Mama

Section 1.1

1

Chapter 1 Prerequisites for Calculus
s Section 1.1 Lines (pp. 1–9)
Quick Review 1.1
1. y 2. 3 3 2x x 3. m 4. m 2 3 3 2 1
2 5 2 3 3 4 ( 3) ( 1) 1 1 5 4 1 4

Section 1.1 Exercises
1. 2. 3. x y x y x y x y 1 0 2 1 1 1 2 8 1 0 4 1 2 2 3

( 3) 2 2 4 ( 3) 0 0 6
y 5

5

4(3 2(x 2x

3) 1) 2

2

4(0)

2

0

2 4.

5. (a, c)

1
B 5 A x

5. (a) 3(2) 6 (b)3(3) 6. (a) 7 7 (b) 1 1 7. d

4

5 5 5 5 Yes (b) m No 6. (a, c)
5

1 4( 1) 13

1 2

( 2) 1

3 1
y

3

2( 1) 5 2 5 Yes 9 (x2 (0 2 x1)2 2)2 y1)2
2

2( 2) No x1)2 1)2

5
x

(y2 (1

y1)2 0)2

5 A B

8. d =

(x2 (1

(y2
1 3 4 2 3

(b) m 7. (a, c)

2 ( 1) 1 ( 2)
y 5 B

1 3

1 3

1

( 1)2 1
25 9 5 3 16 9

A 5 x

9. 4x

3y 3y y

7 4x
4 x 3 7 3(b) m 7 8. (a, c)

3 3 1 2

0 3
y 5 A

0

10.

2x

5y 5y y

3 2x
2 x 5

3
3 5
B

5

x

2

Section 1.1
26. The line contains (0, 0) and (5, 2).
3 2 1 1 5 (undefined) 0

8. continued (b) m

m y 27. 3x

This line has no slope. 9. (a) x (b) y 10. (a) x (b) y 11. (a) x (b) y 12. (a) x (b) y 13. y 14. y y 15. y 16. y y 17. m y y 2y 3x 18. m y y 19. m 1(x 1[x 1(x 2(x2[x 2(x 0) 0 1) 1 ( 1)] 1 1) 1 3 ( 4)] 0 4) 0
4 3

2 3 1 0 2

2 5 2 x 5

0 0

2 5

4y 3x
3 x 4

12 12 3
3 4

4y y

(a) Slope: (c)

(b) y-intercept: 3

[ 10, 10] by [ 10, 10]

28. x y

y x

2 2 1

3 0 3 2 0 2 3 (x 0) 0 2 3 x 2

(a) Slope: (c)

(b) y-intercept: 2

3x 2y
1 2 1 1

0
0 1
[ 10, 10] by [ 10, 10]

0 29. 1

0(x 1
2 2

1)

x 3 y 4

y 4 x3 4 x 3

1 1 4
4 3

0 ( 2)

2 (undefined) 0

y

Vertical line: x 20. m y 4y 4y 3x 21. y 22. y 23. y 24. y
2 2 1 ( 2)

2
3 4 3 4

(a) Slope:

(b) y-intercept: 4 (c)

3 [x 4

( 2)] 2) 2 2 2 2 or y 3 1 4

1

3(x 3x 4y 3x 1x
1 x 2 1 x 3

[ 10, 10] by [ 10, 10]

30. y

2x

4

(a) Slope: 2 x 2 (b) y-intercept: 4 (c)

[ 10, 10] by [ 10, 10]

25. The linecontains (0, 0) and (10, 25). m y
25 10 5 x 2 0 0 25 10 5 2

Section 1.1
31. (a) The desired line has slope (0, 0): y 1(x 0) 0 or y (b) The desired line has slope (0, 0): y 1(x 0) 0 or y x. 1 and passes through x.
1 1

3

39. (a) y

0.680x

9.013

1 and passes through

(b) The slope is 0.68. It represents the approximate average weight gain in pounds per month. (c)

32. (a) The givenequation is equivalent to y 2x 4. The desired line has slope 2 and passes through ( 2, 2): y 2(x 2) 2 or y 2x 2. (b) The desired line has slope ( 2, 2): y
1 (x 2 1 2 1 and passes through 2

[15, 45] by [15, 45]

(d) When x 30, y 0.680(30) She weighs about 29 pounds. 40. (a) y 1,060.4233x

9.013

29.413.

2,077,548.669

2)

2 or y

1 x 2

3.

(b) The slope is 1,060.4233. Itrepresents the approximate rate of increase in earnings in dollars per year. (c)

33. (a) The given line is vertical, so we seek a vertical line through ( 2, 4): x 2. (b) We seek a horizontal line through ( 2, 4): y 4.

34. (a) The given line is horizontal, so we seek a horizontal line through 1,
1 :y 2 1 . 2 1 1, : x 2

[1975, 1995] by [20,000, 35,000]

(b) We seek a vertical line through35. m f(x)
9 3 2 1 7 (x 2 7 2

1.

(d) When x 2000, y 1,060.4233(2000) 2,077,548.669 43,298. In 2000, the construction workers’ average annual compensation will be about $43,298. 41. y 1 (x 3) 4 y x 3 4 y x 1 This is the same as the equation obtained in Example 5.

1)

2

Check: f(5) Since f(x) 36. m

7 (5) 2 7 x 2 3 , 2

7 3 x 2 2 3 16, as expected. 2 7 we have m and b 2

3 . 242. (a) When y When x (b) When y When x

0, we have

4 ( 1) 3 3 4 2 2 2 3 3 f(x) (x 2) ( 1) x 2 2 2 3 Check: f(6) (6) 2 7, as expected. 2 3 3 Since f(x) x 2, we have m and b 2 2 2 3 2 (6) 3 y 4 3 ( 2)

x c y 0, we have d x 0, we have c y 0, we have d

1, so x 1, so y 2, so x 2, so y

c. d. 2c. 2d.

2.

The x-intercept is 2c and the y-intercept is 2d. 43. (a) The given equations...