Math 131 Problem Sheet Section 2.1
In Problems 1 – 8, find the following for each function:
(a) , (b) , (c) , (d)
1. 2. 3.
4. 5. 6.
7. 8.
In Problems 9 – 18, find the following for each function:
(a) , (b) , (c) , (d) (e) (f)
9. 10. 11.
12. 13. 14.
15. 16. 17.
18.
In Problems 19 – 32, find the domain of each function.
19. 20. 21.
22. 23. 24.
25. 26. 27.
28.
In Problems 33 – 42, find the difference quotient, , for each function. Be sure to simplify.
33. 34. 35.
36. 37. 38.
39. 40. 41.
42.
43. For the function , show that , where . [Hint: multiply the numerator and denominator by .]
44. For the function , show that , where .
45. If and , what is the value of A?
46. If and , what is the value of B?
47. If and , what is the value of A?
48. If and , what is the value of B?
49. If and , what is the value of A? Where is not defined?
50. If , , and is undefined, what are the values of A and B?
51. If a rock falls from a height of 20 meters on Earth, its height H (in meters) after x seconds is approximately
(a) What is the height of the rock when second? seconds? seconds? seconds?
(b) When does the rock strike the ground?
52. If a rock falls from a height of 20 meters on the planet Jupiter, its height H (in meters) after x seconds is approximately
(a) What is the height of the rock when second? seconds? seconds?
(b) When does the rock strike the ground?
53. Express the area, A, of a rectangle as a function of the length, x, when the length is twice the width of the rectangle.
54. Express the area, A, of an isosceles right triangle as a function of the length of one of the two equal sides. Let x represent the length of one of the two equal sides.
55. Express the gross salary, G, of a person who earns $5 per hour as a function of the number of hours worked, x.
56. A commissioned salesperson earns $100 base pay plus $10 per item sold. Express the gross salary, G, as a function of the number of items sold, x.
57. The period, T (in seconds), of a simple pendulum is a function of its length, l (in feet), defined by the equation
where feet per second per second is the acceleration due to gravity. (a) Use a calculator to determine the period of a pendulum whose length is 1 foot. (b) By how much does the period increase if the length is increased to 2 feet?
58. An airplane crosses the Atlantic Ocean (3000 miles) with an air speed of 500 miles per hour. The cost, C, (in dollars) per passenger is
,
where x is the ground speed ().
(a) What is the cost per passenger for quiescent (no wind) conditions?
(b) What is the cost per passenger with a head wind of 50 miles per hour?
(c) What is the cost per passenger with a tail wind of 100 miles per hour?
(d) What is the cost per passenger with a head wind of 100 miles per hour?
59. If an object weighs m pounds at sea level, then its weight, W (in pounds), at a height of h miles above sea level is given approximately by
If a woman weighs 120 pounds at sea level, how much will she weigh on Pike’s Peak, which is 14,110 feet above sea level?
60. A function, labeled f, has the property that for all real numbers a and b. Which of the following functions also have this property?
(a) (b) (c) (d)
61. Let f and g be two functions defined on same interval. Suppose we define two functions and as follows:
(a) Show that
(b) Develop a similar formula for
Answers for most odd numbered problems:
1. (a) (b) (c) (d) , 3. (a) 0 (b) (c) (d) , 5. (a) 4 (b) 5 (c) 5 (d) 6,
7. (a) (b) (c) (d) 5, 9. (a) (b) (c) (d) (e) (f) , 11. (a) (b) (c) (d) (e) (f) ,
13. (a) (b) (c) (d) (e) (f) ,
15. (a) (b) (c) (d) (e) (f) , 17. (a)
(b) (c) (d) (e) (f) , 19. All real numbers. 21. All real numbers.
23. 25. 27. 29. 31. 33. 0 35. 37. 39. 41.
43. 45. 47.
49. ; undefined at 3. 51. (a) 15.1 m, 14.07m, 12.94 m, 11.72 m (b) 2.02 sec 53. 55. 57. (a) 1.11 sec (b) 0.46 sec 59. 119.84 pounds.