Question 1

Evaluate the limit   using L'Hôpital's Rule, if necessary.

Accepted Answer

A)    
B)    
C)  1 
D)  0 
E)  Limit does not exist. 
A)    
B)    
C)  1 
D)  0 
E)  Limit does not exist. D

Question 2

The graph below gives membership in an organization during its first year. Estimate the instantaneous rate of change in membership in September.

Accepted Answer

A)  48 members per month 
B)  12 members per month 
C)  89 members per month 
D)  96 members per month 
E)  32 members per month 
A)  48 members per month 
B)  12 members per month 
C)  89 members per month 
D)  96 members per month 
E)  32 members per month A

Question 3

Suppose the managers of a dairy company have modeled weekly production costs as   dollars for u units of dairy products. Weekly shipping cost for u units is given by   dollars
Calculate the total cost to produce and ship 4000 units in 1 week. Round to the nearest cent.

Accepted Answer

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

Question 4

For the function in this problem, find the derivative, by using the definition.

Accepted Answer

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

Question 5

Find the derivative of the function.

Accepted Answer

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

Question 6

Find the derivative of   .

Accepted Answer

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

Question 7

The GDP of a certain country is $617 billion and is increasing by $22 thousand per person. The population of that country is 76 million and is increasing by 0.9 million people per year. How quickly is the GDP increasing?

Accepted Answer

A)  19.8 billion per year 
B)  25.2 billion per year 
C)  68.4 billion per year 
D)  138.8 billion per year 
E)  555.3 billion per year 
A)  19.8 billion per year 
B)  25.2 billion per year 
C)  68.4 billion per year 
D)  138.8 billion per year 
E)  555.3 billion per year

Question 8

Find the derivative of   .

Accepted Answer

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

Question 9

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Accepted Answer

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

Question 10

Find the derivative of   .

Accepted Answer

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

Question 11

Find the derivative of   .

Accepted Answer

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

Question 12

For the function given, find

Accepted Answer

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

Question 13

The future value that accrues when $900 is invested at 7%, compounded continuously, is   , where t is the number of years. At what rate is the money in this account growing when

Accepted Answer

A)  $18.12 per year 
B)  $69.63 per year 
C)  $1812.38 per year 
D)  $965.26 per year 
E)  $126.87 per year 
A)  $18.12 per year 
B)  $69.63 per year 
C)  $1812.38 per year 
D)  $965.26 per year 
E)  $126.87 per year

Question 14

Differentiate the given function.

Accepted Answer

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

Question 15

When t million dollars is invested in technology for a manufacturing plant, the plant needs w(t) workers to maximize production. Labor costs are L(w) million dollars when w workers are employed. When $5 million is invested in technology, 2400 workers are needed to maximize production, and labor needs are increasing by 200 workers per million dollars. It costs $32 million to employ 2400 workers. At 2400 workers, labor costs are increasing by approximately $0.024 per worker. Evaluate w(t) when $5 million is invested in technology, and write a sentence interpreting the value.

Accepted Answer

A)    workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2400 workers to minimize production. 
B)    workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2200 workers to maximize production. 
C)    workers When $5 billion is invested in technology for a manufacturing plant, labor needs are increasing by 200 workers per million dollars. 
D)    workers When $5 billion is invested in technology for a manufacturing plant, labor needs are decreasing by 400 workers per million dollars. 
E)    workers When $5 billion is invested in technology for a manufacturing plant, the plant needs 2600 workers to maximize production. 
A)    workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2400 workers to minimize production. 
B)    workers When $5 million is invested in technology for a manufacturing plant, the plant needs 2200 workers to maximize production. 
C)    workers When $5 billion is invested in technology for a manufacturing plant, labor needs are increasing by 200 workers per million dollars. 
D)    workers When $5 billion is invested in technology for a manufacturing plant, labor needs are decreasing by 400 workers per million dollars. 
E)    workers When $5 billion is invested in technology for a manufacturing plant, the plant needs 2600 workers to maximize production.

Question 16

The profit from the supply of a certain commodity is modeled as   thousand dollars where q is the number of million units produced. How rapidly are profit and average profit changing when 10 million units are produced?

Accepted Answer

A)  $4 thousand per million units   per million units 
B)  $60 thousand per million units   per million units 
C)  $40 thousand per million units   per million units 
D)  $40 thousand per million units   per million units 
E)  $6 thousand per million units   per million units 
A)  $4 thousand per million units   per million units 
B)  $60 thousand per million units   per million units 
C)  $40 thousand per million units   per million units 
D)  $40 thousand per million units   per million units 
E)  $6 thousand per million units   per million units

Question 17

The population (in millions) of the United States between 1970 and 2010 can be modeled as   million people where x is the number of decades after 1970.
The percentage of people in the United States who live in the Midwest between 1970 and 2010 can be modeled as   percent
Where x is the number of decades since 1970.
Write an expression for the number of people who live in the Midwest x decades after 1970.

Accepted Answer

A)    million people 
B)    million people 
C)    million people 
D)    million people 
E)    million people 
A)    million people 
B)    million people 
C)    million people 
D)    million people 
E)    million people

Question 18

Differentiate the given function.

Accepted Answer

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

Question 19

Use L'Hôpital's Rule to find the limit.

Accepted Answer

A)    
B)    
C)    
D)    
E)  Limit does not exist. 
A)    
B)    
C)    
D)    
E)  Limit does not exist.

Question 20

The revenue from the sale of x units of a commodity is r(x) Canadian dollars, and u(r) U.S. dollars is the equivalent value of r Canadian dollars. On September 8, 2009, $1 Canadian was worth $1.0764 U.S., and the rate of change of the U.S. dollar value was $0.925 U.S. per Canadian dollar. On the same day, sales were 470 units, producing revenue of $10,000 Canadian, and revenue was increasing by $4.2 Canadian per unit. Evaluate   of the given expression on September 8, 2009, and write a sentence interpreting the value.

Accepted Answer

A)    Canadian dollars per unit On September 8, 2009, revenue was increasing by 4.2 Canadian dollars per unit. 
B)    Canadian dollars per unit On September 8, 2009, revenue was increasing by 470 Canadian dollars per unit. 
C)    Canadian dollars per unit On September 8, 2009, revenue was decreasing by 8.4 Canadian dollars per unit. 
D)    Canadian dollars per unit On September 8, 2009, revenue was decreasing by 4.2 Canadian dollars per unit. 
E)    Canadian dollars per unit On September 8, 2009, revenue was increasing by 8.4 Canadian dollars per unit. 
A)    Canadian dollars per unit On September 8, 2009, revenue was increasing by 4.2 Canadian dollars per unit. 
B)    Canadian dollars per unit On September 8, 2009, revenue was increasing by 470 Canadian dollars per unit. 
C)    Canadian dollars per unit On September 8, 2009, revenue was decreasing by 8.4 Canadian dollars per unit. 
D)    Canadian dollars per unit On September 8, 2009, revenue was decreasing by 4.2 Canadian dollars per unit. 
E)    Canadian dollars per unit On September 8, 2009, revenue was increasing by 8.4 Canadian dollars per unit.

Evaluate the limit using L'Hôpital's Rule, if necessary.

The graph below gives membership in an organization during its first year. Estimate the instantaneous rate of change in membership in September.

Suppose the managers of a dairy company have modeled weekly production costs as dollars for u units of dairy products. Weekly shipping cost for u units is given by dollars Calculate the total cost to produce and ship 4000 units in 1 week. Round to the nearest cent.

For the function in this problem, find the derivative, by using the definition.

Find the derivative of the function.

Find the derivative of .

The GDP of a certain country is $617 billion and is increasing by $22 thousand per person. The population of that country is 76 million and is increasing by 0.9 million people per year. How quickly is the GDP increasing?

Find the derivative of .

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Find the derivative of .

Find the derivative of .

For the function given, find

The future value that accrues when $900 is invested at 7%, compounded continuously, is , where t is the number of years. At what rate is the money in this account growing when

Differentiate the given function.

The profit from the supply of a certain commodity is modeled as thousand dollars where q is the number of million units produced. How rapidly are profit and average profit changing when 10 million units are produced?

Differentiate the given function.

Use L'Hôpital's Rule to find the limit.

Ingredients of Change: Functions an Limits

Describing Change: Rates

Analyzing Change: Applications of Derivatives

Accumulating Change: Limits of Sums and the Definite Integral

Analyzing Accumulated Change: Integrals in Action

Ingredients of Multivariable Change: Models, Graphs, Rates

Analyzing Multivariable Change: Optimization

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Exam 3: Determining Change: Derivatives

Evaluate the limit using L'Hôpital's Rule, if necessary.

The graph below gives membership in an organization during its first year. Estimate the instantaneous rate of change in membership in September.

Suppose the managers of a dairy company have modeled weekly production costs as dollars for u units of dairy products. Weekly shipping cost for u units is given by dollars Calculate the total cost to produce and ship 4000 units in 1 week. Round to the nearest cent.

For the function in this problem, find the derivative, by using the definition.

Find the derivative of the function.

Find the derivative of .

The GDP of a certain country is $617 billion and is increasing by $22 thousand per person. The population of that country is 76 million and is increasing by 0.9 million people per year. How quickly is the GDP increasing?

Find the derivative of .

For the given pair of functions, write the composite function and its derivative in terms of one input variable.

Find the derivative of .

Find the derivative of .

For the function given, find

The future value that accrues when $900 is invested at 7%, compounded continuously, is , where t is the number of years. At what rate is the money in this account growing when

Differentiate the given function.

The profit from the supply of a certain commodity is modeled as thousand dollars where q is the number of million units produced. How rapidly are profit and average profit changing when 10 million units are produced?

Differentiate the given function.

Use L'Hôpital's Rule to find the limit.

Ingredients of Change: Functions an Limits

Describing Change: Rates

Analyzing Change: Applications of Derivatives

Accumulating Change: Limits of Sums and the Definite Integral

Analyzing Accumulated Change: Integrals in Action

Ingredients of Multivariable Change: Models, Graphs, Rates

Analyzing Multivariable Change: Optimization

Filters