Section 6.2 « Difference Equations to Differential Equations

Section 6.2

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1. (a) $\log(6) = a + b$

(c) $\log(9) = 2b$

2. (a) $\displaystyle{f'(x) = \frac{2}{x}}$

(c) $\displaystyle{g'(x) = \frac{2}{x} + \frac{x}{x^2 + 5}}$

(e) $\displaystyle{f'(x) = \frac{e^{2x}}{x} + 2e^{2x}\log(5x)}$

(g) $\displaystyle{h'(x) = \frac{1}{x\log(x)}}$

(i) $\displaystyle{f'(x) = ex^{e-1}}$

3. (a) $\displaystyle{\int \frac{1}{2x} \ dx = \frac{1}{2}\log|x| + c}$

(c) $\displaystyle{\int \frac{5x}{3x^2 + 1} \ dx = \frac{5}{6}\log(3x^2 + 1) + c}$

(e) $\displaystyle{\int \tan(3x)dx = -\frac{1}{3}\log|\cos(3x)| + c}$

(g) $\displaystyle{\int \csc(x)dx = -\log|\csc(x) + \cot(x)| + c}$

4. (a) $\displaystyle{\int \log(3x)dx = x\log(3x) - x + c}$

(c) $\displaystyle{\int \frac{\log(x)}{x} \ dx = \frac{1}{2}(\log(x))^2 + c}$

(e) $\displaystyle{\int \log(x + 1)dx = (x + 1)\log(x + 1) - x + c}$

(g) $\displaystyle{\int \frac{1}{x\log(x)} \ dx = \log|\log(x)| + c}$

6. (b) $\log(2) \approx P_{200}(2) = 0.6907$ , rounded to 4 decimal places

(c) $\log(1.5) \approx P_7(1.5) = 0.4058$ , rounded to 4 decimal places

10. (a) $\displaystyle{\lim_{x \to \infty}\log(\log(x)) = \infty}$

(d) $x = e^{20} = 485,165,195.4$ , rounded to one decimal place

(e) $x = e^{\left(e^{20}\right)} = 1.50655428 \times 10^{210,704,567}$ , rounded to 8 decimal places

11. Length $\displaystyle{= \int_1^{10} \sqrt{1 + \frac{1}{x^2}} \ dx = 9.4172}$ , rounded to four decimal places

12. $\displaystyle{x(t) = 100e^{\alpha t}}$ , where $\alpha =\dfrac{\log(2)}{5}$

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