Solution Manual For Calculus 12th Edition by
Ron Larson (Author), Bruce H. Edwards
(Author)
1. Evaluate (\displaystyle \lim_{x \to 2} \frac{x^2 - 4}{x-2})
Solution: Factor numerator: ((x-2)(x+2)/(x-2) = x+2). Limit as (x\to2) is (4).
2. Evaluate (\displaystyle \lim_{x \to 0} \frac{\sin 3x}{x})
Solution: Use (\lim_{x\to0} \frac{\sin x}{x} = 1):
(\lim_{x\to0} \frac{\sin 3x}{x} = 3).
3. (\displaystyle \lim_{x \to \infty} \frac{4x^2 + 7}{2x^2 - 5x} )
Solution: Divide top and bottom by (x^2): (\frac{4 + 7/x^2}{2 - 5/x} \to 2)
4. Determine (\displaystyle \lim_{x \to 1} \frac{x^3 -1}{x-1})
Solution: Factor numerator: ((x-1)(x^2+x+1)/(x-1) = x^2+x+1). Limit: (1+1+1=3)
5. (\displaystyle \lim_{x \to 0} \frac{e^x -1}{x})
Solution: Standard limit: (1)
Derivatives
6. Find (\frac{d}{dx} (5x^3 - 2x + 7))
Solution: (15x^2 - 2)
7. Find (\frac{d}{dx} (\sqrt{x^5 + 1}))
Solution: Chain rule: (\frac{1}{2\sqrt{x^5+1}} \cdot 5x^4 = \frac{5x^4}{2\sqrt{x^5+1}})
8. Derivative of (f(x) = x^2 \cos x)
Solution: Product rule: (2x \cos x - x^2 \sin x)
9. (\frac{d}{dx} \frac{2x+3}{x-1})
Solution: Quotient rule: (\frac{(2)(x-1)-(2x+3)(1)}{(x-1)^2} = \frac{-5}{(x-1)^2})
10. Find (\frac{dy}{dx}) if (y^2 + x^2 = 49)
Solution: (2y \frac{dy}{dx} + 2x = 0 \implies \frac{dy}{dx} = -\frac{x}{y})
,Integrals
11. (\int (3x^2 - 4x + 1) dx)
Solution: (x^3 - 2x^2 + x + C)
12. (\int_0^2 (4x - 1) dx)
Solution: (2x^2 - x \Big|_0^2 = 8-2 = 6)
13. (\int \frac{1}{x} dx)
Solution: (\ln|x| + C)
14. (\int e^{3x} dx)
Solution: (\frac{1}{3} e^{3x} + C)
15. (\int \sin 2x dx)
Solution: (-\frac{1}{2} \cos 2x + C)
Applications
16. A balloon rises at 5 m/s. Find its height after 10 s.
Solution: (h = 5t = 50) m
17. A rectangle has area 50. Express perimeter as a function of width (x).
Solution: Length (L = 50/x), perimeter (P = 2(x + 50/x) = 2x + 100/x)
18. Find the slope of (y = x^3 - 6x^2 + 9x) at (x=2)
Solution: (y' = 3x^2 -12x +9), (y'(2) = 12 -24 +9 = -3)
19. A particle moves along (s(t) = t^3 -6t^2 +9t). Find velocity at (t=3)
Solution: (v = s' = 3t^2 -12t +9), (v(3) = 27-36+9=0)
20. Volume of a cube: (V = x^3). Find rate of change when (x=2) and (\frac{dx}{dt}=3)
Solution: (dV/dt = 3x^2 dx/dt = 343 = 36)
Limits / L’Hospital
21. (\lim_{x \to 0} \frac{\ln(1+x)}{x})
Solution: Standard limit: 1
22. (\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 - x} = \frac{5}{2})
, 23. (\lim_{x \to 0} \frac{1 - \cos x}{x^2})
Solution: Standard limit: (1/2)
24. (\lim_{x \to 1} \frac{x^2 -1}{x-1} = 2)
25. (\lim_{x \to 0} \frac{\tan x}{x} = 1)
Limits
26. Evaluate (\displaystyle \lim_{x \to 4} \frac{\sqrt{x}-2}{x-4})
Solution: Multiply numerator and denominator by (\sqrt{x}+2):
[
\frac{\sqrt{x}-2}{x-4} \cdot \frac{\sqrt{x}+2}{\sqrt{x}+2} = \frac{x-4}{(x-4)(\sqrt{x}+2)} =
\frac{1}{\sqrt{x}+2} \to \frac{1}{4}
]
27. (\displaystyle \lim_{x \to 0} \frac{e^x - 1 - x}{x^2})
Solution: Use series expansion: (e^x \approx 1 + x + x^2/2)
(\frac{x^2/2}{x^2} = 1/2)
28. (\displaystyle \lim_{x \to \infty} \frac{3x^3 + 2x}{6x^3 - x^2} = \frac{1}{2})
29. (\displaystyle \lim_{x \to 0} \frac{\sin x - x}{x^3})
Solution: Series: (\sin x = x - x^3/6 + \dots \implies (\sin x - x)/x^3 = -1/6)
30. (\displaystyle \lim_{x \to 2} \frac{x^3 - 8}{x-2})
Solution: Factor numerator: ((x-2)(x^2 +2x +4)/(x-2) = x^2+2x+4 \to 4+4+4=12)
Derivatives
31. Find (\frac{d}{dx} (\ln(5x+1)))
Solution: Chain rule: (\frac{5}{5x+1})
32. Find derivative of (y = e^{2x} \cos x)
Solution: Product rule: (y' = 2e^{2x}\cos x - e^{2x}\sin x = e^{2x}(2\cos x - \sin x))
33. Derivative of (y = \tan(x^2))
Solution: Chain rule: (y' = \sec^2(x^2) \cdot 2x = 2x \sec^2(x^2))
34. Derivative of (y = x^4 \ln x)
Solution: Product rule: (y' = 4x^3 \ln x + x^3)
Ron Larson (Author), Bruce H. Edwards
(Author)
1. Evaluate (\displaystyle \lim_{x \to 2} \frac{x^2 - 4}{x-2})
Solution: Factor numerator: ((x-2)(x+2)/(x-2) = x+2). Limit as (x\to2) is (4).
2. Evaluate (\displaystyle \lim_{x \to 0} \frac{\sin 3x}{x})
Solution: Use (\lim_{x\to0} \frac{\sin x}{x} = 1):
(\lim_{x\to0} \frac{\sin 3x}{x} = 3).
3. (\displaystyle \lim_{x \to \infty} \frac{4x^2 + 7}{2x^2 - 5x} )
Solution: Divide top and bottom by (x^2): (\frac{4 + 7/x^2}{2 - 5/x} \to 2)
4. Determine (\displaystyle \lim_{x \to 1} \frac{x^3 -1}{x-1})
Solution: Factor numerator: ((x-1)(x^2+x+1)/(x-1) = x^2+x+1). Limit: (1+1+1=3)
5. (\displaystyle \lim_{x \to 0} \frac{e^x -1}{x})
Solution: Standard limit: (1)
Derivatives
6. Find (\frac{d}{dx} (5x^3 - 2x + 7))
Solution: (15x^2 - 2)
7. Find (\frac{d}{dx} (\sqrt{x^5 + 1}))
Solution: Chain rule: (\frac{1}{2\sqrt{x^5+1}} \cdot 5x^4 = \frac{5x^4}{2\sqrt{x^5+1}})
8. Derivative of (f(x) = x^2 \cos x)
Solution: Product rule: (2x \cos x - x^2 \sin x)
9. (\frac{d}{dx} \frac{2x+3}{x-1})
Solution: Quotient rule: (\frac{(2)(x-1)-(2x+3)(1)}{(x-1)^2} = \frac{-5}{(x-1)^2})
10. Find (\frac{dy}{dx}) if (y^2 + x^2 = 49)
Solution: (2y \frac{dy}{dx} + 2x = 0 \implies \frac{dy}{dx} = -\frac{x}{y})
,Integrals
11. (\int (3x^2 - 4x + 1) dx)
Solution: (x^3 - 2x^2 + x + C)
12. (\int_0^2 (4x - 1) dx)
Solution: (2x^2 - x \Big|_0^2 = 8-2 = 6)
13. (\int \frac{1}{x} dx)
Solution: (\ln|x| + C)
14. (\int e^{3x} dx)
Solution: (\frac{1}{3} e^{3x} + C)
15. (\int \sin 2x dx)
Solution: (-\frac{1}{2} \cos 2x + C)
Applications
16. A balloon rises at 5 m/s. Find its height after 10 s.
Solution: (h = 5t = 50) m
17. A rectangle has area 50. Express perimeter as a function of width (x).
Solution: Length (L = 50/x), perimeter (P = 2(x + 50/x) = 2x + 100/x)
18. Find the slope of (y = x^3 - 6x^2 + 9x) at (x=2)
Solution: (y' = 3x^2 -12x +9), (y'(2) = 12 -24 +9 = -3)
19. A particle moves along (s(t) = t^3 -6t^2 +9t). Find velocity at (t=3)
Solution: (v = s' = 3t^2 -12t +9), (v(3) = 27-36+9=0)
20. Volume of a cube: (V = x^3). Find rate of change when (x=2) and (\frac{dx}{dt}=3)
Solution: (dV/dt = 3x^2 dx/dt = 343 = 36)
Limits / L’Hospital
21. (\lim_{x \to 0} \frac{\ln(1+x)}{x})
Solution: Standard limit: 1
22. (\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 - x} = \frac{5}{2})
, 23. (\lim_{x \to 0} \frac{1 - \cos x}{x^2})
Solution: Standard limit: (1/2)
24. (\lim_{x \to 1} \frac{x^2 -1}{x-1} = 2)
25. (\lim_{x \to 0} \frac{\tan x}{x} = 1)
Limits
26. Evaluate (\displaystyle \lim_{x \to 4} \frac{\sqrt{x}-2}{x-4})
Solution: Multiply numerator and denominator by (\sqrt{x}+2):
[
\frac{\sqrt{x}-2}{x-4} \cdot \frac{\sqrt{x}+2}{\sqrt{x}+2} = \frac{x-4}{(x-4)(\sqrt{x}+2)} =
\frac{1}{\sqrt{x}+2} \to \frac{1}{4}
]
27. (\displaystyle \lim_{x \to 0} \frac{e^x - 1 - x}{x^2})
Solution: Use series expansion: (e^x \approx 1 + x + x^2/2)
(\frac{x^2/2}{x^2} = 1/2)
28. (\displaystyle \lim_{x \to \infty} \frac{3x^3 + 2x}{6x^3 - x^2} = \frac{1}{2})
29. (\displaystyle \lim_{x \to 0} \frac{\sin x - x}{x^3})
Solution: Series: (\sin x = x - x^3/6 + \dots \implies (\sin x - x)/x^3 = -1/6)
30. (\displaystyle \lim_{x \to 2} \frac{x^3 - 8}{x-2})
Solution: Factor numerator: ((x-2)(x^2 +2x +4)/(x-2) = x^2+2x+4 \to 4+4+4=12)
Derivatives
31. Find (\frac{d}{dx} (\ln(5x+1)))
Solution: Chain rule: (\frac{5}{5x+1})
32. Find derivative of (y = e^{2x} \cos x)
Solution: Product rule: (y' = 2e^{2x}\cos x - e^{2x}\sin x = e^{2x}(2\cos x - \sin x))
33. Derivative of (y = \tan(x^2))
Solution: Chain rule: (y' = \sec^2(x^2) \cdot 2x = 2x \sec^2(x^2))
34. Derivative of (y = x^4 \ln x)
Solution: Product rule: (y' = 4x^3 \ln x + x^3)
