Final Exam
In This Chapter
Nothing helps you understand math like good old-fashioned practice, and that’s the purpose of this chapter. You can use it however you like, but I suggest one of the following three strategies:
1. As you finish reading each chapter, skip back here and work on the practice problems from that chapter.
2. If you’re using this book as a refresher for a class you’ve already taken, complete this test before you start reading the book. Then, go back and read the chapters containing problems you missed. After you’ve reviewed those topics, try these problems again.
3. Save this chapter until the end, and use it to see how much you remember of each topic when you haven’t seen it for a while.
Because these problems are just meant for practice, and not meant to teach new concepts, only the answers are given at the end of the chapter, usually without explanation or justification (unlike the problems in the “You’ve Got Problems” sidebars throughout the book). However, these practice problems are designed to mirror those examples, so you can always go back and review if you forgot something or need extra practice.
Are you ready? There’s a lot of practice ahead of you—as some problems have multiple parts, there are actually more than 100 practice problems in this chapter! (But no one said you have to do them all at one sitting.)
1. Put the linear equation in standard form: –3(x + 2y) – 4y + 8 = x – 1.
2. Determine the equation of the line that passes through the point (–5,3) and has slope ; write the equation in standard form.
3. Calculate the slope of the line that passes through points (2,–3) and (–5,–8).
4. Line n passes through the point (2,–1) and is perpendicular to the line 3x – 5y = 2. Write the equation of n in standard form.
5. Simplify the expression .
6. Factor the expression completely: 32x2 – 98y2.
7. Solve the equation 2x2 – 16x = 22 by completing the square.
8. Solve the equation by factoring: x2 – 256 = 0.
9. Solve the equation using the quadratic formula: 3x2 + 4x + 1.
Chapter 3
10.
11. Determine what kind of symmetry, if any, is evident in the graph of y = x5 – x3 + x – 5.
12. Find the inverse function, f–1(x), if f(x) = 5x – 3; verify that f(x) and are inverses by demonstrating that .
13. Given the graph of function j(x) and a table of values for the following function k(x), calculate j –1(k(2)). Assume that j(x) has inverse function j –1(x); also assume that all points indicated on the graph of j(x) have integer coordinates.
14. Put the parametric equations x = 2t + 6, into rectangular form.
Chapter 4
15. If calculate cscθ and cotθ.
16. Evaluate using a coterminal angle and the unit circle.
17. Factor and simplify the trigonometric expression 1 – tan4θ.
18. Solve the equation and provide all solutions on the interval [0,2π).
20. Given f(x) as defined here, calculate .
21. Evaluate the limits on the graph:
(a)
(b)
(c)
(d)
Chapter 6
22. Evaluate the limits using substitution:
(a)
(b)
23. Evaluate the limits using the factoring method:
(a)
(b)
24. Evaluate using the conjugate method.
25. Evaluate the limit of as x approaches each value for which g(x) is undefined.
26. Evaluate the following limits:
(a)
(b)
(c)
(d)
27. Given h(x) = tan x, evaluate the following limits:
(a)
(b)
Chapter 7
28. Determine whether or not the function f(x), as defined here, is continuous at x = 4:
29. Find the value of c that makes the function g(x) everywhere continuous:
30. Find all the x-values for which the function is discontinuous and classify each instance of discontinuity.
31. Does the Intermediate Value Theorem guarantee the following function values for f(x) = 3x2 – 12x + 4 on the closed interval [0,5]? Why or why not?
(a) 10
(b) 20
Chapter 8
32. Use the difference quotient to find the derivative of f(x) = x3 – 2x and use it to evaluate f ′(–3).
33. Determine g ′(1) if g(x) = 3x2 – 8x + 2 using the alternative formula for the difference quotient.
Chapter 9
34. Find the derivative of each expression with respect to x:
(a)
(b) (3x2 + 4)(9x – 5)
(c)
(d) (x2 – 7x + 2)10
(e)
35. Given the function h(x) = 3x4 – 9x2 + 2, calculate the following values:
(a) The average rate of change of h(x) on the x-interval [–1,3]
(b) The instantaneous rate of change of h(x) when x = 2
36. Given f(x) = tan (cos x), calculate .
37. Assume functions j(x) and k(x) are continuous and differentiable for all real numbers. The following table lists values of the functions and their derivatives for specific x-values.
Given this information, calculate the following:
(a)
(b)
Chapter 10
38. Identify the equation of the tangent line to f(x) = x2 sin x when x = π. Hint: use the Product Rule to differentiate f(x).
39. Calculate the slope of the tangent line to the graph of x2 – 7xy – 4y2 + y – 9 = 0 at the point (–3,0).
40. Calculate the slope of the normal line to the graph of j(x) = tan x cos x when x = 0.
41. Given g(x) = x3 – 4, evaluate (g –1)′(–3).
42. If h(x) = –2x3 – 5x + 3, calculate (h –1)′(–1).
43. Given the parametric equations x = cosθ and y = 2θ, determine
Chapter 11
44. If f(x) = x3 – 16x, find f ′(x), determine its critical numbers, and determine if f(x) changes direction at each.
45. If some function g(x) has derivative , use a wiggle graph to determine the interval(s) on which g(x) is decreasing.
46. What are the absolute maximum and minimum values of on the closed interval [–4,3]?
47. On what interval is f(x) = x3 – 8x2 + 9x – 12 concave up?
48. Use the Second Derivative Test to classify the relative extrema of the function .
Chapter 12
49. A goldfish swims back and forth inside a large fish tank featuring a plastic, bubbling, sunken treasure chest ornament. At time t, the horizontal position of the goldfish (relative to the treasure chest) is inches. (If s(t) > 0, the fish is right of the treasure chest, and a negative s(t) means the fish is left of it.)
Based on this information, answer the following questions:
(a) At what time(s) is the fish 3.5 inches left of the treasure chest?
(b) What is the speed of the fish at t = 4.2 seconds?
(c) What is the fish’s average velocity between t = 0 and t = 5?
(d) On what interval(s) does the fish have positive acceleration between t = 0 and t = 2 seconds?
50. Nick throws a baseball straight up from an initial height of 3 feet, with a velocity of 25 ft/sec. Given this information, answer the following questions:
(a) What is the velocity of the ball t = 1 second after Nick throws it?
(b) When does the ball reach its maximum height?
(c) What is the maximum height of the ball?
(d) Assuming no one catches the throw, how long does the ball remain in the air before it hits the ground?
Chapter 13
51. Given x0 = 0, apply Newton’s Method to calculate x1 and approximate the root of f(x) = e3x – 2.
52. Evaluate .
53. Given the function f(x) = 6x2 – 2x + 3, find the x-value that satisfies the Mean Value Theorem on the interval [–1,1].
54. Erin and Sara are coworkers and exit their office at precisely the same time, 5 P.M. Erin walks due south from the office at a constant speed of 3.5 miles per hour. Sara bikes due west at a constant speed of 12 miles per hour. At what rate is the distance between Erin and Sara increasing at exactly 5:15 P.M.?
55. A farmer owns a plot of land whose western boundary is a river. He wishes to design a rectangular pasture but will only use fence for three of its sides, trusting the river to define the remaining side of the pasture, as illustrated here.
What are the dimensions of the largest pasture he can create using 2,500 feet of fence?
Chapter 14
56. Approximate the area under the curve on the interval [4,8] using the following:
(a) Right sums with n = 8 rectangles
(b) Midpoint sums with n = 4 rectangles
(c) Trapezoidal Rule with n = 4 trapezoids
(d) Simpson’s Rule with n = 6 subintervals
Chapter 15
57. Evaluate: .
58. Integrate: .
59. Calculate the area beneath the curve on the interval [4,8] using a definite interval.
60. Calculate the derivative: .
61. Integrate using u-substitution:
(a)
(b)
(c)
Chapter 16
62. Calculate the area between the functions .
63. Calculate the value guaranteed by the Mean Value Theorem for Integration on the function on the interval [0,1].
64. The velocity of a particle moving horizontally along the x-axis is modeled by the equation v(t) = t3 – 7t + 6, measured in inches per second. Use this information to answer the following questions:
(a) What is the total displacement of the particle between t = 0 and t = 3?
(b) What total distance does the particle travel between t = 0 and t = 3?
65. Given , evaluate the following:
(a) f(1)
(b) f ′(1)
66. Write an integral expression representing the length of each graph segment described here, and then use a computational tool (such as a graphing calculator) to compute each integral:
(a) f(x) = tan x, between
(b) The parametric curve defined by x = e2t and y = ln (4t + 2) on the t-interval [0,3]
Chapter 17
67. Solve the differential equation x2dy = –2dx.
68. A popular new song is predicted to sell at a rate of million purchases per day. In fact, it sells 1.85 million copies by the end of the first day alone! Use this information to answer the following questions:
(a) What equation, y(t), models the sales of this CD? Note: calculate C accurate to four decimal places.
(b) Approximately how many songs will have been sold exactly 730 days (2 years) after release? Note: round your answer to four decimal places.
69. By ignoring any standards of cleanliness, and choosing to live a life of squalor, you have inadvertently invented a new kind of chemical weapon forged out of soggy Cheetos, stagnant milk-filled cereal bowls, and a chocolate Easter bunny of indeterminate age.
Here’s the downside. The government has quarantined you inside your filthy house until the nasty mixture disintegrates a bit. In a truly disturbing development, they’ve determined that (like nuclear waste), your food weapon has a half-life, and they’re reasonably sure the half-life is four days.
Assuming this is true, how long will it take the 3,000 grams of dangerous procrastination-produced glop to decay to a safer (but equally stinky) 10 grams?
Chapter 18
70. Estimate the value of without a calculator by using a linear approximation tocentered at x = 9.
71. Draw the slope field for by calculating slopes at each point indicated in the following coordinate plane:
Sketch the specific solution to the differential equation that contains the point (0,1).
72. If you begin at the point and proceed units to the right along a line with slope , what are the coordinates of your destination?
73. Use Euler’s Method with three steps of width to approximate if , given that the solution graph passes through (–1,1).
Solutions
Chapter 2
1. 4x + 10y = 9
2. x + 2y = 1
3.
4. 5x + 3y = 7
5.
6. 2(4x + 7y)(4x – 7y)
7.
8. x = –16 or x = 16
9. x = –1 or x = –
Chapter 3
10. –3
11. no symmetry
12.
13. 5
14.
Chapter 4
15.
16.
17. sec2θ(1 + tanθ)(1 – tanθ)
18.
19. –16
20. –1
21a. does not exist because , but ∞ is not a finite number
21b. 4
21c. does not exist because
21d. 0
Chapter 6
22a.
22b. 4a2 – 6a + 1
23a.
23b.
24.
25. does not exist
26a.
26b. , which means it does not exist
26c. 0
26d. : split the two terms into separate limits, apply factoring and substitution methods, and add the results
27a. –1, because
27b. does not exist, because the graph of y = tan x has a vertical asymptote at x = π/2 (see Figure 4.4)
Chapter 7
28. Because is continuous at x = 4
29. c = 2
30. (infinite discontinuity) and x = 7 (point discontinuity)
31a. yes, because f(0) = 4, f(5) = 19, and 4 ≤ 10 ≤ 19
31b. no, because 20 does not fall between the function values of the endpoints f(0) = 4 and f(5) = 19
Chapter 8
32. f ′(x) = 3x2 – 2, f ′(–3) = 25
33. g′(1) = –2
Chapter 9
34a.
34b. 81x2 – 30x + 36
34c.
34d. 10 (x2 – 7x + 2)9 (2x – 7)
34e. . Note: use the Product Rule and take the derivative of (2x – 3)4 with the Chain Rule
35a. 42
35b. 60
36.
37b. Apply the Quotient Rule:
Chapter 10
38. y = –π2x + π3. Note: f(π) = 0 and f ′(π) = – π2
39.
40. j ′(x) = –tan x sin x + cos x sec2 x, so j ′(0) = 0 + 1 = 1, which means tangent and normal slopes are both 1 (as 1 is the reciprocal of itself)
41.
42.
43.
Chapter 11
44. f(x) changes from increasing to decreasing at , because f′(x) changes from positive to negative there; similarly, f(x) changes from decreasing to increasing at , because f′(x) changes from negative to positive there
45. g(x) decreases on (–4,–3) and (1,5)
46. maximum = 0, minimum = –9
47.
48. is a relative minimum because is a relative maximum because g"(–6) = –23 < 0.
Chapter 12
49a. t = 0.2596, t = 1.3756, and t = 1.7194 seconds
49b.
49c. 0.6739 in/sec
49d. (0,0.3962) and (1.1746,1.9670)
50a. v(t) = s′(t) = –32t + 25, so
v(1) = –32(1) + 25 = –7 ft/sec (the ball is falling at a rate of 7 ft/sec, so this tells you the ball has already reached its maximum height before t = 1)
50b. v(t) = 0 when –32t + 25 = 0, or
50c. maximum height: s(0.78125) ≈ 12.7656 feet
50d. Solve equation –16t2 + 25t + 3 = 0 for t and select only the positive solution: t ≈ 1.674 seconds.
Chapter 13
51.
52.
53. x = 0
54. After 15 minutes, Erin has traveled miles and Sara has traveled miles, so allow those to be the lengths of two sides of the right triangle illustrated here:
Apply the Pythagorean Theorem to determine that the length D of the hypotenuse, the distance between them, is 3.125 miles and then differentiate the Pythagorean Theorem with respect to t:
Solve for to get 12.5 miles per hour.
55. w = 625 feet, l = 2500 – 2(625) = 1250 feet. Note: pasture perimeter is 2w + l = 2500 so l = 2500 – 2w, plug this into the primary equation A = lw to get A = (2500 – 2w)w and optimize
Chapter 14
56a.
56b.
56c.
56d.
Chapter 15
57.
58.
59.
60. 3y 2 cos y3
61a. .
61b.
61c.
Chapter 16
62.
63.
64a.
64b.
65a.
65b.
66a. . Note: you may need to enter sec4 x as (1/cos(x))4
66b.
Chapter 17
67.
68a. . Note: and y(1) = 1.85, so you need to solve the equation 1.85 = for C
68b. y(730) ≈ 4.5831, so 4,583,100 copies sold in two years
69. 32.915 days. Note: y(t) = 3000e –0.173287t
Chapter 18
70. 3.01667. Note: linear approximation is
71.
72.
73. . Note: the coordinates of the three steps are ,