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Section 6.2 »

Section 6.1

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1. (a) f'(x) = 6e^{2x}

(c) h'(z) = 3z(15z^3 - 30z + 2)e^{5z^3}

(e) \displaystyle{g'(z) = \frac{3}{2}(1 - x)e^{-x}}

(g) \displaystyle{f'(s) = \frac{(1 + 6s)e^{-2s} + 6}{(e^{-2s} + 2)^2}}

2. (a) \displaystyle{\int 3e^{2x}dx = \frac{3}{2}e^{2x} + c}

(c) \displaystyle{\int 4te^{3t}dt = \frac{4}{3}te^{3t} - \frac{4}{9}e^{3t} + c}

(e) \displaystyle{\int z^2 e^z dz = z^2e^z - 2ze^z + 2e^z + c}

(g) \displaystyle{\int e^x\cos(x)dx = \frac{1}{2}(\cos(x) + \sin(x))e^x + c}

3. The maximum value of f is 4e^{-2} at x = 2 .

5. (a) \displaystyle{\lim_{x \to \infty}xe^{-x} = 0}

(c) \displaystyle{\lim_{t \to 0}\frac{e^{-t} - 1}{t} = -1}

8. (a) \displaystyle{\int_0^\infty e^{-x}dx = 1}

(c) \displaystyle{\int_0^\infty xe^{-x}dx = 1}

(e) \displaystyle{\int_0^\infty x^2e^{-x}dx = 2}

10. (a) For -\infty < x < \infty ,

\displaystyle{e^{-x^2} = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{n!} = 1 - x^2 + \frac{x^4}{2} - \frac{x^6}{3!} + \frac{x^8}{4!} - \cdots .}

(cb For -\infty < x < \infty ,

\displaystyle{\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)n!} = \frac{2x}{\sqrt{\pi}} - \frac{2x^3}{3\sqrt{\pi}} + \frac{x^5}{5\sqrt{\pi}} - \frac{x^7}{21\sqrt{\pi})} + \frac{x^9}{108\sqrt{\pi}} - \cdots .}

(c) \mathrm{erf}(1) \approx P_{13}(1) = 0.84271 , rounded to 5 decimal places

11. (a) 26.4 million, rounded to one decimal place

(b) 2072

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