t g(t) h(t) k(t)

Problem 1. Determine whether g(t), h(t), and k(t) could correspond to a linear function or an exponential function, or neither. If it is linear or exponential find the formula for the function, and then evaluate it at t = 10. t g(t) h(t) k(t) 0 12 20 20 1 10 19 22 2 8 18.05 24.2 3 6 17.1475 26.62 1

Problem 2. Suppose a town has a population of 10,000. Fill in the values of the population in the table if: a. Each year the town s population grows at a rate of 500 people per year. b. Each year the town s population grows at a rate of 5% per year. Year 0 1 2 3 a. 10000 b. 10000 2

Problem 3. A certain hand-held calculator is being sold by the manufacturer at a price of $90 per unit. The fixed cost for production is $120,000, and each unit costs $30 to make. Let q be the number of units sold. a. Write an equation for each function: 1. Revenue function, R(q). 2. Cost function, C(q). 3. Profit function, P (q). b. How many units does the manufacturer need to sell to break even? 3

Problem 4. Solve for t in each of the following equations. a. 3e 4t = 2e 2t b. 6(5 t ) = 8(2 t ) c. ln (2t 1) ln (2t + 1) = 0 4

Problem 5. You open an IRA account with an initial deposit of $8,000 which will accumulate tax-free at 4% per year, compounded continuously. a. How much (round to the nearest penny) will you have in your account after 10 years? b. How long does it take your initial investment to double? 5

Problem 6. A fishery stocks a pond with 2,000 young trout. The number of the original trout still alive after t years is given by P (t) = 2000e 0.4t. a. How many trout are left after six months? b. At what time will there be 200 trout left? 6

Problem 7. Following the birth of a child a parent wants to make an initial investment, P 0, that will grow to $100,000 by the child s eighteenth birthday. Assuming that the annual rate of return on the investment is 7%, compounded continuously, what should the initial investment be? 7

Problem 8. It is determined that the value of a certain computer declines exponentially. A computer purchased 2 years ago for $5,000 is worth only $2,500 today. What will the value of the computer be 2 years from now? 8

Problem 9. How long does it take an investment to double at %8.5 interested that is compounded a. annually? b. continuously? 9

Problem 10. If the quantity of a certain radioactive substance is decreased by %5 in 10 hours, find the substance s half-life. 10

Problem 11. Use the following graph to sketch the following: a. The line segment corresponding to f(b) f(a), label as A. b. The line whose slope is given by f(b) f(a), label as B. b a c. The line whose slope is given by f (c), label as C. 11

Problem 12. The population of a town in millions is given by P (t) = 1.2(1.01) t, where t is the number of years since the start of 1998. Find the following. a. The population in 2000. b. The average rate of growth between 1998 and 2000. c. How fast is the population growing at the start of 1998? 12

Problem 13. Draw a possible graph for the following functions, also write at least one (x, y) point on the graph. a. s(t) = mt 4, where m > 0. b. s(t) = mt + 4, where m < 0. c. s(t) = 5a t, where a > 1. d. s(t) = 3a t, where a < 1. 13

Problem 14. Draw a possible graph of y = f(x) given the following information about its derivative. 1. f (x) < 0 for x < 1 and x > 3. 2. f (x) > 0 for 1 < x < 3. 3. f (x) = 0 at x = 1 and x = 3. 14

Problem 15. Using the following graph to fill in the tables with the appropriate intervals or points. f (x) > 0 f (x) < 0 f (x) = 0 f (x) > 0 f (x) < 0 f (x) < 0 and f (x) = 0 f (x) > 0 and f (x) = 0 15

Problem 16. The cost and revenue functions for a company are shown in the following figure. Use the marginal revenue and marginal cost to answer the following questions. a. Should the company produce the 100 th unit? Why? b. Should the company produce the 300 th unit? Why? c. The maximum profit occurs where? Why? 16

Problem 17. The following graph shows the cost and revenue functions that are associated with a certain product. Fill in the appropriate values of q in the table. R > C C > R C = R R > C C > R C = R a. What is the value of q that will maximize profit? b. After producing 30 units, should the manufacturer produce more? Why? c. After producing 60 units, should the manufacturer produce more? Why? 17

Problem 18. Sketch the graph of the first and second derivatives of the function given below. 18

Problem 19. Referring to the graph of the derivative, f (x), below, fill in the table with the appropriate intervals or points. f(x) is increasing on: f(x) is decreasing on: f(x) has infection points at: f(x) has horizontal tangent lines at: 19

Problem 20. Values of the function W (t) are given in the table below. t 1.0 1.4 1.8 2.2 2.6 3.0 W (t) 25 28 35 45 50 60 a. Estimate 3 1 W (t) dt using: 1. Left-hand sums 2. Right-hand sums b. For the estimate in part (a) what is n (i.e. the number of rectangles) and what is t? 20

Problem 21. Estimate the definite integral a. Left-hand sums. b. Right-hand sums. 3 1 6 dx with n = 4 by using: x 21

Problem 22. Suppose that r = 10(1.5 t ), where r is the rate in which the world s oil is being consumed. Using three subdivisions find an approximate value for the total quantity of oil used between t = 0 in 1995 and t = 3 in 1998 using a left-hand sum. 22

Problem 23. Suppose that the velocity of an object is given by v(t) = t 2 + 8t + 10, where t is in seconds. Estimate the distance traveled by the object during the first five seconds using n = 5. a. Use a left-hand sum. b. Use a right-hand sum. 23

Problem 24. Find the points on the graph y = x 2 4x + 10 where the tangent line is horizontal. 24

Problem 25. Find an equation of the tangent line to the graph of y = e 4x + 4 at x = 0. 25

Problem 26. Assume the demand function for a certain product is q = 1000e 0.02p, where q is the quantity and p is the price. a. Write the revenue, R, as a function of price. b. Find the rate of change of revenue with respect to price. c. Find R(10) and R (10). What is the economic significance of each answer? d. Determine the price for which R = 0. What is the economic significance of the answer? 26

Problem 27. Using any method you wish, find dy dx a. y = 4 x 3 + 4 3 x 4x + 2 b. y = 8 x 4 2 c. y = ( e 2x 4 ) 3 d. y = ln (x 2 2) + 3 x e. y = 4e5x + 3e 2x e 3x for each of the following. 27

Problem 28. Using any method you wish, find dy dx a. y = (2x 3 + 4) ln (5x) b. y = 5x 4 e 4x for each of the following. 28

Problem 29. Find the following and then sketch the graph of the function f(x) = x 5 15x 3. a. The intervals where f is increasing and decreasing. b. The local maximum and minimum points. c. The inflection points. d. The intervals where f is concave up and concave down. 29

Problem 30. The total revenue from the sale of x units of a product is R(x) = 100 x 2. The total cost of producing x units is C(x) = 1 3 x3 6x 2 + 89x + 100. Find the number of units that must be produced and sold in order to maximize profit. 30

Problem 31. An appliance firm determines to sell q radios. The price per unit is given by p = 400 q. The firm also determines that the total cost of producing q radios is given by C(q) = 1000 + q 2. How many radios must be produced and sold to maximize profit? 31

Problem 32. A company estimates that it can sell 1,000 units per week if it sets the unit price at $15. Its weekly sales will rise by 100 units for each $0.20 decrease in price per unit. Find the production level (i.e. the price per unit and the number of units) that will maximize revenue. 32

Problem 33. An apartment complex can fill 100 units when the rent is $400 per month. It is estimated that for each $10 per month decrease in rent that 5 more units will become occupied. The complex has a monthly maintenance cost of $100 for each unit rented. What monthly rent should be charged to maximize profit? 33

Problem 34. Find the area between y = 4x and y = x 2 + 3, also sketch the region bounded by the graphs. 34

Problem 35. Find the area between y = 2x 2 3 and y = 5x, also sketch the region bounded by the graphs. 35

Problem 36. Evaluate the following integrals. ( a. x 3 + 2 ) x 3 8 dx b. (8x 2 + 6e 2x ) dx c. d. ( 3 x + 2 ) x + 1 dx e 1 ( x 1 ) dx x 36

Problem 37. The marginal revenue for the price of tickets is given by R = 10q 50 dollars per ticket, where q is the number of tickets. Find the total revenue from the sale of tickets for the first 20 tickets. 37

Problem 38. What should an annuity, A, per year be so that the amount of a continuous money flow over 20 years at an interest rate of %8, compounded continuously, will be $30,000? 38

Problem 39. Find the present value of an investment over a 10 year period if there is a continuous money flow of $1,500 per year and the current interest rate is %6 compounded continuously. 39