Deck 2: Describing Change: Rates

Question 1

Calculate the average rate of change in disposable personal income between 1995 and 2000. Round your answer to the nearest dollar.

A) 2390 dollars per person per year
B) 1195 dollars per person per year
C) 3348 dollars per person per year
D) 5150 dollars per person per year
E) 1030 dollars per person per year

Answer 1

2390 dollars per person per year

Question 2

The percentage of students meeting national mathematics benchmarks on the ACT increased from 39% in 2001 to 42% in 2010. Calculate the average rate of change. Round your answer to two decimal places.

A) 0.23 percentage points per year
B) 0.33 percentage points per year
C) 0.99 percentage points per year
D) 0.08 percentage points per year
E) 3.20 percentage points per year

Answer 2

0.33 percentage points per year

Question 3

A graph of a model for the sales of services between 2004 and 2008 by Kelly Services, Inc., a leading global provider of staffing services, is shown below. Use the graph to calculate the average rate of change in Kelly's sales of services between 2005 and 2007.

A) 241 million dollars per year
B) 274 million dollars per year
C) 5500 million dollars per year
D) 550 million dollars per year
E) 5 million dollars per year

Answer 3

241 million dollars per year

Question 4

Given the statement: In five trading days, the stock price rose $2.90.
Choose the sentence that best expresses how rapidly, on average, the quantity has changed over the given interval.

A) The stock price increased by an average rate of $1.72 per day.
B) The stock price increased by an average rate of $0.58 per day.
C) The stock price increased by an average rate of $1.45 per day.
D) The stock price increased by an average rate of $2.90 per day.
E) The stock price increased by an average rate of $5.00 per day.

Question 5

The function p gives the number of meters from an airport that a plane has flown after t minutes. What are the units on .

A) kilometers per minute
B) meters per hour
C) meters per minute
D) kilometers per hour
E) minutes per meter

Question 6

The function B gives the balance, in dollars, in a mutual fund t years after the initial investment. Assume that no deposits or withdrawals are made during the investment period. What is the financial interpretation of .

A) interest rate
B) interest value
C) fixed rate
D) floating rate
E) simple interest

Question 7

The function w gives the number of words per minute (wpm) that a student in a keyboard class can type after t weeks in the course. Is it possible for to be negative? Explain.

A) can be negative if the number of words typed per minute is decreasing at week 5.
B) can be negative if the number of words typed per minute is increasing at week 5.
C) can be negative if the number of words typed per minute is decreasing at day 5.
D) can be negative if the number of words typed per minute is increasing at day 5
E) cannot be negative because the lowest number of words typed in a minute is 5 .

Question 8

The function D gives the time, in years, that it takes for an investment to double if interest is continuously compounded at r%. Write a sentence of interpretation for

A) If the interest rate for an investment at 4% compounded continuously is changed to 5% compounded continuously, the doubling time will decrease by approximately 2.74 years.
B) If the interest rate for an investment at 4% compounded continuously the doubling time will decrease by approximately 2.74 years.
C) If the interest rate for an investment at 2.74% compounded continuously the doubling time will decrease by approximately 4 years.
D) If the interest rate for an investment at 4% compounded continuously is changed to 5% compounded continuously, the doubling time will increase by approximately 2.74 years.
E) If the interest rate for an investment at 2.74% compounded continuously is changed to 3.74% compounded continuously, the doubling time will increase by approximately 4 years.

Question 9

The function D gives the time, in years, that it takes for an investment to double if interest is continuously compounded at r%. Write a sentence of interpretation for

A) At 8% interest compounded continuously, an investment will double its value in 7.2 years.
B) At 8% interest compounded continuously, an investment will triple its value in 7.2 years.
C) At 8% interest the investment will be same after 7.2 years.
D) At 7.2% interest compounded continuously, an investment will double its value in 8 years.
E) At 7.2% interest compounded continuously, an investment will triple its value in 8 years.

Question 10

The function g gives the fuel efficiency, in miles per gallon, of a car traveling v miles per hour. Write a sentence of interpretation for the following statement. .

A) The fuel efficiency of a car traveling 35 is increasing by 0.25 mpg per mph.
The fuel efficiency is neither increasing nor decreasing for a car traveling 52 mph.
B) The fuel efficiency of a car traveling 52 is increasing by 0.25 mpg per mph.
The fuel efficiency is neither increasing nor decreasing for a car traveling 35 mph.
C) The fuel efficiency of a car traveling 35 is increasing by 0.25 mpg per mph.
The fuel efficiency is increasing for a car traveling 52 mph.
D) The fuel efficiency of a car traveling 35 is increasing by 0.25 mpg per mph.
The fuel efficiency is decreasing for a car traveling 52 mph.
E) The fuel efficiency of a car traveling 35 is increasing by 0.25 mpg per mph.
The fuel efficiency is increasing or decreasing for a car traveling 35 mph.

Question 11

Calculate the average rate of change of the function over the given interval. Round your answer to two decimal places.

A) -4.14
B) -5.00
C) 3.89
D) 4.14
E) 5.00

Question 12

The function P gives the profit in dollars that a fraternity makes selling x T-shirts. Is it possible for to be negative? Explain.

A) can be negative if the cost of the 80 shirts is greater than the revenue from the sales of 80 shirts.
B) can be negative if the cost of the 80 shirts is less than the revenue from the sales of 80 shirts.
C) can be negative if the cost of the 80 shirts is greater than or equal to the revenue from the sales of 80 shirts.
D) cannot be negative as the cost of the 80 shirts is less than the revenue from the sales of 80 shirts.
E) cannot be negative as the cost of the 80 shirts is greater than the revenue from the sales of 80 shirts.

Question 13

A graph of a model for the sales of services between 2004 and 2008 by Kelly Services, Inc., a leading global provider of staffing services, is shown below. Calculate the percentage change in Kelly's sales between 2006 and 2007.

A) increased by 2.823%.
B) decreased by 2.823%.
C) increased by 2.746%.
D) decreased by 0.028%.
E) increased by 0.028%.

Question 14

A graph of a model for the sales of services between 2004 and 2008 by Kelly Services, Inc., a leading global provider of staffing services, is shown below. Calculate the percentage change in Kelly's sales between 2004 and 2007.

A) $822 million
B) $4860 million
C) $411 million
D) $10542 million
E) $5271 million

Question 15

The population in a certain country was 364 thousand in 1930 and 3.5 million in 2005. Calculate the percentage change. Round your answer to one decimal place.

A) 86.2%
B) 861.5%
C) 313.6%
D) 11.6%
E) 1.2%

Question 16

The function f gives the weekly profit, in thousand dollars, that an airline makes on its flights from Boston to Washington D.C. when the ticket price is p dollars. Choose the sentence that best expresses .

A) When the ticket price is $100, the weekly profit to the airline on flights from Boston to Washington is increasing by $3 thousand per dollar (of ticket price).
B) When the ticket price is $3, the weekly profit to the airline on flights from Boston to Washington is decreasing by $100 thousand per dollar (of ticket price).
C) When the ticket price is $3, the weekly profit to the airline on flights from Boston to Washington is increasing by $100 thousand per dollar (of ticket price).
D) When the ticket price is $100, the weekly profit to the airline on flights from Boston to Washington is decreasing by $3 thousand per dollar (of ticket price).
E) When the ticket price is $100, the weekly profit to the airline on flights from Boston to Washington is decreasing by $100 thousand per dollar (of ticket price).

Question 17

Use the figure to answer the questions. Is the graph steeper at point C or at point E?

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

Question 18

An Airline industry posted a profit of $17.8 million at the end of 2009 compared with a loss of $121.7 million in 2008. Calculate change.

A) $139.5 million
B) $121.7 million
C) $103.9 million
D) $69.8 million

Question 19

Identify which points have lines drawn through them that are not tangent to the graph.

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

Question 20

Given the statement: The company lost $15,000 during the past 5 months.
Choose the sentence that best expresses how rapidly, on average, the quantity has changed over the given interval.

A) The company lost on average $15,000 per month.
B) The company lost on average $75,000 per month.
C) The company lost on average $7,500 per month.
D) The company lost on average $8,000 per month.
E) The company lost on average $3,000 per month.

Question 21

Determine whether the function is continuous or differentiable for given input values.

A) The function is continuous on the input interval shown, but it is not differentiable at x = 4.
B) The function is continuous on the input interval shown, but it is not differentiable at x = 6.
C) The function is not continuous on the input interval shown, but it is not differentiable at x = 6.
D) The function is not continuous on the input interval shown, but it is not differentiable at x = 4.
E) The function is both continuous and differentiable at x = 4.

Question 22

Using the following table, estimate

A) 127 newspapers per year
B) 499 newspapers per year
C) 542 newspapers per year
D) 335 newspapers per year
E) 64 newspapers per year

Question 23

Find the slope of the tangent to the graph of f (x) at any point. estimate to the nearest tenth

A) 12.6
B) 22.2
C) 12.1
D) 10.1
E) 11.1

Question 24

The future value of a certain savings account with no activity besides compounding of interest is modeled as dollars where t is the number of years since $1400 was invested.
Numerically estimate to the nearest cent the rate of change of the future value when .

A) $89.82 per year
B) $188.08 per year
C) $173.62 per year
D) $18.65 per year
E) $451.40 per year

Question 25

The figure shows the terminal speed, in meters per second, of a raindrop as a function of the size of the drop measured in terms of its diameter. Estimate the slope of secant line connecting the points for diameters of 1 mm and 5 mm.

A) 1.275 m/s per mm
B) 2.217 m/s per mm
C) 3.325 m/s per mm
D) 5 m/s per mm
E) 4 m/s per mm

Question 26

When advertising an existing commodity, companies are interested in changes in revenue totals at various levels of advertising. The figure shows the advertising threshold effect for a certain commodity. Determine the input value where the line tangent to the graph is not defined.

A)
B)
C)
D)
E)

Question 27

Find the slope of the tangent to the graph of f (x) at any point.

A)
B)
C)
D)
E)

Question 28

At the indicated x-value, find the slope of the tangent line.

A) 75
B) 245
C) 26
D) 33
E) 47

Question 29

Find the slope of the tangent at

A) 15
B) 21
C) 33
D) 42
E) 0

Question 30

The figure shows the terminal speed, in meters per second, of a raindrop as a function of the size of the drop measured in terms of its diameter. Estimate the derivative of the speed for a diameter of 2 mm.

A) 1.7
B) 3.7
C) 0.3
D) 3.2
E) 4.7

Question 31

For a certain brand of bicycle, Canadian dollars gives the profit from the sale of x mountain bikes. On June 27, 2009, P Canadian dollars were worth American dollars. Assume that this conversion applies today. Calculate the profit in Canadian and in American dollars from the sale of 600 mountain bikes. Round your answer to two decimal places.

A) $ 256.27 Canadian
$ 256.27 U. S.
B) $ 391.58 Canadian
$ 391.58 U. S.
C) $ 391.58 Canadian
$ 256.27 U. S.
D) $ 256.27 Canadian
$ 391.58 U. S.
E) $ 1.01 Canadian
$ 0)00 U. S.

Question 32

A chemical reaction begins when a certain mixture of chemicals reaches 95°C. The reaction activity is measured in units (U) per 100 microliters (100µL) of the mixture. Measurements during the first 18 minutes after the mixture reaches 95°C are listed in Table below. Chemical Reaction The Logistic model for the above data is approximately given as .
Estimate the average rate of change of the reaction activity between 5 minutes and 15 minutes. Round your answer to three decimal places.

A) 0.141
B) 0.071
C) 0.227
D) 0.113
E) 7.074

Question 33

Find the derivative of the given function using the algebraic method.

A)
B)
C)
D)
E)

Question 34

For the function given, find

A)
B)
C)
D)
E)

Question 35

If find .

A)
B)
C)
D)
E)

Question 36

Find the derivative of the given function using the algebraic method.

A)
B)
C)
D)
E)

Question 37

The relation u gives the number of people unemployed in a country t months after the election of a new president. Choose the sentence that best expresses u(t) describing the unemployment situation.

A) Three years after the election, the number of people unemployed was decreasing by 800,000 per month.
B) Three years after the election, the number of people unemployed was 800,000 per month.
C) Three years after the election, the number of people unemployed was increasing by 36 per month.
D) Three years after the election, the number of people unemployed was decreasing by 36 per month.
E) Three years after the election, the number of people unemployed was increasing by 800,000 per month.

Question 38

For a certain brand of bicycle, Canadian dollars gives the profit from the sale of x mountain bikes. On June 27, 2009, P Canadian dollars were worth American dollars. Assume that this conversion applies today. Write a function for profit in American dollars from the sale of x mountain bikes.

A)
B)
C)
D)
E)

Question 39

The future value of a certain savings account with no activity besides compounding of interest is modeled as dollars where t is the number of years since $1600 was invested.
Calculate the percentage rate of change of the future value when .

A) 3.97% per year
B) 90.80% per year
C) 9.08% per year
D) 25.41% per year
E) 22.87% per year

Question 40

Suppose the function G(t) represents a test grade (out of 100 points) as a function of hours studied. If points, approximate, to one decimal place, the slope of the tangent line when .

A) 87.2 points per hour
B) 3.8 points per hour
C) 21.9 points per hour
D) -0.5 points per hour
E) -7.1 points per hour

Question 41

By imagining tangent lines at points state whether the slopes are positive, zero, or negative at these points.

A)
B)
C)
D)
E)

Question 42

An object is dropped off a building. Ignoring air resistance we know from physics that its height above ground t seconds after being dropped is given by Height feet. Determine the object's velocity after 2.5 seconds.

A) -80 feet per second
B) -50 feet per second
C) -20 feet per second
D) -16 feet per second
E) -32 feet per second

Question 43

Clinton County, Michigan, is mostly flat farmland partitioned by straight roads (often gravel) that run either north/south or east/west. A tractor driven north on Lowell Road from the Schafers farm's mailbox is miles north of Howe Road t minutes after leaving the farm's mailbox. How far is the Schafers' mailbox from Howe Road.

A) 0.4 miles
B) 0.28 miles
C) 0.68 miles
D) 0.12 miles
E) 2.8 miles

Question 44

The amount of airline fuel consumed by an Airline company each year between 2001 and 2009 can be modeled as billion gallons where t is the number of years since 2001. Calculate the amount of fuel consumed in 2009.

A) 2.999 billion gallons
B) 1.541 billion gallons
C) 2.189 billion gallons
D) 10.829 billion gallons
E) 2.351 billion gallons

Question 45

If find .

A)
B)
C)
D)
E)

Question 46

Sketch a slope graph of the function.

A)
B)
C)
D)
E)

Question 47

The figure shows cattle prices (for choice 600-pound steer calves) from October 1994 through May 1995. Identify the input value where the derivative fail to exist.

A) m = 4
B) m = 3
C) m = 0
D) m = 7
E) m = 2