Question 1

​Use the graph of   to identify at which of the indicated points the derivative   changes from positive to negative. ​
​   ​

Accepted Answer

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

Question 2

​Find the x-values of all relative maxima of the given function. ​
​   ​

Accepted Answer

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

Question 3

​If the total revenue function for a blender is   ,find the maximum revenue. ​

Accepted Answer

A)  ​$100 
B)  ​$2,500 
C)  ​$40 
D)  ​$250 
E)  ​$2,000 
A)  ​$100 
B)  ​$2,500 
C)  ​$40 
D)  ​$250 
E)  ​$2,000

Question 4

A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint:   where k > 0 is the proportionality constant.]   ​

Accepted Answer

A)    inches and   inches 
B)    inches and   inches 
C)    inches and   inches 
D)    inches and   inches 
E)    inches and   inches 
A)    inches and   inches 
B)    inches and   inches 
C)    inches and   inches 
D)    inches and   inches 
E)    inches and   inches

Question 5

​Find the limit: ​
​   ​

Accepted Answer

A)  ​-∞ 
B)  ​∞ 
C)  ​0 
D)  ​-1 
E)  ​1 
A)  ​-∞ 
B)  ​∞ 
C)  ​0 
D)  ​-1 
E)  ​1

Question 6

​Use analytic methods to find the limit as   for the given function. ​
​   ​

Accepted Answer

A)  ​2500 
B)  ​2425 
C)  ​-2425 
D)  ​-2500 
E)  ​does not exist 
A)  ​2500 
B)  ​2425 
C)  ​-2425 
D)  ​-2500 
E)  ​does not exist

Question 7

​Find the limit. ​
​   ​

Accepted Answer

A)  ​0 
B)  ​   
C)  ​   
D)  ​∞ 
E)  ​-∞ 
A)  ​0 
B)  ​   
C)  ​   
D)  ​∞ 
E)  ​-∞

Question 8

​Find the points of inflection and discuss the concavity of the function. ​​
​   ​

Accepted Answer

A)  ​inflection point at   ; concave downward on   ; concave upward on   
B)  ​inflection point at   ; concave upward on   ; concave downward on   
C)  ​​inflection point at   ; concave downward on   ; concave upward on   
D)  ​inflection point at   ; concave upward on   ; concave downward on   
E)  ​none of the above 
A)  ​inflection point at   ; concave downward on   ; concave upward on   
B)  ​inflection point at   ; concave upward on   ; concave downward on   
C)  ​​inflection point at   ; concave downward on   ; concave upward on   
D)  ​inflection point at   ; concave upward on   ; concave downward on   
E)  ​none of the above

Question 9

A business has a cost (in dollars)of   for producing x units.What is the limit of   as x approaches infinity? ​

Accepted Answer

A)  ​∞ 
B)  ​$0.40 
C)  ​$900.40 
D)  ​$900.00 
E)  ​$899.60 
A)  ​∞ 
B)  ​$0.40 
C)  ​$900.40 
D)  ​$900.00 
E)  ​$899.60

Question 10

​You are in a boat 2 miles from the nearest point on the coast.You are to go to point Q located 3 miles down the coast and 1 mile inland (see figure).You can row at a rate of 1 miles per hour and you can walk at a rate of 2 miles per hour.Toward what point on the coast should you row in order to reach point Q in the least time? ​   ​

Accepted Answer

A)  ​3 miles 
B)  ​8 miles 
C)  ​2 miles 
D)  ​1 mile 
E)  ​5 miles 
A)  ​3 miles 
B)  ​8 miles 
C)  ​2 miles 
D)  ​1 mile 
E)  ​5 miles

Question 11

​Find the speed v,in miles per hour,that will minimize costs on a 125-mile delivery trip.The cost per hour for fuel is   dollars,and the driver is paid   dollars per hour.(Assume there are no costs other than wages and fuel.) ​

Accepted Answer

A)  ​75 mph 
B)  ​125 mph 
C)  ​200 mph 
D)  80​ mph 
E)  ​40 mph 
A)  ​75 mph 
B)  ​125 mph 
C)  ​200 mph 
D)  80​ mph 
E)  ​40 mph

Question 12

​A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist,where   Confirm your results analytically.​ ​
​   ​
​

Accepted Answer

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

Question 13

​Find the limit. ​
​   ​

Accepted Answer

A)  ​   
B)  ​12 
C)  ​-12 
D)  ​   
E)  ​   
A)  ​   
B)  ​12 
C)  ​-12 
D)  ​   
E)  ​

Question 14

​Use the graph f '' to sketch the graph of ​f​. ​   ​

Accepted Answer

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

Question 15

​If the total cost function for a product is   dollars,determine how many units x should be produced to minimize the average cost per unit? ​

Accepted Answer

A)  ​450 units 
B)  ​400 units 
C)  ​141 units 
D)  ​200 units 
E)  ​97 units 
A)  ​450 units 
B)  ​400 units 
C)  ​141 units 
D)  ​200 units 
E)  ​97 units

Question 16

​Analyze and sketch a graph of the function   ​

Accepted Answer

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

Question 17

​Analyze and sketch a graph of the function   ​

Accepted Answer

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

Question 18

​Sketch the graph of the function   using any extrema,intercepts,symmetry,and asymptotes. ​

Accepted Answer

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

Question 19

​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 20

​Find any vertical asymptotes for the given function. ​
​   ​

Accepted Answer

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

Use the graph of to identify at which of the indicated points the derivative changes from positive to negative.

Find the x-values of all relative maxima of the given function.

If the total revenue function for a blender is ,find the maximum revenue.

Find the limit:

Use analytic methods to find the limit as for the given function.

Find the limit.

Find the points of inflection and discuss the concavity of the function.

A business has a cost (in dollars)of for producing x units.What is the limit of as x approaches infinity?

Find the speed v,in miles per hour,that will minimize costs on a 125-mile delivery trip.The cost per hour for fuel is dollars,and the driver is paid dollars per hour.(Assume there are no costs other than wages and fuel.)

A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist,where Confirm your results analytically.

Find the limit.

Use the graph f '' to sketch the graph of f.

If the total cost function for a product is dollars,determine how many units x should be produced to minimize the average cost per unit?

Analyze and sketch a graph of the function

Analyze and sketch a graph of the function

Sketch the graph of the function using any extrema,intercepts,symmetry,and asymptotes.

Find any horizontal asymptotes for the given function.

Find any vertical asymptotes for the given function.

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

​Use the graph of to identify at which of the indicated points the derivative changes from positive to negative. ​ ​ ​

​Find the x-values of all relative maxima of the given function. ​ ​ ​

​If the total revenue function for a blender is ,find the maximum revenue. ​

​Find the limit: ​ ​ ​

​Use analytic methods to find the limit as for the given function. ​ ​ ​

​Find the limit. ​ ​ ​

​Find the points of inflection and discuss the concavity of the function. ​​ ​ ​

A business has a cost (in dollars)of for producing x units.What is the limit of as x approaches infinity? ​

​Find the speed v,in miles per hour,that will minimize costs on a 125-mile delivery trip.The cost per hour for fuel is dollars,and the driver is paid dollars per hour.(Assume there are no costs other than wages and fuel.) ​

​A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist,where Confirm your results analytically.​ ​ ​ ​ ​

​Find the limit. ​ ​ ​

​Use the graph f '' to sketch the graph of ​f​. ​ ​

​If the total cost function for a product is dollars,determine how many units x should be produced to minimize the average cost per unit? ​

​Analyze and sketch a graph of the function ​

​Analyze and sketch a graph of the function ​

​Sketch the graph of the function using any extrema,intercepts,symmetry,and asymptotes. ​

​Find any horizontal asymptotes for the given function. ​ ​ ​

​Find any vertical asymptotes for the given function. ​ ​ ​

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

Use the graph of to identify at which of the indicated points the derivative changes from positive to negative.

Find the x-values of all relative maxima of the given function.

If the total revenue function for a blender is ,find the maximum revenue.

Find the limit:

Use analytic methods to find the limit as for the given function.

Find the limit.

Find the points of inflection and discuss the concavity of the function.

A business has a cost (in dollars)of for producing x units.What is the limit of as x approaches infinity?

Find the speed v,in miles per hour,that will minimize costs on a 125-mile delivery trip.The cost per hour for fuel is dollars,and the driver is paid dollars per hour.(Assume there are no costs other than wages and fuel.)

A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist,where Confirm your results analytically.

Find the limit.

Use the graph f '' to sketch the graph of f.

If the total cost function for a product is dollars,determine how many units x should be produced to minimize the average cost per unit?

Analyze and sketch a graph of the function

Analyze and sketch a graph of the function

Sketch the graph of the function using any extrema,intercepts,symmetry,and asymptotes.

Find any horizontal asymptotes for the given function.

Find any vertical asymptotes for the given function.