Textbook Chapter 1

56 Chapter 1 Prerequisites for Calculus
Chapter 1 Review Exercises
10. y 5 3 x 1 9
3
The collection of exercises marked in red could be used as a chapter In Exercises 39 and 40, write a piecewise formula for the function.
12. with x-intercept 3 and y-intercept 5 y 5 3 x 5
find the (b) domain and (c) range of each.
13. the line y f (x), where ƒ has the following values: y 1 2 x 3 43. f (x) 2 x 2 , g(x) x 2
44. f (x) x, g(x) 1 x
x 2 2 4
In Exercises 45–48, a parametrization is given for a curve.
f x 4 2 1
(a) Graph the curve. Identify the initial and terminal points, if
test.
In Exercises 1–14, write an equation for the specified line.
39. y
40. y
1. through (1, 6) with slope 3 y 3x 9
1
5
(2, 5)
2. through (1, 2) with slope 12 y 1 2 x 3 2
3. the vertical line through (0, 3) x 0
x
0 1 2
4. through (3, 6) and (1, 2) y 2x
x
5. the horizontal line through (0, 2) y 2
f (x) { 1x, 0 x 1
0 4
6. through (3, 3) and (2, 5) y 2 2 x, 1 x 2
5 x 2 1
In Exercises 41 and 42, find
5
7. with slope 3 and y-intercept 3 y 3x 3
(a) ( f g)(1) (b) (g f )(2) (c) ( f f )(x) (d) (g g)(x)
8. through (3, 1) and parallel to 2x y 2 y 2x 5
9. through (4, 12) and parallel to 4x 3y 12 y 4 3 x 2 0 41. f (x) 1
x , g(x) 1
(a) 1
(c) x, x 0
x 1
(b)
2 2.5 2 5 1
(d)
3
1 x 2
2
10. through (2, 3) and perpendicular to 3x 5y 1
42. f (x) 2 x, g(x) 3 x 1
11. through (1, 2) and perpendicular to 1 2 x 1 3 y 1 y 2 3 x 8 3 (a) 2 (b) 1 (c) x (d) 3 3 x 1
1
In Exercises 43 and 44, (a) write a formula for ƒ g and g ƒ and
any. Indicate the direction in which the curve is traced.
14. through (4, 2) with x-intercept 3 y 2 7 x 6 7
(b) Find a Cartesian equation for a curve that contains the parametrized
In Exercises 15–18, determine whether the graph of the function is
symmetric about the y-axis, the origin, or neither.
curve. What portion of the graph of the Cartesian
equation is traced by the parametrized curve?
15. y x 15 Origin 16. y x 25 y-axis
45. x 5 cos t, y 2 sin t, 0 t 2p
17. y x 2 2x 1 Neither 18. y e x 2 y-axis
46. x 4 cos t, y 4 sin t, p2 t 3p2
In Exercises 19–26, determine whether the function is even, odd, or
neither.
19. y x 2 1 Even 20. y x 5 x 3 x Odd
21. y 1 cos x Even 22. y sec x tan x Odd
4
1
23. y x
x3
Odd
2x
24. y 1 sin x Neither
25. y x cos x Neither 26. y x 4 1 Even
In Exercises 27–38, find the (a) domain and (b) range, and (c) graph
the function.
27. y x 2 28. y 2 1 x
29. y 16 x 2 30. y 3 2x 1
31. y 2e x 3 32. y tan (2x p)
33. y 2 sin (3x p) 1 34. y x 25
35. y ln (x 3) 1 36. y 1 3 2 x
x, 4 x 0
37. y { x, 0 x 4
x 2, 2 x 1
38. y x, 1 x 1
{
x 2, 1 x 2
47. x 2 t, y 11 2t, 2 t 4
48. x 1 t, y 4 2t, t 2
In Exercises 49–52, give a parametrization for the curve.
49. the line segment with endpoints (2, 5) and (4, 3)
Possible answer: x 2 6t, y 5 2t, 0 t 1
50. the line through (3, 2) and (4, 1)
Possible answer: x 3 7t, y 2 t, t
51. the ray with initial point (2, 5) that passes through (1, 0)
Possible answer: x 2 3t, y 5 5t, 0 t
52. y x (x 4), x 2
Possible answer: x t, y t(t 4), t 2
Group Activity In Exercises 53 and 54, do the following.
(a) Find f 1 and show that ( f f 1 )(x) ( f 1 f )(x) x.
(b) Graph ƒ and ƒ 1 in the same viewing window.
53. f (x) 2 3x 54. f (x) (x 2) 2 , x 2
In Exercises 55 and 56, find the measure of the angle in radians and
degrees.
0.6435 radians or 36.8699°
55. sin 1 (0.6)
1.1607 radians or 66.5014°
56. tan 1 (2.3)
57. Find the six trigonometric values of u cos 1 (37). Give exact
answers.
58. Solve the equation sin x 0.2 in the following intervals.
(a) 0 x 2p (b) x
59. Solve for x: e 0.2x 4 x 5 ln 4
58. (a) x 3.3430 and x 6.0818
(b) x 3.3430 2kp and x 6.0818 2kp, k any integer