Math 122 Exam 1 Review Name: __________________________ 1. From the following table, A. Find f(2) B. Find the value(s) of x for which f ( x) 1 . If there is more than one, list them in increasing order, separated by commas. x f(x) 1 1 2 3 3 7 4 6 5 4 6 1 2. Let y f ( x) 2 x2 2 . A. Find the value of y when x is zero. B. Find f(-1). 3. Could the following table represent a linear function? Answer yes or no. t d 0 0 1 15 2 30 3 45 4. The average weight in pounds of American men in their sixties (in 1979) as a function of their heights in inches is given in the following table. The formula that expresses the weight w in terms of the height h is given by w = _____+_____h height (h) weight (w) 68 168 69 172 70 176 71 180 72 184 73 188 5. Suppose that y f (t ) is the distance in miles traveled in t hours by a car moving at 65 miles per hour. Give a formula for the function f (t ) . Page 1 6. Find a formula for the linear function f. x 0 1 10 15 f ( x) A) f ( x) 5x 10 B) f ( x) x 10 C) f ( x) 15x 10 D) f ( x) 15x 10 2 ? 3 ? 4 ? 7. A car is worth $15,000 when it is 1 year old, and it is worth $11,000 when it is three years old. A. Write the value of the car, V (in dollars), as a function of the age of the car, a (in years). Assume this is a linear function. B. How much does the car depreciate in value each year? C. How much was the car worth when it was first purchased? 8. The equation of the line through the points (2, 5) and (-3, -5) is: 9. The bill for electricity is $175 when 35 kilowatt hours are used and $325 when 85 kilowatt hours are used. A. The base cost (without using any electricity) is $______. B. Each additional kilowatt hour used costs $_____. 10. The population of Los Angeles, California was 2,811,801 in 1970 and was 3,448,613 in 1994. The change in the population of Los Angeles between 1970 and 1994 was _____ people. 11. The following table gives the number of students taking an applied calculus course at a community college. Find the change in the number of students taking the course between 2000 and 2004. Year students 1998 312 1999 358 2000 384 2001 431 Page 2 2002 492 2003 569 2004 636 12. Consider the following graph. Between point A and point B, the graph is: (mark all that apply) A) B) C) D) decreasing increasing concave up concave down 13. Find the average rate of change of s(t ) -16t 2 50t 15 between t = 1 and t = 3. Round to two decimal places. A) -14.00 B) -28.00 C) 36 D) -10.88 14. Values of a linear cost function are given in the following table. Find a formula for the cost function. q 0 100 200 300 100 105 110 115 C ( q) 15. A textbook company had fixed costs of $25,000 and variable costs of $15 for a certain book. The company sells the books for $30 each. Find a formula for the profit function, (q) . 16. A textbook company had fixed costs of $35,000 and variable costs of $25 for a certain book. The company sells the books for $50 each. What is the break-even point for the company (to the nearest book)? Page 3 17. The following figure gives both supply and demand curves for a certain product. If the price is $50 per item, how many items will the producers supply? A) B) C) D) 2400 1000 1700 4000 18. Production costs for manufacturing T-shirts consist of a fixed cost of $16,000 plus variable costs of $4 per shirt. Each T-shirt sells for $12 dollars. What is the marginal cost? A) $4 B) $12 C) $8 D) $16,000 19. Production costs for manufacturing T-shirts consist of a fixed cost of $18,000 plus variable costs of $4 per shirt. Each T-shirt sells for $12 dollars. How many T-shirts must be sold for the company to break even? A) 1,500 B) 2,250 C) 4,500 D) 3,000 Page 4 20. The demand and supply curves for a certain product are given in terms of price, p, by D( p) 370 9 p and S ( p) 6 p 140 . What is the equilibrium price? A) $3 B) $64 C) $34 D) $230 21. The demand and supply curves for a certain product are given in terms of price, p, by D( p) 400 12 p and S ( p) 6 p 140 . What is the equilibrium quantity? A) 30 B) 40 C) 6 D) 260 22. The cost of producing q items in a tortilla factory is given by C(q) 3000 0.71q dollars. The revenue from sales of q items is R(q) 0.89q dollars. A. For what values of q does the tortilla factory make a profit? B. Write a formula for profit as a function of q. 23. A population is growing according to the formula P 250(1.055)t , where P is the population at year t. What is the initial population? A) 356 B) 264 C) 106 D) 250 24. A population is growing according to the formula P 375(1.065)t , where P is the population at year t. What is the annual growth rate? A) 3.75% B) 3.99% C) 6.5% D) 13.42% Page 5 25. A population is growing according to the formula P 225(1.05)t . What is the population in year 8? A) 332 B) 1,890 C) 970 D) 226 26. A population is growing according to the formula P 350(1.055)t , where P is the population in year t. How many years will it take for the population to exceed 1000? A) 119 B) 19 C) 20 D) 118 27. A town has 1800 people initially. Find the formula for the population of the town, P, in terms of the number of years, t, if the town grows at an annual rate of 4% a year. A) P 1800 4t B) P 1800(0.04)t C) P 1800(1.04)t D) P 1000 (1.04)t 28. A town has 2400 people initially. Find the formula for the population of the town, P, in terms of the number of years, t, if the town shrinks by 90 people a year. A) P 2400 90t B) P 2400(0.1)t C) P 2400(0.9)t D) P 2400 (0.9)t 29. Using the table, find the value of a if f is linear. x f ( x) 0 50 10 100 20 a 30. Using the table, find the value of a if f is exponential. x f ( x) 0 50 10 100 20 a Page 6 31. A bar of soap starts out weighing 100 grams. Write a formula for the quantity S grams of soap remaining after t days if the decrease is 5 grams per day. 32. A bar of soap starts out at 125 grams. Write a formula for the quantity S grams of soap remaining after t days if the decrease is 5% per day 33. Does P 2(0.9)t represent exponential growth or decay? 34. Does P 4.3e-0.29t represent exponential growth or decay? 35. A bakery has 300 pounds of flour. If they use 5% of the available flour each day, how many pounds do they have left after 10 days? Round to the nearest pound. 36. A substance has a half-life of 44 years. What percent of the original amount of the substance will remain after 20 years? Round to the nearest percent. 37. Solve 7 3t 3 2t for t. Round to 3 decimal places. 38. Solve 90 14t 45 12t for t. Round to two decimal places. 39. What interest rate, compounded annually, is equivalent to a 3% rate compounded continuously? Round to two decimal places. 40. Write the function a(t ) 20(1.1)t in the form Aekt . Round k to 3 decimal places. 41. Write the function b(t ) 10e-1.02t in the form P0 (a)t . Round a to 3 decimal places. 2 42. Simplify the expression 8eln( a ) as much as possible. 43. Solve 4ex3 9e9 x using logs. Round your answer to 3 decimal places. Page 7 44. What is the doubling time of prices which are increased by 14% per year? Round to the nearest hundredth of a year. 45. If the size of a bacteria colony doubles in 6 hours, how many hours will it take for the number of bacteria to be 5 times the original amount? Round to 3 decimal places. 46. You win the lottery and are offered a choice of $90,000 now or $20,000 at the end of each year for five years. Assuming a 5% annual interest rate and ignoring taxes, which is the better option? A) The first B) The second 47. A standard cup of coffee contains about 100 mg of caffeine, and caffeine leaves the body at a rate of about 17% an hour. How many mg of caffeine are left in the body after 5 hours if this rate is hourly? Round to 2 decimal places. 48. A standard cup of coffee contains about 100 mg of caffeine, and caffeine leaves the body at a rate of about 17% an hour. How many mg of caffeine are left in the body after 4 hours if this rate is continuous? Round to 2 decimal places. 49. An exponentially decaying substance was weighed every hour and the results are given below. What is the half-life of the substance? Round to the nearest tenth. Time 9 am 10 am 11 am 12 noon 1 pm Weight (in grams) 12.000 10.750 9.630 8.627 7.728 50. Which is worth more after 5 years: $1200 invested at 10% annual interest or $1500 invested at 8% annual interest? A) The 8% investment B) The 10% investment 51. A cigarette contains about 0.4 mg of nicotine. The half-life of nicotine in the body is about 2 hours. How many hours does it take, after smoking a cigarette, for the level of nicotine in a smoker's body to be reduced to 0.08 mg? Round to 2 decimal places. Page 8 52. If the size of a bacteria colony doubles in 8 hours, how many hours will it take for the number of bacteria to be 5 times the original amount? Round to 2 decimal places. 53. Use the following table to find f ( g (2)) . x 0 1 2 3 4 2 4 6 3 5 f ( x) 5 3 2 1 0 g ( x) 54. Given the function m( z) z 2 , find and simplify m( z h) m( z) . 55. Let A) B) C) D) E) f ( x) 4x 1 and g ( x) x2 5 . What is f ( g ( x)) ? 4x2 21 16x2 8x 6 x2 4 x 6 4x3 x2 20x 5 4x3 20x 1 56. What is the equation for the graph obtained by shifting the graph of y x3 vertically upward by 2 units, followed by vertically stretching the graph by a factor of 5? A) 5x3 10 B) 5x3 2 C) 10x2 2 D) 5x3 30 x2 60 x 40 57. What is the equation for the graph obtained by shifting the graph of y x3 vertically upward by 3 units, followed by vertically stretching the graph by a factor of 2, followed by reflecting the graph across the x-axis? A) 2 x3 6 B) 2x3 3 C) 6 x2 3 D) 2x3 18x2 54x 54 58. Given the equation q( x) x3 , find and simplify q(2x a) q( x) . Page 9 59. Given the equation q( x) x3 , find and simplify q( x2 ) q( x a) . 60. For g ( x) 3x2 5x and h( x) 4x 1 , find and simplify h( g ( x)) . 61. For g ( x) 5x2 3x and h( x) 4x 1 , find and simplify g (h( x)) . 62. Which of the following are power functions? Select all that apply. A) 6 y 3x B) y 3 4x C) y 1 x D) y (6 x )3 63. Use shifts of power functions to find a possible formula for the following graph. A) B) C) D) y ( x 4)3 10 y ( x 4)3 10 y ( x 10)3 4 y ( x 10)3 4 64. Simplify (7)2 by hand, leaving your answer as a fraction where appropriate. Page 10 65. Write a function that represents the following situation. The gravitational force, F, between two bodies is inversely proportional to the square of the distance, d, between them. A) F kd 2 B) 1 F k 2 d C) 1 F2 k d 2 D) F kd 66. The number of species, S, on an island is proportional to the square root of the area, A, of the island. An island which has an area of 4 square miles contains 20 species. How many species would be expected on an island of 16 square miles? 67. The illumination, I, of a candle is inversely proportional to the square of its distance, d, from the object it illuminates. Write a formula that expresses this relationship. 68. Coulomb's law says that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the distance between the objects. Let q1 and q2 be the charge on the two objects. Let d be the distance between the objects and F be the electrical force between them. Translate Coulomb's Law into mathematical symbols. A) qq F 1 22 d B) kd F (q1 q2 ) 2 C) k (q2 q1 )2 F d2 D) kq q F 12 2 d 69. As x - , the function f ( x) 17 x4 _____. A) B) C) neither of these Page 11 70. As x , the function f ( x) 5x7 _____. A) B) - C) neither of these Page 12 Answer Key 1. A. 3 B. 1, 6 2. A. -2 B. 0 3. yes 4A. -104 4B. 4 5. f (t ) 65t 6. A 7. A. V 17,000 2000a B. $2000 C. $17,000 8. y 1 2x 9. A. 70.00 B. 3.00 10. 636,812 11. 252 12. A, C 13. A 14. C(q) 0.05q 100 15. (q) 15q 25,000 16. 1400 17. A 18. A 19. B 20. C 21. B 22. A. q > 16,667 items 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. B. (q) 0.18q 3000 dollars D C A C C A 150 200 S 100 5t S 125(0.95)t decay decay 180 73% Page 13 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. -2.090 -4.50 3.05% 20e0.095t 10(0.361)t 8a2 0.274 5.29 years 13.932 A 39.39 50.66 6.3 hours A 4.64 18.58 6 2zh h2 A A A 9x3 12ax2 6a2 x a3 x6 x3 3ax2 3a2 x a3 12x2 20 x 1 80x2 28x 2 A, D A 1/49 B 40 k I 2 D 68. D 69. A 70. A Page 14