A) Find the Rate of Change of the Temperature at One Hour

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MATH 176

Test 2 Review

Computer Part

Direction: You may use a computer or a graphing calculator to answer the following questions. Use separate paper to show all work.

1. The cost of producing x units of a product is given by dollars where x is number of units of a product. Find the marginal cost of producing 30 units and interpret your answer in a complete sentence.

2. The temperature of a person during an illness is given by in degrees Fahrenheit at time t in hours.

a) Find the rate of change of the temperature at one hour.

b) Interpret your answer in sentence from a) above.

3. Graph the two functions and .

a) What can you tell about these two graphs?

b) Find the domain and range of these two functions.

For the following functions 4 – 11, find

a) sketch each function.

b) the domain of each function.

c) the range of each function.

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11.

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12. The cost of a 30-second commercial during the Super Bowl can be modeled by thousand dollars where x is the number of years after 2000.

a) What did a commercial cost in the 2004 Super Bowl?

b) Use algebra to find when will the cost commercial be $300,000?

13. Find the limit of each of the following function.

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14. The size of a certain insect population at time t in days is approximated by .

a) What is the population after 10 days?

Age

(Months)

/ Length
(Centimeters)
3 / 51
6 / 61
9 / 68
27 / 88
38 / 96

b) Use algebra to find when will the insect population reach 650?

15. The length of an average boy t months after birth is given in the following table.

a) Find a log function model to fit the data.

b) What is the length for an average boy in a year (12 months)?

Year

/ CPI
1990 / 171.2
1991 / 189.2
1992 / 207.3
1993 / 220.5
1994 / 231.4
1995 / 241.2
1996 / 246.0
1997 / 250.5

16. The consumer price index (CPI) values for refuse collection are shown in the table on the right.

a) Find a log model for the data since 1988.

b) What will happen to CPI values beyond 1997 under this model?

c) What year would have CPI value reaches to 275?

Year

/ Percent with Internet Connection
1997 / 22.2
1998 / 32.7
1999 / 39.1
2000 / 44.4
2001 / 53.9
2002 / 55.0
2003 / 56.0

17. The table at the right gives the percent of the U.S. population with Internet connections for the years 1997 to 2003.

a) Find the logistic function model since 1995.

b) As time increases, the total percentage number of the population will never exceed at what value?

c) Use the model to predict the percent of the U.S. population with Internet connections in 2008.

Year

/ Number of Cases
1992 / 954
1993 / 927
1994 / 821
1995 / 687
1996 / 515
1997 / 329
1998 / 235
1999 / 179
2000 / 120
2001 / 101

d) In what year there will be 57% of the U.S. population has Internet connections?

18. The following data is the estimation pediatric Aids incidence in the United States since 1992.

a) Find the logistic function model to fit the data since 1990.

b) As time increases, the total number of pediatric Aids will never exceed at what value?

c) Estimate how many pediatric AIDS cases of the United States were in 2005.

19. Swimming World lists the time in seconds that an average athlete takes to swim 100 meters free style at age x years. The function is a representative as follows.

seconds at age x years.

Use first derivative to find the age at which the minimum swim time occurs.

20. A manufacturing company has found that it can stock no more than 1 week’s worth of perishable raw material for its manufacturing process. When purchasing this material, however, the company receives a discount based on the size of the order. Company managers have modeled function of the cost data as follows:

dollars per week during the xth quarter after January 1, 2000.

xmin: 0 xmax: 24 xscl: 5 ymin: 500 ymax: 600 yscl: 100

a) Based on the graph of C(x), will cost ever decrease from January 2000 to January 2025? Why or why not? Explain clearly.

b) Find the rate of change of the cost function.

c) Use the first derivative test to find any relative minimum or relative maximum. Interpret the meaning of this in complete sentence.

d) Interpret the meaning from c) above in complete sentence. Does this make sense?

Does this answer the same or different from the answer a) above. Explain why.

21. The number of hits on a certain website (in thousands) is approximated by thousands hits where x is the number of years after 2000. How fast is the number of hits growing in 2004?

22. The population of a city, in thousands, over the past decade has decreased and can be modeled by , where t is the number of years since 1995.

a) Evaluate P(6) and interpret the answers in complete sentence.

b) Evaluate and interpret the answers in complete sentence.

23. Per capita spending on health care in the United States has increased steadily since 1970 and can be modeled by dollars where x is the number of years since 1970.

a) Estimate per capita health care spending in 2005.

b) Determine the rate of which health care spending was changing in 2008.

24. The revenue (in millions of dollars) of the Polo Ralph Lauren Corporation from 1990 through 2001 is given as follows.

million dollars of revenue t years after 1990.

a) Find the first and second derivatives of this function.

b) Use the first derivative test to find the relative max and interpret your answer in a complete sentence.

Use window: xmin: –10, xmax: 20, ymin: –500, ymax: 500

25. The percentage of foreign-born residents in the United States from 1900 through 2000 is approximated given as follows.

percent

where t is measured in decades.

Use window: xmin: –10, xmax: 10, ymin: –10, ymax: 10

Use the first derivative to find the absolute extrema over [0, 10] and interpret your answer.

26. The total outstanding mortgage debt in the United States after 1980 is shown at the following equation.

billion dollars

t years after 1980

for M(t): xmin: 0 xmax: 40 xscal: 10 ymin: 10,000 ymax: 40,000 yscal: 1,000

for M’(x): xmin:0 xmax: 40 xscal: 10 ymin: – 1,300 ymax: 10,000 yscal: 1,000

for : xmin: 0 xmax: 40 xscal: 10 ymin: –100 ymax: 100 yscal: 50

a) Find the derivative of the equation.

b) Find the derivative at this function at 1998. Interpret your answer in sentence.

c) Find the inflection point(s).

d) Describe the concavity of this function.

27. For the function on [–2, 4]

a) Find the first derivative.

b) Find the relative minimum and relative maximum.

c) Find the interval where the function is increasing and decreasing.

d) Find the absolute max and absolute min.

e) Find the second derivative.

f) Find the inflection points.

g) Find the interval where the function is concave up and concave down.

28. The percentage of the U.S. population that lives in poverty is estimated by

percentage

where x is the number of year after 1990.

Find the absolute extrema over [8, 15] and interpret your answers.

29. Find the price p that maximizes revenue if the revenue is given by the function

30. Demand for tickets to a theme park is given by the function where p is the price in dollars.

a) Find the revenue function. (hint: )

b) Use the first derivative test to find the price which will maximize revenue.

Use window: xmin: –50, xmax: 100, ymin: –5000, ymax: 30000

31. Shauna owns a tattoo parlor. If she charges $p for a basic tattoo, the weekly demand is .

a) What is the revenue function? Remember: the revenue is the number of items sold times the price per item. .

b) What price would give the greatest revenue and how much revenue would it be?

c) If it cost Shauna $25 per tattoo, what is the cost function?

d) What is the profit function based on a) and c) above?

e) What price will generate the maximum profit and how much profit would it be??

32. Mark’s restaurant can produce once chicken sandwich for $2. The sandwiches sell for $5 each, and, at this price, his customers buy 1200 sandwiches each month. Because of rising costs from suppliers, the restaurant is planning to raise the price of the sandwich. Based on the results of previous price increases, Mark estimates that he will sell 120 fewer sandwiches each month for each $1 increase in the price.

a) Let x be the new price of a sandwich, find the function of how many of sandwiches would sell each month?

b) Find the domain of the function from a) above.

c) Find the revenue function. (remember:)

d) Find the cost function.

e) Find the profit function.

f) Use the first derivative to find what price should the sandwiches sold to maximize Mark’s profit?

g) What is the maximum profit?

33. Emily’s babysitting charges $8 per hour and, at that rate, averages 20 babysitting jobs each week. For each additional $1 charge per hour, the number of jobs per week declines by two.

a) Let x be the new amount she charges per hour. Find the function number of babysitting jobs.

b) Find the domain of the .

c) Find the revenue function.

d) Use the first derivative to find what should she charge per hour to maximize revenue?

e) What is the maximum revenue? Write the complete sentence.

f) If Emily spends $2 per job on supplies, find the cost function.

g) Find the profit function.

h) Use the first derivative to find what should she charge per hour to maximize her profits?

i) What is the maximum profit? Write the complete sentence.

34. For

a) Find the slope of the tangent line to f (x) at x = 1.

b) Find the equation of the tangent line based on a) above.

The following problems are optional:

35. The percentage of the U.S. population living in the metropolitan area can be modeled by where x is the number of year after 2000.

a) Find the first derivative of this function.

b) At what rate was the percentage changing in 2010.

MATH 176

Test 2 Review

Non Computer or Graphing Calculator Part

Direction: No graphing calculator or computer is allowed for this part of the test.

1. For the function [ –2, 4]

a) find the relative min. and max.

b) find the absolute min. and max. of the function on the given interval.

c) find the inflection point(s).

d) find the interval for the function that is concave up and concave down.

2. An economy’s consumer price index is described by the function

where t is the year since 1995.

a) Find the point of inflection of this function.

b) Write the interval for concave up and concave down of this function.

3. For the following graph,

a) find the relative minimum and relative max.

b) find the absolute minimum and absolute maximum.

c) find the inflection points.

a b c d e r f s g t h

4. The graph of the function is shown on the right.

a) Draw a sign graph of.

b) Draw .

c) Find relative extreme.

d) Find the inflection points.

e) the interval of this function is increasing and decreasing.

f) The interval of this function is concave up and concave down.

5. Consider the function

Find the inflection point for this function.

6. Consider the function

a) Find the relative maximum and relative minimum of f(x).

b) Find the absolute maximum and the absolute minimum of f(x) at [–3, 8].

c) Find the inflection point of this function.

d) Describe the concavity of this function.

7. For the function

a) Find the relative min and relative max of the function.

b) Find the inflection points of the function

c) Find the interval where the function is concave up and concave down.

8. If the total revenue function for a blender is dollars where x is sales of how many units. Find maximum revenue of blender sales.

For problems 9 – 24, find the first derivative of each following function.

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