema,and asymptotes as sketching aids. ​
​   ​

Accepted Answer

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

Question 14

Match the function   with one of the following graphs. ​

Accepted Answer

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

Question 15

​A function and its graph are given.Use the second derivative to locate all x-values of points of inflection on the graph of   Check these results against the graph shown. ​
​   ​   ​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   ,   
E)  ​   ,   ,   
A)  ​   
B)  ​   
C)  ​   
D)  ​   ,   
E)  ​   ,   ,

Question 16

​Find the length and width of a rectangle that has perimeter 56 meters and a maximum area. ​​

Accepted Answer

A)  ​7, 21 
B)  ​1, 27 
C)  ​14, 14 
D)  ​15, 13 
E)  ​18,10 
A)  ​7, 21 
B)  ​1, 27 
C)  ​14, 14 
D)  ​15, 13 
E)  ​18,10

Question 17

Suppose the number y of medical degrees conferred in the United States can be modeled by   ,for   ,where t is the time in years,with   corresponding to 1972.Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing. ​

Accepted Answer

A)  The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2005, and decreasing during 1989 to 1997. 
B)  The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2005, and decreasing during 1988 to 1996. 
C)  The number of medical degrees is increasing from 1972 to 1989 and 2001 to 2005, and decreasing during 1989 to 2001. 
D)  The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2005, and decreasing during 1990 to 2001. 
E)  The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2005,and decreasing during 1989 to 1995. 
A)  The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2005, and decreasing during 1989 to 1997. 
B)  The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2005, and decreasing during 1988 to 1996. 
C)  The number of medical degrees is increasing from 1972 to 1989 and 2001 to 2005, and decreasing during 1989 to 2001. 
D)  The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2005, and decreasing during 1990 to 2001. 
E)  The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2005,and decreasing during 1989 to 1995.

Question 18

​Analytically determine the location of any vertical asymptotes. ​   ​
​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​no vertical asymptotes 
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​no vertical asymptotes

Question 19

​Analyze and sketch a graph of the function   ​

Accepted Answer

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

Question 20

​State the signs of   and   on the interval (0,2). ​
​   ​

Accepted Answer

A) f ' 0 C) f ' > 0 f '' 0 f '' > 0 E) f ' < 0 f '' < 0 A) f ' 0 C) f ' > 0 f '' 0 f '' > 0 E) f ' < 0 f '' < 0

Analyze and sketch a graph of the function

Average costs.Suppose the average costs of a mining operation depend on the number of machines used,and average costs,in dollars,are given by , ,where x is the number of machines used.How many machines give minimum average costs?

This problem contains a function and its graph,where Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically.

For the given function,find all critical numbers.

Locate the absolute extrema of the function on the closed interval

Find any horizontal asymptotes for the given function.

The revenue R for a company selling x units is Use differentials to approximate the change in revenue if sales increase from to units.

Determine the open intervals on which the graph of is concave downward or concave upward.

For the function (a)Find the critical numbers of f (if any); (b)Find the open intervals where the function is increasing or decreasing; and (c)Apply the First Derivative Test to identify all relative extrema.

Find all relative minima of the given function.

Graph a function on the interval having the following characteristics. Absolute maximum at Absolute minimum at Relative minimum at Relative maximum at

The measurement of the edge of a cube is found to be 18 inches,with a possible error of 0.7 inch.Use differentials to estimate the propagated error in computing (a)the volume of the cube and (b)the surface area of the cube.Give your answers to two decimal places.

Sketch the graph of the equation given below.Use intercepts,extrema,and asymptotes as sketching aids.

Match the function with one of the following graphs.

A function and its graph are given.Use the second derivative to locate all x-values of points of inflection on the graph of Check these results against the graph shown.

Find the length and width of a rectangle that has perimeter 56 meters and a maximum area.

Analytically determine the location of any vertical asymptotes.

Analyze and sketch a graph of the function

State the signs of and on the interval (0,2).

Functions, graphs, and Limits

Differentiation

Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions

Probability and Calculus

Series and Taylor Polynomials

Differential Equations

Filters

Exam 3: Applications of the Derivative

​Analyze and sketch a graph of the function ​

​Average costs.Suppose the average costs of a mining operation depend on the number of machines used,and average costs,in dollars,are given by , ,where x is the number of machines used.How many machines give minimum average costs? ​

​This problem contains a function and its graph,where Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically. ​ ​ ​ ​ ​

​For the given function,find all critical numbers. ​ ​ ​

​Locate the absolute extrema of the function on the closed interval ​

Find any horizontal asymptotes for the given function. ​ ​

​The revenue R for a company selling x units is Use differentials to approximate the change in revenue if sales increase from to units. ​

​Determine the open intervals on which the graph of is concave downward or concave upward.

For the function ​ (a)Find the critical numbers of f (if any); (b)Find the open intervals where the function is increasing or decreasing; and (c)Apply the First Derivative Test to identify all relative extrema. ​

​Find all relative minima of the given function. ​ ​ ​

Graph a function on the interval having the following characteristics. Absolute maximum at Absolute minimum at Relative minimum at Relative maximum at ​

​The measurement of the edge of a cube is found to be 18 inches,with a possible error of 0.7 inch.Use differentials to estimate the propagated error in computing (a)the volume of the cube and (b)the surface area of the cube.Give your answers to two decimal places. ​

​Sketch the graph of the equation given below.Use intercepts,extrema,and asymptotes as sketching aids. ​ ​ ​

Match the function with one of the following graphs. ​

​A function and its graph are given.Use the second derivative to locate all x-values of points of inflection on the graph of Check these results against the graph shown. ​ ​ ​ ​

​Find the length and width of a rectangle that has perimeter 56 meters and a maximum area. ​​

​Analytically determine the location of any vertical asymptotes. ​ ​ ​

​Analyze and sketch a graph of the function ​

​State the signs of and on the interval (0,2). ​ ​ ​

Functions, graphs, and Limits

Differentiation

Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions

Probability and Calculus

Series and Taylor Polynomials

Differential Equations

Filters

Analyze and sketch a graph of the function

Average costs.Suppose the average costs of a mining operation depend on the number of machines used,and average costs,in dollars,are given by , ,where x is the number of machines used.How many machines give minimum average costs?

This problem contains a function and its graph,where Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically.

For the given function,find all critical numbers.

Locate the absolute extrema of the function on the closed interval

Find any horizontal asymptotes for the given function.

The revenue R for a company selling x units is Use differentials to approximate the change in revenue if sales increase from to units.

Determine the open intervals on which the graph of is concave downward or concave upward.

For the function (a)Find the critical numbers of f (if any); (b)Find the open intervals where the function is increasing or decreasing; and (c)Apply the First Derivative Test to identify all relative extrema.

Find all relative minima of the given function.

Graph a function on the interval having the following characteristics. Absolute maximum at Absolute minimum at Relative minimum at Relative maximum at

The measurement of the edge of a cube is found to be 18 inches,with a possible error of 0.7 inch.Use differentials to estimate the propagated error in computing (a)the volume of the cube and (b)the surface area of the cube.Give your answers to two decimal places.

Sketch the graph of the equation given below.Use intercepts,extrema,and asymptotes as sketching aids.

Match the function with one of the following graphs.

A function and its graph are given.Use the second derivative to locate all x-values of points of inflection on the graph of Check these results against the graph shown.

Find the length and width of a rectangle that has perimeter 56 meters and a maximum area.

Analytically determine the location of any vertical asymptotes.

Analyze and sketch a graph of the function

State the signs of and on the interval (0,2).