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C H A P T E R 2 Differentiation Section 2.1

The Derivative and the Tangent Line Problem . . . . . 95

Section 2.2

Basic Differentiation Rules and Rates of Change . . 109

Section 2.3

Product and Quotient Rules and Higher-Order Derivatives . . . . . . . . . . . . . . . 120

Section 2.4

The Chain Rule . . . . . . . . . . . . . . . . . . . . 134

Section 2.5

Implicit Differentiation . . . . . . . . . . . . . . . . 147

Section 2.6

Related Rates . . . . . . . . . . . . . . . . . . . . . 161

Review Exercises

. . . . . . . . . . . . . . . . . . . . . . . . . . 173

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

C H A P T E R Differentiation Section 2.1

2

The Derivative and the Tangent Line Problem

1. (a) At 共x1 , y1兲, slope ⫽ 0.

2. (a) At 共x1 , y1兲, slope ⬇ 32.

At 共x2 , y2兲, slope ⬇ 25.

At 共x2 , y2兲, slope ⬇ ⫺ 52.

5 (b) At 共x1 , y1兲, slope ⬇ ⫺ 2 .

(b) At 共x1 , y1兲, slope ⬇ 34.

At 共x2 , y2兲, slope ⬇ 2.

3. (a), (b) y

f ) 4) 4

f )1) )x 1

5

At 共x2 , y2兲, slope ⬇ 4 .

f )1)

1)

x

1

f ) 4)

5 )4, 5)

4

f ) 1)

f )4)

3

3 2

f )1) ) 1, 2)

2

1

x 1

(c) y ⫽ ⫽

5⫺2 f 共4 兲 ⫺ f 共1 兲 ⫽ ⫽1 4⫺1 3 5 ⫺ 4.75 f 共4 兲 ⫺ f 共3 兲 ⬇ ⫽ 0.25 4⫺3 1

y 6 5

4. (a)

2

3

4

5

6

f共4兲 ⫺ f 共1 兲 共 x ⫺ 1兲 ⫹ f 共1) 4⫺1

Thus,

f 共4兲 ⫺ f 共 1兲 f 共4兲 ⫺ f 共3兲 > . 4⫺1 4⫺3

(b) The slope of the tangent line at共1, 2兲 equalsf ⬘共1兲. This slope is steeper than the slope of the line through 共1, 2 兲 and 共 4, 5兲. Thus, f 共4 兲 ⫺ f 共1 兲 < f ⬘共1兲. 4⫺1

3 共x ⫺ 1兲 ⫹ 2 3

⫽ 1 共x ⫺ 1兲 ⫹ 2 ⫽x⫹1 5. f 共x兲 ⫽ 3 ⫺ 2x is a line. Slope ⫽ ⫺2

7. Slope at 共 1, ⫺ 3 兲 ⫽ ⌬lim x 0

g共 1 ⫹ ⌬x兲 ⫺ g共1 兲 ⌬x

共1 ⫹ ⌬x 兲2 ⫺ 4 ⫺ 共⫺3兲 ⌬x 1 ⫹ 2 共⌬x兲 ⫹ 共⌬x兲2 ⫺ 1 ⫽ ⌬lim x 0 ⌬x 关 共 兲兴 ⌬x ⫽ ⌬lim 2 ⫹ ⫽2 x 0 ⫽ ⌬lim x 0

6. g 共x兲 ⫽ 23 x ⫹ 1 is a line. Slope ⫽ 2

3

8. Slope at 共2, 1兲 ⫽ ⌬lim x 0

g共 2 ⫹ ⌬x兲 ⫺ g共2 兲 ⌬x

⫽ ⌬lim x 0

5 ⫺ 共2 ⫹ ⌬x兲 2 ⫺ 1 ⌬x

⫽ ⌬lim x 0

5 ⫺ 4 ⫺ 4共 ⌬x 兲 ⫺ 共⌬x兲2 ⫺ 1 ⌬x

共⫺4 ⫺ ⌬x 兲 ⫽ ⫺4 ⫽ ⌬lim x 0

9. Slope at 共 0, 0兲 ⫽ ⌬tlim0

f 共 0 ⫹ ⌬t兲 ⫺ f 共0 兲 ⌬t

10. Slope at 共⫺2, 7兲 ⫽ ⌬tlim0

h 共⫺2 ⫹ ⌬t 兲 ⫺ h 共⫺2兲 ⌬t

⫽ ⌬tlim0

3共 ⌬t兲 ⫺ 共⌬t 兲2 ⫺ 0 ⌬t

⫽ ⌬tlim0

共⫺2 ⫹ ⌬t 兲2 ⫹ 3 ⫺ 7 ⌬t

⫽ ⌬tlim0

4 ⫺ 4共⌬t兲 ⫹ 共⌬t 兲2 ⫺ 4 ⌬t

⫽ ⌬tlim0 共3 ⫺ ⌬t兲 ⫽ 3

⫽ ⌬tlim0 共⫺4 ⫹ ⌬t 兲 ⫽ ⫺4

95

96

Chapter 2

Differentiation

11. f 共 x兲 ⫽ 3 f⬘ 共x兲 ⫽ lim ⌬x 0 ⫽ ⌬lim x 0

13. f 共x 兲 ⫽ ⫺5x

12. g共x 兲 ⫽ ⫺5 f 共x ⫹ ⌬x 兲 ⫺ f 共x兲 ⌬x

g⬘共x兲 ⫽ lim ⌬x 0

3⫺3 ⌬x

0⫽0 ⫽ ⌬lim x 0

g共x ⫹ ⌬x兲 ⫺ g共x兲 ⌬x ⫺5 ⫺ 共⫺5兲 ⌬x

⫽ ⌬lim x 0

⫽ ⌬lim x 0

0 ⫽0 ⌬x

⫽ ⌬lim ⫺5 ⫽ ⫺5 x 0

f 共 x ⫹ ⌬x兲 ⫺ f 共 x兲 ⌬x

h ⬘共s兲 ⫽ ⌬slim0

关3 共x ⫹ ⌬x 兲 ⫹ 2 兴 ⫺ 关3x ⫹ 2兴 ⌬x

⫽ ⌬slim0

⫽ ⌬lim x 0

3⌬x ⌬x

⫽ ⌬slim0

3⫽3 ⫽ ⌬lim x 0

1 16. f 共x 兲 ⫽ 9 ⫺ x 2

⫽ ⌬lim x 0

f 共 x ⫹ ⌬x兲 ⫺ f 共 x兲 ⌬x

关9 ⫺ 共1兾2 兲共x ⫹ ⌬x 兲兴 ⫺ 关9 ⫺ 共1兾2兲 x兴 ⌬x

冢 12冣 ⫽ ⫺12

⫽ ⌬lim ⫺ x 0

17. f 共x兲 ⫽ 2 x 2 ⫹ x ⫺ 1 f⬘共 x兲 ⫽ ⌬lim x 0

f 共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x

⫽ ⌬lim x 0

关 2共x ⫹ ⌬x兲 2 ⫹ 共 x ⫹ ⌬x 兲 ⫺ 1兴 ⫺ 关2x 2 ⫹ x ⫺ 1 兴 ⌬x

⫽ ⌬lim x 0

共 2x2 ⫹ 4x ⌬x ⫹ 2共 ⌬x兲 2 ⫹ x ⫹ ⌬x ⫺ 1兲 ⫺ 共2x 2 ⫹ x ⫺ 1 兲 ⌬x

⫽ ⌬lim x 0

4x ⌬x ⫹ 2 共⌬x 兲2 ⫹ ⌬x ⫽ ⌬lim 共4x ⫹ 2 ⌬x ⫹ 1 兲 ⫽ 4x ⫹ 1 x 0 ⌬x

18. f 共 x兲 ⫽ 1 ⫺ x 2 f ⬘共x 兲 ⫽ lim ⌬x 0

f 共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x

⫽ ⌬lim x 0

关1 ⫺ 共 x ⫹ ⌬x兲2兴 ⫺ 关1 ⫺ x 2兴 ⌬x

⫽ ⌬lim x 0

1 ⫺ x 2 ⫺ 2x ⌬x ⫺ 共⌬x兲2 ⫺ 1 ⫹ x2 ⌬x

⫽ ⌬lim x 0

⫺2x ⌬x ⫺ 共⌬x 兲 2 ⫽ lim 共 ⫺2x ⫺ ⌬x兲 ⫽ ⫺2x ⌬x 0 ⌬x

⫺5共x ⫹ ⌬x 兲 ⫺ 共⫺5x 兲 ⌬x

2 15. h 共s兲 ⫽ 3 ⫹ s 3

⫽ ⌬lim x 0

f ⬘共x兲 ⫽ lim ⌬x 0

f 共x ⫹ ⌬x兲 ⫺ f 共x 兲 ⌬x

⫽ ⌬lim x 0

14. f 共x兲 ⫽ 3x ⫹ 2 f ⬘共x兲 ⫽ ⌬lim x 0

f ⬘共x 兲 ⫽ lim ⌬x 0

h共s ⫹ ⌬s 兲 ⫺ h共s 兲 ⌬s 3 ⫹ 3共s ⫹ ⌬s兲 ⫺ 共 3 ⫹ 32s兲 ⌬s 2

2 3

⌬s 2 ⫽ ⌬s 3

Section 2.1

The Derivative and the Tangent Line Problem

19. f 共 x兲 ⫽ x 3 ⫺ 12x f 共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x

f ⬘共x 兲 ⫽ lim ⌬x 0

关共x ⫹ ⌬x 兲3 ⫺ 12共x ⫹ ⌬x 兲兴 ⫺ 关 x 3 ⫺ 12x 兴 ⌬x x 3 ⫹ 3x 2 ⌬x ⫹ 3x共 ⌬x 兲2 ⫹ 共⌬x 兲3 ⫺ 12x ⫺ 12 ⌬x ⫺ x 3 ⫹ 12x ⫽ ⌬lim x 0 ⌬x 2 2 3 共 兲 共 兲 3x ⌬x ⫹ 3x ⌬x ⫹ ⌬x ⫺ 12 ⌬x ⫽ ⌬lim x 0 ⌬x 2 共3x ⫹ 3x ⌬x ⫹ 共⌬x 兲2 ⫺ 12兲 ⫽ 3x2 ⫺ 12 ⫽ ⌬lim x 0 ⫽ ⌬lim x 0

20. f 共x 兲 ⫽ x 3 ⫹ x2 f 共x ⫹ ⌬x 兲 ⫺ f 共x兲 ⌬x

f ⬘共x兲 ⫽ ⌬lim x 0

关共 x ⫹ ⌬x兲3 ⫹ 共x ⫹ ⌬x兲2 兴 ⫺ 关x 3 ⫹ x2兴 ⌬x 3 2 x ⫹ 3x ⌬x ⫹ 3x 共⌬x兲2 ⫹ 共⌬x 兲3 ⫹ x2 ⫹ 2x ⌬x ⫹ 共⌬x 兲2 ⫺ x3 ⫺ x 2 ⫽ ⌬lim x 0 ⌬x ⫽ ⌬lim x 0

⫽ ⌬lim x 0

3x2 ⌬x ⫹ 3x 共⌬x 兲2 ⫹ 共⌬x 兲3 ⫹ 2x ⌬x ⫹ 共⌬x 兲2 ⌬x

共 3x 2 ⫹ 3x ⌬x ⫹ 共⌬x 兲2 ⫹ 2x ⫹ 共⌬x 兲兲 ⫽ 3x 2 ⫹ 2x ⫽ ⌬lim x 0 21. f 共x兲 ⫽

1 x⫺1

f⬘共 x兲 ⫽ lim ⌬x 0

22. f 共x兲 ⫽ f 共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x

1 x2

f⬘共x 兲⫽ lim ⌬x 0

1 1 ⫺ 共x ⫹ ⌬x兲2 x2 ⫽ ⌬lim x 0 ⌬x

1 1 ⫺ x ⫹ ⌬x ⫺ 1 x ⫺ 1 ⫽ ⌬lim x 0 ⌬x

共 x ⫺ 1兲 ⫺ 共x ⫹ ⌬x ⫺ 1 兲 ⫽ ⌬lim x 0 ⌬x共x ⫹ ⌬x ⫺ 1 兲共x ⫺ 1 兲 ⫺⌬x ⫽ ⌬lim x 0 ⌬x 共x ⫹ ⌬x ⫺ 1 兲共x ⫺ 1 兲 ⫺1 ⫽ ⌬lim x 0 共 x ⫹ ⌬x ⫺ 1兲共 x ⫺ 1 兲

⫽ ⌬lim x 0

x 2 ⫺ 共x ⫹ ⌬x 兲2 ⌬x 共 x ⫹ ⌬x兲2x 2

⫽ ⌬lim x 0

⫺2x ⌬x ⫺ 共⌬x兲2 ⌬x 共x ⫹ ⌬x 兲2x 2

⫽ ⌬lim x 0

⫺2x ⫺ ⌬x 共x ⫹ ⌬x兲2 x 2

⫽ 1 ⫽⫺ 共x ⫺ 1兲2

⫺2x x4

⫽⫺

23. f 共x兲 ⫽ 冪x ⫹ 1 f ⬘共x兲 ⫽ ⌬lim x 0 ⫽ ⌬lim x 0 ⫽ ⌬lim x 0 ⫽ ⌬lim x 0

f 共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x 冪x ⫹ ⌬x ⫹ 1 ⫺ 冪x ⫹ 1

⌬x

⭈

冪

冪

⫹1⫹ x⫹1 冢冪xx ⫹⫹ ⌬x ⌬x ⫹ 1 ⫹ 冪x ⫹ 1冣

共x ⫹ ⌬x ⫹ 1兲 ⫺ 共x ⫹ 1 兲 ⌬x 关冪x ⫹ ⌬x ⫹ 1 ⫹ 冪 x ⫹ 1 兴 1 冪x ⫹ ⌬x ⫹ 1 ⫹ 冪x ⫹ 1

f 共x ⫹ ⌬x 兲 ⫺ f共x 兲 ⌬x

⫽

1 冪x ⫹ 1 ⫹ 冪x ⫹ 1

⫽

1 2冪 x ⫹ 1

2 x3

97

98

Chapter 2

24. f 共x兲 ⫽

Differentiation

4

25. (a) f 共x兲 ⫽ x2 ⫹ 1

冪x

f共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x 4 4 ⫺ 冪x ⫹ ⌬x 冪x ⫽ ⌬lim x 0 ⌬x

f⬘ 共x兲 ⫽ lim ⌬x 0

f ⬘共x兲 ⫽ ⌬lim x 0

冢

冪x ⫹ 冪x ⫹ ⌬x 4冪x ⫺ 4冪x ⫹ ⌬x ⭈ 冪 x ⫹ 冪x ⫹ ⌬x ⌬x冪x 冪x ⫹ ⌬x

⫽ ⌬lim x 0

4x ⫺ 4共x ⫹ ⌬x 兲 ⌬x冪 x冪x ⫹ ⌬x 共冪x ⫹ 冪 x ⫹ ⌬x兲

⫽ ⌬lim x 0

⫺4 冪x 冪x ⫹ ⌬x 共冪x ⫹ 冪 x ⫹ ⌬x兲 ⫺4

共

冪 x 冪x 冪x ⫹ 冪x

兲

⫽

关共 x ⫹ ⌬x兲2 ⫹ 1 兴 ⫺ 关x2 ⫹ 1 兴 ⌬x 2x ⌬x ⫹ 共⌬x兲2 ⫽ ⌬x lim0 ⌬x ⫽ ⌬x lim0 共 2x ⫹ ⌬x兲 ⫽ 2x ⫽ ⌬lim x 0

⫽ ⌬lim x 0

⫽

f 共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x

冣

At 共2, 5 兲, the slope of the tangent line ism ⫽ 2共2 兲 ⫽ 4. The equation of the tangent line is y ⫺ 5 ⫽ 4共x ⫺ 2 兲 y ⫺ 5 ⫽ 4x ⫺ 8

⫺2 x冪x

y ⫽ 4x ⫺ 3. (b)

8

(2, 5) 5

5 2

26. (a) f 共 x兲 ⫽ x2 ⫹ 2x ⫹ 1 f⬘共 x兲 ⫽ ⌬lim x 0

(b)

5

f 共 x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

⫽ ⌬lim x 0

关共x ⫹ ⌬x兲 2 ⫹ 2 共x ⫹ ⌬x兲 ⫹ 1兴 ⫺ 关x2 ⫹ 2x ⫹ 1 兴 ⌬x

⫽ ⌬lim x 0

2x ⌬x ⫹ 共⌬x兲 2 ⫹ 2 ⌬x ⌬x

( 3, 4)

6

3 1

共 2x ⫹ ⌬x ⫹ 2兲 ⫽ 2x ⫹ 2 ⫽ ⌬lim x 0 At 共⫺3, 4 兲, the slope of the tangent line ism ⫽ 2共⫺3 兲 ⫹ 2 ⫽ ⫺4. The equation of the tangent line is y ⫺ 4 ⫽ ⫺4 共x ⫹ 3 兲 y ⫽ ⫺4x ⫺ 8.

27. (a) f 共 x兲 ⫽ x3 f⬘共 x兲 ⫽ ⌬lim x 0

(b)

10

(2, 8)

f 共x ⫹ ⌬x 兲 ⫺ f 共x 兲 ⌬x 5

共x ⫹ ⌬x兲 3 ⫺ x 3 ⫽ ⌬lim x 0 ⌬x ⫽ ⌬lim x 0

3x2 ⌬x ⫹ 3x 共⌬x 兲2 ⫹ 共⌬x兲 3 ⌬x

共 3x 2 ⫹ 3x ⌬x ⫹ 共⌬x兲2 兲 ⫽ 3x 2 ⫽ ⌬lim x 0 At 共2, 8 兲, the slope of the tangent ism ⫽ 3共2 兲2 ⫽ 12. The equation of the tangent line is y ⫺ 8 ⫽ 12共 x ⫺ 2兲 y ⫽ 12x ⫺ 16.

5

4

Section 2.1 28. (a) f 共 x兲 ⫽ x3 ⫹ 1

The Derivative and the Tangent Line Problem (b)

f 共 x ⫹ ⌬x兲 ⫺ f 共x兲 f ⬘共x 兲 ⫽ lim ⌬x 0 ⌬x 关共x ⫹ ⌬x兲 3 ⫹ 1 兴 ⫺ 共x 3 ⫹ 1兲 ⫽ ⌬lim x 0 ⌬x x 3 ⫹ 3x 2 共 ⌬x兲 ⫹ 3x 共⌬x 兲2 ⫹ 共⌬x 兲 3 ⫹ 1 ⫺ x 3 ⫺ 1 ⫽ ⌬lim x 0 ⌬x 2 2兴 关 共 兲 共 兲 ⌬x ⫹ ⌬x 3x ⫹ 3x ⫽ 3x 2 ⫽ ⌬lim x 0

4

(1, 2) 6

6

4

At 共1, 2 兲, the slope of the tangent line ism ⫽ 3共 1兲 2 ⫽ 3. The equation of the tangent line is y ⫺ 2 ⫽ 3 共x ⫺ 1兲 y ⫽ 3x ⫺ 1. 29. (a) f 共x 兲 ⫽ 冪x f ⬘共x 兲 ⫽ lim ⌬x 0

(b)

3

f共 x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

(1, 1) 1

⫽ ⌬lim x 0 ⫽ ⌬lim x 0 ⫽ ⌬lim x 0

冪 x ⫹ ⌬x ⫺ 冪x

⌬x

5

冪x ⫹ ⌬x ⫹ 冪x

⭈冪

1

x ⫹ ⌬x ⫹ 冪x

共x ⫹ ⌬x 兲 ⫺ x ⌬x 共冪x ⫹ ⌬x ⫹ 冪 x兲 1 冪 x ⫹ ⌬x ⫹ 冪x

⫽

1 2 冪x

At 共1, 1 兲, the slope of the tangent line is m⫽

1 1 ⫽ . 2 冪1 2

The equation of the tangent line is y⫺1⫽ y⫽

1 共 x ⫺ 1兲 2 1 1 x⫹ . 2 2

30. (a) f 共x兲 ⫽ 冪x ⫺ 1 f ⬘共x 兲 ⫽ lim ⌬x 0 ⫽ ⌬lim x 0 ⫽ ⌬lim x 0 ⫽ ⌬lim x 0

(b)

(5, 2)

f 共x ⫹ ⌬x 兲 ⫺ f 共 x兲 ⌬x

冪x ⫹ ⌬x ⫺ 1 ⫺ 冪x ⫺ 1

⌬x

2

⭈

冢

冪x ⫹ ⌬x ⫺ 1 ⫹ 冪x ⫺ 1 冪x ⫹ ⌬x ⫺ 1 ⫹ 冪x ⫺ 1

共 x ⫹ ⌬x ⫺ 1兲 ⫺ 共x ⫺ 1兲 ⌬x共 冪x ⫹ ⌬x ⫺ 1 ⫹ 冪x ⫺ 1 兲 1 冪x ⫹ ⌬x ⫺ 1 ⫹ 冪x ⫺ 1

⫽

1 2冪x ⫺ 1

At 共5, 2 兲, the slope of the tangent line is 1 1 ⫽ 2 冪5 ⫺ 1 4 The equation of the tangent line is m⫽

1 共x ⫺ 5 兲 4 1 3 y⫽ x⫹ 4 4

y⫺2⫽

4

冣

10

4

99

100

Chapter 2

Differentiation

31. (a) f 共x兲 ⫽ x ⫹

4 x

(b)

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

f⬘共 x兲 ⫽ ⌬lim x 0

共 x ⫹ ⌬x 兲 ⫹ ⫽ ⌬lim x 0

(4, 5) 12

冢

4 4 ⫺ x⫹ x x ⫹ ⌬x ⌬x

冣

12

6

⫽ ⌬lim x 0

x 共x ⫹ ⌬x 兲共x ⫹ ⌬x兲 ⫹ 4x ⫺ x2 共x ⫹ ⌬x兲 ⫺ 4 共x ⫹ ⌬x 兲 x共 ⌬x兲共x ⫹ ⌬x兲

⫽ ⌬lim x 0

x 3 ⫹ 2x 2共 ⌬x兲 ⫹ x共⌬x 兲2 ⫺ x 3 ⫺ x 2共⌬x 兲 ⫺ 4 共⌬x 兲 x共⌬x 兲共x ⫹ ⌬x兲

⫽ ⌬lim x 0

x 2共⌬x兲 ⫹ x共 ⌬x 兲2 ⫺ 4 共⌬x 兲 x共 ⌬x兲共 x ⫹ ⌬x兲

⫽ ⌬lim x 0

x 2 ⫹ x 共⌬x兲 ⫺4 x 共x ⫹ ⌬x兲

⫽

10

4 x2 ⫺ 4 ⫽1⫺ 2 x x2

At 共4, 5 兲, the slope of the tangent line is m⫽1⫺

3 4 ⫽ . 16 4

The equation of the tangent line is 3 y ⫺ 5 ⫽ 共x ⫺ 4兲 4 3 y ⫽ x ⫹ 2. 4

32. (a) f 共x 兲 ⫽

1 x⫹1

f ⬘共x 兲 ⫽ lim ⌬x 0

33. From Exercise 27 we know that f⬘共x兲 ⫽ 3x2 . Since the slope of the given line is 3, we have f 共x ⫹ ⌬x兲 ⫺ f 共x 兲 ⌬x

1 1 ⫺ x ⫹ ⌬x ⫹ 1 x⫹1 ⫽ ⌬lim x 0 ⌬x ⫽ ⌬lim x 0

共x ⫹ 1 兲 ⫺ 共 x ⫹ ⌬x ⫹ 1兲 ⌬x 共x ⫹ ⌬x ⫹ 1 兲共x ⫹ 1兲

⫺ ⫽ ⌬lim x 0 ⫽⫺

1 共 x ⫹ ⌬x ⫹ 1兲共x ⫹ 1 兲

1 共x ⫹ 1兲 2

At 共0, 1 兲, the slope of the tangent line is m⫽

⫺1 ⫽ ⫺1. 共 0 ⫹ 1兲2

The equation of the tangent line isy ⫽ ⫺x ⫹ 1. (b)

3

(0, 1) 6

3

3

3x2 ⫽ 3 x ⫽ 1. Therefore, at the points共 1, 1 兲 and共 ⫺1, ⫺1兲 the tangent lines are parallel to3x ⫺ y ⫹ 1 ⫽ 0. These lines have equations y ⫺ 1 ⫽ 3共x ⫺ 1 兲 y ⫽ 3x ⫺ 2

and

y ⫹ 1 ⫽ 3共 x ⫹ 1兲 y ⫽ 3x ⫹ 2.

Section 2.1

The Derivative and the Tangent Line Problem

101

34. Using the limit definition of derivative,f⬘共 x兲 ⫽ 3x 2. Since the slope of the given line is 3, we have 3x2 ⫽ 3 x2 ⫽ 1 x ⫽

1.

Therefore, at the points共 1, 3兲 and 共⫺1, 1兲 the tangent lines are parallel to3x ⫺ y ⫺ 4 ⫽ 0. These lines have equations y ⫺ 3 ⫽ 3共x ⫺ 1 兲

and

y ⫺ 1 ⫽ 3共 x ⫹ 1兲

y ⫽ 3x

y ⫽ 3x ⫹ 4. 36. Using the limit definition of derivative,

35. Using the limit definition of derivative, f⬘ 共x兲 ⫽

⫺1 . 2x冪x

f⬘共x兲 ⫽

1 Since the slope of the given line is⫺ 2 , we have

1 Since the slope of the given line is⫺2 , we have

⫺

⫺1 . 2共 x ⫺ 1兲 3Ⲑ 2

⫺1 1 ⫽⫺ 2共x ⫺ 1 兲3Ⲑ2 2

1 1 ⫽⫺ 2 2x冪 x

1 ⫽ 共x ⫺ 1兲 3Ⲑ2

x ⫽ 1.

1⫽x⫺1

Therefore, at the point 共1, 1兲 the tangent line is parallel to x ⫹ 2y ⫺ 6 ⫽ 0. The equation of this line is

x ⫽ 2.

At the point 共2, 1 兲, the tangent line is parallel to x ⫹ 2y ⫹ 7 ⫽ 0. The equation of the tangent line is

1 y ⫺ 1 ⫽ ⫺ 共 x ⫺ 1兲 2

1 y ⫺ 1 ⫽ ⫺ 共 x ⫺ 2兲 2

1 1 y⫺1⫽⫺ x⫹ 2 2

1 y ⫽ ⫺ x ⫹ 2. 2

1 3 y⫽⫺ x⫹ . 2 2 37. f 共x兲 ⫽ x

f ⬘ 共x 兲 ⫽ 1

39. f 共x兲 ⫽ 冪x

f⬘共x兲

38. f 共x兲 ⫽ x 2

Matches (b).

40. f⬘ does not exist at x ⫽ 0. Matches (c).

Matches (a).

共decreasing slope as x

⬁兲

41. g 共5兲 ⫽ 2 because the tangent line passes through 共5, 2兲. g⬘共 5兲 ⫽

f⬘共 x兲 ⫽ 2x Matches (d).

42. h共⫺1 兲 ⫽ 4 because the tangent line passes through 共⫺1, 4 兲.

1 2 2⫺0 ⫽ ⫽⫺ 5 ⫺ 9 ⫺4 2

43. The slope of the graph off is 1 f ⬘共 x兲 ⫽ 1.

h⬘共⫺1 兲 ⫽

44. The slope of the graph of f is 0 f⬘ 共x兲 ⫽ 0.

y

1 6⫺4 2 ⫽ ⫽ 4 2 3 ⫺ 共 ⫺1兲 45. The slope of the graph of f is negative for x < 4, positive for x > 4, and 0 at x ⫽ 4.

y y

4

2 4

3 1

2

f

f x x

3

2

1

1

2

3

2

1

1

2

x 6

4

2

2 2

1 4

1 2

f

2

2

6 8

4

6

102

Chapter 2

Differentiation

46. The slope of the graph off is⫺1 for x < 4, 1 for x > 4, and undefined at x ⫽ 4.

y

y 3 2

48. Answers will vary. Sample answer: y ⫽ x

47. Answers will vary. Sample answer: y ⫽ ⫺x

y

4

4

3

3

2

2 1

1

f

x

1 4

x 1

2

3

4

5

3

2

1

6

49. f共x兲 ⫽ 5 ⫺ 3x and c ⫽ 1

1

1

x 4

4

3

3

4

4

51. f 共 x兲 ⫽ ⫺x 2 and c ⫽ 6

f 共x兲 ⫽ x3

3

f

1

8 x

3

6

1

3

2

4

2

2

56. (a) If f⬘共 c兲 ⫽ 3 and f is odd, then f⬘ 共⫺c兲 ⫽ f ⬘共c 兲 ⫽ 3. 57. Let 共x0, y0 兲 be a point of tangency on the graph of f. By the limit definition for the derivative,f ⬘ 共x兲 ⫽ 4 ⫺ 2x. The slope of the line through 共2, 5 兲 and 共x 0, y 0兲 equals the derivative of f atx0 :

4

(b) If f⬘共c兲 ⫽ 3 and f is even, then ⬘f 共⫺c兲 ⫽ ⫺f⬘共 c 兲 ⫽ ⫺3. 58. Let 共x0, y 0 兲 be a point of tangency on the graph of f. By the limit definition for the derivative,f⬘ 共x 兲 ⫽ 2x. The slope of the line through 共1, ⫺ 3兲 and共x0 , y0 兲 equals the derivative of f atx0: ⫺3 ⫺ y0 ⫽ 2x 0 1 ⫺ x0

5 ⫺ y0 ⫽ 共2 ⫺ x 0 兲共 4 ⫺ 2x0 兲

⫺3 ⫺ y 0 ⫽ 共1 ⫺ x 0 兲2x0

5 ⫺ 共4x0 ⫺ x 02 兲 ⫽ 8 ⫺ 8x 0 ⫹ 2x 02

⫺3 ⫺ x 02 ⫽ 2x 0 ⫺ 2x 02

0 ⫽ x02 ⫺ 4x 0 ⫹ 3

x 02 ⫺ 2x 0 ⫺ 3 ⫽ 0 x 0 ⫽ 1, 3

Therefore, the points of tangency are共1, 3 兲 and共3, 3兲, and the corresponding slopes are 2 and⫺2 . The equations of the tangent lines are: y ⫺ 5 ⫽ 2共x ⫺ 2 兲

y ⫺ 5 ⫽ ⫺2 共x ⫺ 2兲

y ⫽ 2x ⫹ 1

共x0 ⫺ 3 兲共x 0 ⫹ 1 兲 ⫽ 0

x 0 ⫽ 3, ⫺1

Therefore, the points of tangency are共3, 9 兲 and(⫺1, 1兲 , and the corresponding slopes are 6 and⫺2. The equations of the tangent lines are: y ⫹ 3 ⫽ 6共x ⫺ 1 兲

y ⫽ ⫺2x ⫹ 9

y ⫹ 3 ⫽ ⫺2共x ⫺ 1兲

y ⫽ 6x ⫺ 9

y ⫽ ⫺2x ⫺ 1

y y 7 10

6

(2, 5)

6

(3, 3)

4

(1, 3) ( 1, 1)

1 2

(3, 9)

8

4 2

1

2

3

6

x

x

6

5 ⫺ y0 ⫽ 4 ⫺ 2x 0 2 ⫺ x0

0 ⫽ 共x 0 ⫺ 1 兲共 x0 ⫺ 3 兲

3

3

x 6

2

2

2

f

3

1 1

4

2

f

2

10

2

4

y

12

1

3

52. f 共 x兲 ⫽ 2 冪x and c ⫽ 9

y

1

2

55. f 共0兲 ⫽ 0; f ⬘共0 兲 ⫽ 0; f ⬘共x兲 > 0 if x⫽0

f 共x 兲 ⫽ x 2 ⫹ 4

2

5

1 2

y

3

2

3

54. f 共0 兲 ⫽ 4, f⬘共0 兲 ⫽ 0; f ⬘共x兲 < 0 for x < 0, f⬘ 共x兲 > 0 for x > 0

f 共x兲 ⫽ ⫺3x ⫹ 2

2

3

2

50. f 共x 兲 ⫽ x 3 and c ⫽ ⫺2

53. f共0 兲 ⫽ 2 and f⬘共x 兲 ⫽ ⫺3, ⫺⬁ < x < ⬁

3

2

8

6

4

2

2 2 4

4

(1, 3)

6

x

Section 2.1

The Derivative and the Tangent Line Problem

59. (a) g⬘共0 兲 ⫽ ⫺3 (b) g⬘ 共3兲 ⫽ 0 (c) Because g ⬘共1兲 ⫽ ⫺ 3 , g is decreasing (falling) atx ⫽ 1. 8

(d) Because g⬘共⫺4兲 ⫽ 37 , g is increasing (rising) atx ⫽ ⫺4. (e) Because g ⬘共4兲 and g⬘共6 兲 are both positive,g共 6兲 is greater thang 共 4兲 , andg共6 兲 ⫺ g共4 兲 > 0. (f) No, it is not possible. All you can say is that g is decreasing (falling) at x ⫽ 2.

60. (a) f 共x兲 ⫽ x 2

(b) g ⬘共x兲 ⫽ ⌬lim x 0

f⬘ 共x兲 ⫽ ⌬lim x 0

f 共x ⫹ ⌬x 兲 ⫺ f 共x兲 ⌬x

⫽ ⌬lim x 0

共x ⫹ ⌬x 兲2 ⫺ x 2 ⌬x

⫽ ⌬lim x 0

x 2 ⫹ 2x 共⌬x 兲 ⫹ 共⌬x 兲2 ⫺ x2 ⌬x

⫽ ⌬lim x 0

⌬x 共2x ⫹ ⌬x 兲 ⌬x

⫽ ⌬lim x 0

共x ⫹ ⌬x兲3 ⫺ x 3 ⌬x

⫽ ⌬lim x 0

x 3 ⫹ 3x 2共 ⌬x 兲 ⫹ 3x共 ⌬x兲2 ⫹ 共 ⌬x兲3 ⫺ x 3 ⌬x

⫽ ⌬lim x 0

⌬x 共3x 2 ⫹ 3x 共 ⌬x兲 ⫹ 共⌬x 兲2 兲 ⌬x

共3x 2 ⫹ 3x 共 ⌬x兲 ⫹ 共⌬x 兲 2兲 ⫽ 3x2 ⫽ ⌬lim x 0

共2x ⫹ ⌬x兲 ⫽ 2x ⫽ ⌬lim x 0

At x ⫽ ⫺1, g⬘共 ⫺1兲 ⫽ 3 and the tangent line is

At x ⫽ ⫺1, f⬘ 共⫺1 兲 ⫽ ⫺2 and the tangent li...