calculus exam.

TAKE HOME EXERCISES ARE DUE WEDNESDAY 07/19/17. NO EXCEPTIONS.

WRITE ALL WORK ON SEPARATE PAPER. WRITE CLEARLY AND PERFECTLY.

Name___________________________________

1) Solve the differential equation dx dt

= -3x, x(0) = 7.

2) Find the consumer’s surplus for the following demand function at the given point.

D(x) = (x – 3)2; x = 3 2

3) The Bay of Fundy in North America has some of the largest varying tidal depths on the planet. If the depth of the water there is at lowtide at 11:00AM and high tide is at 6:00PM, and lowtide is 3m depth and high tide is 17m. Use a sine model to answer the following questions.

a) How deep is the tide at 1:00AM? b) How fast is the tide changing, and in what direction, at 8:00PM?

4) Evaluate the following integral.

3xe2x∫ dx

5) Evaluate the integral.

(8×3 + 6×2 ) x4 + x3 + 9dx∫

6) Evaluate the following integral.

x 3 – x∫ dx

7) The Cobb-Douglas function for a new product is given by N(x, y) = 15×0.6 y0.4 where x is the number of units of labor and y is the number of units of capital required to produce N(x, y) units of the product. Each unit of labor costs $40, and each unit of capital costs $80. If $400,000 has been budgeted for the production of this product, determine how this amount should be allocated in order to maximize production, and find the maximum production.

8) Calculate the derivative of f(x) = sin(x2 – 1)

9) The rate of flow of a continuous income stream (in thousands of dollars per day) is given by f(t) = 1 t + 1

.

Find the total income produced during the first thirty days of operation.

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10) At the beginning of an advertising campaign for a new product in a city of 500,000 people, no one is aware of the product. After 10 days, 100,000 people are aware of the product. If N = N(t) is the number of people (in thousands) who are aware of the product t days after the beginning of the advertising campaign, solve the following differential equation for N(t): dN dt

= k(500 – N); N(0) = 0; N(10) = 100.

11) Find and classify all critical points for f(x, y) = 2×3 – 24xy + 4y3

12) Suppose that the labor cost for a building is approximated by C(x,y) = 10×2 + 3y2 – 240x – 180y + 24,000, where x is the number of days of skilled labor and y is the number of days of semiskilled labor required. Find the x and y that minimize cost C.

13) Use Newton’s Law of Cooling to find the temperature in the following case. A loaf of bread is removed from an oven at 375° F and cooled in a room whose temperature is 65° F. If the bread cools to 210° F in 15 minutes, how much longer will it take the bread to cool to 100° F?

14) After a person takes a pill, the drug contained in the pill is assimilated into the bloodstream. The rate of assimilation minutes after taking the pill is R(t) = te-0.4t. Find the total amount of the drug that is assimilated into the bloodstream during the first 15 minutes after the pill is taken. Round your answer to 2 decimal places.

15) Evaluate the integral.

1

0

14

5 (s + t) dt ds∫∫

16) Find critical points for f(x, y) = 5×2 – 5y2 + 2xy + 34x + 38y + 12.

17) Find the local extrema for f(x, y) = x3 – 12x + y2 .

18) Suppose that the labor cost for a building is approximated by C(x,y) = 10×2 + 3y2 – 240x – 180y + 24,000, where x is the number of days of skilled labor and y is the number of days of semiskilled labor required. Find the x and y that minimize cost C.

19) The marginal price for a weekly demand of x bottles of cough medicine in a drug store is given by

p'(x) = -13,300 (5x + 40)2

. Find the price-demand equation if the weekly demand is 125 when the price of a bottle of

cough medicine is $4. What is the weekly demand (to the nearest bottle) when the price is $3?

20) Use Lagrange multipliers to maximize f(x, y) = 5xy subject to x + y = – 6.

21) Use Lagrange multipliers to maximize the product of two numbers if their sum must be 26.

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22) The Cobb-Douglas function for a new product is given by N(x, y) = 15×0.6 y0.4 where x is the number of units of labor and y is the number of units of capital required to produce N(x, y) units of the product. Each unit of labor costs $40, and each unit of capital costs $80. If $400,000 has been budgeted for the production of this product, determine how this amount should be allocated in order to maximize production, and find the maximum production.

23) Use the Lagrange multiplier method to maximize f(x, y, z) = xy + z subject to x2 + y2 + z2 = 1.

24) The rectangular box below, with an open top and one partition, is to be constructed from 18 square inches of cardboard. Find the dimensions that will result in a box with the largest possible volume by using the method of Lagrange multipliers.

25) The revenue from the sale of electric blankets is seasonal, with maximum revenue in the winter. Let the revenue received from the sale of heaters be approximated by R(x) = 18 cos 2πx + 600, where x is time in years, measured from January 1. Find and interpret R'(x) for August 1st.

26) Find the amount of an annuity if $360 per month is paid into it for a period of 7 years, earning interest at a rate of 8% per year compounded continuously.

27) Determine if each improper integral converges or diverges. If the integral converges, find the value.

a. ∞

1

dx x1/2

∫ b. 1

0

dx x1/2

∫ c. ∞

1

dx x2

∫ d. 1

0

dx x2

∫

28) Solve the differential equation y(t) = y(6t5 + 3 t2), y(0) = 2.

29) Find the sum of the first 25 terms of the geometric sequence 250, 250(1.05), 250(1.05)2, . . .

30) Evaluate the integral with the order reversed. 2

-2

4 – y2

0 dx dy∫∫

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