Deck 3: Determining Change: Derivatives

Question 1

Find the derivative of

A)
B)
C)
D)
E)

Answer 1

Question 2

Find the derivative of .

A)
B)
C)
D)
E)

Answer 2

Question 3

The population of Aurora, a Nevada ghost town, can be modeled as where output is measured in people and t is the number of years since 1859. Write a model for the rate of change of the population of Aurora.

A) where output is measured in people per year gives the rate of change of the population of Aurora, Nevada t years since 1859.
B) where output is measured in people per year gives the rate of change of the population of Aurora, Nevada t years since 1859.
C) where output is measured in people per year gives the rate of change of the population of Aurora, Nevada t years since 1859.
D) where output is measured in people per year gives the rate of change of the population of Aurora, Nevada t years since 1859.
E) where output is measured in people per year gives the rate of change of the population of Aurora, Nevada t years since 1859.

Answer 3

where output is measured in people per year gives the rate of change of the population of Aurora, Nevada t years since 1859.

Question 4

Find the derivative of .

A)
B)
C)
D)
E)

Question 5

Find the derivative of

A)
B)
C)
D)
E)

Question 6

A publishing company estimates that when a new book by a best-selling American author first hits the market, its sales can be predicted by the equation , where represents the total number (in thousands) of copies of the book sold in the United States by the end of the xth week. The number of copies of the book sold abroad by the end of the xth week can be modeled by thousand copies of the book How many books will be sold by the end of the first year (that is, 52 weeks)?

A) 497,278 books
B) 503,778 books
C) 6,500 books
D) 490,778 books
E) 4,907 books

Question 7

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

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

Question 8

Find of by using the definition of the derivative.

A)
B)
C)
D)
E)

Question 9

Find the derivative of

A)
B)
C)
D)
E)

Question 10

Find the derivative of .

A)
B)
C)
D)
E)

Question 11

Find the derivative of .

A)
B)
C)
D)
E)

Question 12

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

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 13

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 14

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 15

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

A)
B)
C)
D)
E)

Question 16

The managers of a company have modeled some cost data and found that if they produce x storm windows each hour, the cost (in dollars) to produce one window is given by the function , for . The company sells its storm windows for $190 each. Assuming that every window made will be sold, give a formula for the profit made from the sale of one storm window when the company is producing x windows each hour.

A)
B)
C)
D)
E)

Question 17

For the function given, find

A)
B)
C)
D)
E)

Question 18

Calculate the derivative of the function with the appropriate formula.

A)
B)
C)
D)
E)

Question 19

Find the derivative of .

A)
B)
C)
D)
E)

Question 20

Differentiate the given function.

A)
B)
C)
D)
E)

Question 21

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.

A)
B)
C)
D)
E)

Question 22

Evaluate where given , , , .

A) -13.5
B) -10
C) 9
D) 19
E) -19

Question 23

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

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

Question 24

Using the product rule write the rate-of-change function.

A)
B)
C)
D)
E)

Question 25

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

A)
B)
C)
D)
E)

Question 26

Differentiate the given function.

A)
B)
C)
D)
E)

Question 27

Find the derivative of .

A)
B)
C)
D)
E)

Question 28

Using the product rule write the rate-of-change function.

A)
B)
C)
D)
E)

Question 29

Differentiate the given function.

A)
B)
C)
D)
E)

Question 30

Suppose the percentage of children living with their grandparents between 1970 and 2000 can be modeled by the equation percent, t years after 1970. How rapidly on average did the percentage of children living with their grandparents grow between 1974 and 1993?

A) 24.01 % per year
B) 14.06 % per year
C) 16.98 % per year
D) 9.95 % per year
E) 2.92 % per year

Question 31

Find the derivative of .

A)
B)
C)
D)
E)

Question 32

Find the derivative of .

A)
B)
C)
D)
E)

Question 33

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?

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 34

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

A)
B)
C)
D)
E)

Question 35

Differentiate the given function.

A)
B)
C)
D)
E)

Question 36

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.

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 37

For each of the composite functions, identify an inside function and an outside function and write the derivative with respect to x of the composite function.

A)
B)
C)
D)
E)

Question 38

Find the derivative of .

A)
B)
C)
D)
E)

Question 39

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.

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.

Question 40

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.

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

Question 41

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?

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 42

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

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

Question 43

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

A)
B)
C) 0
D)
E)

Question 44

The number of private donations received by nongovernment disaster relief organizations can be modeled as thousand donations where x is the number of hours since a major disaster has struck. At what time is the rate of change of donations zero?
Round to the nearest thousandth.

A) 12.346 hours
B) 0.111 hours
C) 1.111 hours
D) 0.992 hours
E) 11.111 hours

Question 45

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

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

Question 46

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

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

Question 47

Find .

A) -3
B)
C) 0
D) 3
E) inf

Question 48

Determine the following limit.

A) 0
B) Does not exist
C) 10
D) -10
E) -20