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Mat210 Test 3 – Practice Problems
1. The rate of change of the number of U.S. Coast Guard personnel on active duty from 1992 through 1995 can be modeled by
[pic]
thousand personnel per year x years since 1992. (Source: based on data from the U.S.
Department of Transportation)
a) Sketch a graph of g(x) from x = 0 through x = 3.
b) What does the fact that a portion of g(x) lies below the x-axis tell you about the number of personnel on active duty?
c) What does the accumulated area of the region between g(x) and the x-axis represent?
d) Use five midpoint rectangles to estimate the accumulated area of the region between g(x) and the horizontal axis from x = 0 through x = 3. Interpret your answer.
2. During the summer tourist season, a souvenir stand operator expects daily profits to change at a rate of
[pic]
dollars per day x days after Memorial Day.
a) Write the definite integral representing the total profit made during the first 30 days of the tourist season.
b) Find the total profit made during the first 30 days of the tourist season.
3. During a track test of a 1997 Jaguar XK8, the car's speed was given by
[pic]
miles per second after t seconds. (Source: based on data from Road & Track, August
1997)
Evaluate [pic]and interpret your answer.
4. Between 1970 and 1989, the amount of outstanding consumer credit grew at a rate of
[pic]
billion dollars per year x years since 1970. Outstanding consumer credit was 799.5
billion dollars in 1989. (Source: based on data from the Federal Reserve System)
a) Find a model for the amount of outstanding consumer credit.
b) Use your model to estimate the amount of outstanding consumer credit in 1985.
c) According to the model, by how much did outstanding consumer credit grow between 1970 and 1974?
5. The rate of change of the value of deposits in U.S. banks from 1935 through 1985 can be modeled by
[pic]
million dollars per year x years after 1935. (Source: Federal Deposit Insurance
Corporation)
a) Write the general antiderivative of v(x).
b) According to the model, how much did the value of deposits in U.S. banks increase from 1970 through 1980?
c) Is it possible to find the value of deposits in 1976 based on the given information? Why or why not?
6. Consider the functions
[pic]
a) Sketch the functions f(x) and g(x) on the same axes.
b) Find the input values at which f(x) and g(x) intersect.
c) Shade the region between f(x) and g(x) between a = 1 and b = 11.
d) Write the integral notation representing the difference in the area of the region between f(x) and the horizontal axis and the area of the region between g(x) and the horizontal axis from a = 1 and b = 11. Evaluate this integral.
e) Calculate the total area of the shaded regions.
7. integral:[pic].
8. For the year ending December 31, 1996, Gateway 2000, Inc., the second largest direct marketer of PCs in the U.S. and known for its black-and-white cowhide-pattern boxes, posted a net profit of $250,700,000. Assume that profits will increase by 45% per year and that Gateway will invest 5% of its annual profits at an annual rate of return of 8% compounded continuously. (Source: Gateway 2000, Inc.)
a) Write a function for the annual rate at which money flows into this investment.
b) What is the future value of the investment at the end of 2001?
9. For the year ending January 31, 1997, The Gap, Inc., posted an annual net profit of $452.9 million. Assume that these profits can be reinvested at an annual rate of return of 12% compounded continuously. Also assume that The Gap will maintain this level of annual net profit investment continuously for the next 10 years. (Source: The Gap, Inc.)
a) What is the future value of The Gap's 10-year net profit?
b) What is the present value of The Gap's 10-year net profit?
AND THE SUPPLEMENTAL NOTES AND HOMEWORK PROBLEMS FROM SECTION 5.5
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