# Differences

This shows you the differences between two versions of the page.

 — de2de:chapter-6 [2012/06/25 11:33] (current)dcs created 2012/06/25 11:33 dcs created 2012/06/25 11:33 dcs created Line 1: Line 1: + ====Differential Equations to Difference Equations==== + ===Answers for selected problems=== + + **Chapter 6** + + ---- + + **Section 6.1** + + 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 $-\infty < x < \infty$, + $+ e^{-x^2} = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{n!} = 1 - x^2 + \frac{x^4}{2} - \frac{x^6}{3!} + \frac{x^8}{4!} - \cdots . +$ + + %%(c)%% For $-\infty < x < \infty$, + $+ \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)n!} = \frac{2x}{\sqrt{\pi}} - \frac{2x^3}{3\sqrt{\pi}} + \frac{x^5}{5\sqrt{\pi}} - \frac{x^7}{21\sqrt{\pi})} + \frac{x^9}{108\sqrt{\pi}} - \cdots . +$ + + %%(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) $2072$ + + ---- + + **Section 6.2** + + 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 $\alpha =\dfrac{\log(2)}{5}$ + + ---- + + **Section 6.3** + + 1. (a) $\displaystyle{\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e}$ + + %%(c)%% $\displaystyle{\lim_{n \to \infty}\left(1 - \frac{2}{n}\right) = e^{-2}}$ ​ + + (e) $\displaystyle{\lim_{n \to \infty}\left(1 + \frac{2}{n^2}\right) = 1}$ + + (f) $\displaystyle{\lim_{n \to \infty}\left(1 - \frac{4}{n} + \frac{1}{n^2}\right) = e^{-4}}$ ​ + + + 3. (a) $\$6946.20$​ + + + (b)$ %%$%%6961.83$ + + %%(c)%% $%%$%%6967.91 $+ + (d)$ %%$%%6969.48$ + + (e) $%%$%%6969.74 $+ + + 4. The$ 5.5 $% interest compounded quarterly is the most advantageous. + + 6.$ 24,765 $years + + + 7. The bone is between$ 23,257 $years and$ 26,609 $years old. + + + 8.$ x(t) = x_0e^{-0.034657t} $+$ 66.4 $minutes until 10% remains;$ 86.4 $minutes until 5% remains ​ + + + 9.$ x(t) = x_0e^{-0.000028881t} $+$ 79,726 $years until 10% remains;$ 103,726 $years until 5% remains ​ + + + 10.$ 69 $years + + + 11. (a)$ y(t) = 179.3e^{0.012562t} $+ + (b) Prediction for 1980:$ 230.5 $million + + Prediction for 1990:$ 261.4 $million + + Prediction for 2000:$ 296.3 $million ​ + + 13. (a)$ x(t) = 75.995e^{0.0190825t} $, in millions + + Prediction for 1920:$ 111.3 $million + + Prediction for 1930:$ 134.7 $million + + Prediction for 1950:$ 197.3 $million + + Prediction for 1970:$ 289.0 $million + + Prediction for 1990:$ 423.3 $million + + Prediction for 2000:$ 512.3 $million + + The population in 1936 would be twice the population of 1900. + + (b)$ \displaystyle{x(t) = \frac{10595}{75.995 + 63.424e^{-0.048106t}}}$,​ in millions + + Prediction for 1930:$ 116.5 $million + + Prediction for 1950:$ 129.7 $million + + Prediction for 1970:$ 135.5 $million + + Prediction for 1990:$ 137.9 $million + + Prediction for 2000:$ 138.5 $million + + This model predicts a limiting population of$ M = 139.4 $million. Since this is less than twice the population in 1900, this model predicts that the population will never double from its 1900 level. ​ + + + 14. Hint:$ \displaystyle{\ddot{x}(t) = \alpha \dot{x}(t) - \frac{2\alpha}{M}x(t)\dot{x}(t)}$+ + This tells us that the rate of growth of the population is increasing when the population is less than one-half of its limiting size and decreasing when the population exceeds one-half of its limiting size. + + ---- + + **Section 6.4** + + 1. (a)$ \displaystyle{\int \frac{x-1}{x} \ dx = x - \log|x| + c}$+ + %%(c)%%$ \displaystyle{\int \frac{3x^2}{x-2} \ dx = \frac{3}{2}x^2 + 6x + 12\log|x-2| + c}$+ + (e)$ \displaystyle{\int \frac{4x+1}{2x^2+x-3} \ dx = \log|2x^2 + x - 3| + c}$+ + + 2. (a)$ \displaystyle{\int \frac{1}{(x+2)(x-4)} \ dx = \frac{1}{6}\log|x-4| - \frac{1}{6}\log|x+2| + c}$+ + %%(c)%%$ \displaystyle{\int \frac{3x}{(2x+3)(x+1)} \ dx = \frac{9}{2}\log|2x+3| - 3\log|x+1| + c}$+ + (e)$ \displaystyle{\int \frac{x}{x^2+x-6} \ dx = \frac{2}{5}\log|x-2| + \frac{3}{5}\log|x+3| + c}$+ + (g)$ \displaystyle{\int \frac{3}{x^2+5x+6} \ dx = 3\log|x+2| - 3\log|x+3| + c}$+ + + 3. (a)$ \displaystyle{\frac{1}{(x-1)^2} \ dx = -\frac{1}{x-1} + c}$+ + %%(c)%%$ \displaystyle{\int \frac{x}{x^2+2x+1} \ dx = \frac{1}{x+1} + \log|x+1| + c}$+ + (e)$ \displaystyle{\int \frac{5}{(x+2)^3} \ dx = -\frac{5}{2(x+2)^2} + c}$+ + (g)$ \displaystyle{\int \frac{3x^2}{(x+1)^2(x-3)} \ dx = \frac{3}{4(x+1)} + \frac{21}{16}\log|x+1| + \frac{27}{16}\log|x-3| + c}$+ + + 4. (a)$ \displaystyle{\int \frac{1}{(3x+2)^2} \ dx = -\frac{1}{3(3x+2)} + c}$+ + %%(c)%%$ \displaystyle{\int \frac{9x^2-4x}{3x^3-2x^2+5} \ dx = \log|3x^3 - 2x^2 + 5| + c}$+ + (e)$ \displaystyle{\int_{-1}^1 \frac{1}{x^2-4} \ dx = -\frac{1}{2}\log(3)}$​ + + (g)$ \displaystyle{\int \frac{4x+5}{(x-2)^2(x+5)} \ dx = -\frac{13}{7(x-2)} + \frac{15}{49}\log|x-2| - \frac{15}{49}\log|x+5| + c}$+ + + 5.$ \displaystyle{x(t) = \frac{1-e^{2t}}{1+e^{2t}}}$+ + ---- + + **Section 6.5** + + 1. (a)$ \displaystyle{f'​(x) = \frac{x}{1+x^2} + \tan^{-1}(x)}$​ + + %%(c)%%$ \displaystyle{g'​(x) = \frac{3}{x\sqrt{1-9x^2}} - \frac{\sin^{-1}(3x)}{x^2}}$​ + + + 2. (a)$ \displaystyle{\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}}$​ + + %%(c)%%$ \displaystyle{\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}}$​ + + (e)$ \displaystyle{\sin^{-1}\left(\sin\left(\frac{3\pi}{4}\right)\right) = \frac{\pi}{4}}$​ + + (f)$ \displaystyle{\sin\left(\sin^{-1}\left(-\frac{1}{\sqrt{2}}\right)\right) = -\frac{1}{\sqrt{2}}}$​ + + 3. (a)$ \displaystyle{\int \frac{1}{1 + 2x^2} \ dx = \frac{1}{\sqrt{2}}\tan^{-1}(\sqrt{2}x) + c}$+ + %%(c)%%$ \displaystyle{\int \frac{3}{x^2 + 4} \ dx = \frac{3}{2}\tan^{-1}\left(\frac{x}{2}\right) + c}$+ + (e)$ \displaystyle{\int \frac{x}{x^2+4x+5} \ dx = \frac{1}{2}\log(x^2 + 4x + 5) - 2\tan^{-1}(x + 2) + c}$+ + (g)$ \displaystyle{\int_{-1}^1 \frac{1}{1 + x^2} \ dx = \frac{\pi}{2}}$​ + + + 4. (a)$ \displaystyle{\int \frac{1}{x^3+x} \ dx = \log|x| - \frac{1}{2}\log(1 + x^2) + c}$+ + %%(c)%%$ \displaystyle{\int \frac{1}{x^2(x^2+1)} \ dx = -\frac{1}{x} - \tan^{-1}(x) + c}$+ + (e)$ \displaystyle{\int \sin^{-1}(x)dx = x\sin^{-1}(x) + \sqrt{1 - x^2} + c}$+ + (g)$ \displaystyle{\int \frac{5}{\sqrt{1-9x^2}} \ dx = \frac{5}{3}\sin^{-1}(3x) + c}$+ + (i)$ \displaystyle{\int \frac{3x}{\sqrt{1-x^2}} \ dx = -3\sqrt{1-x^2} + c}$+ + + 8.$ \displaystyle{\int_{-\infty}^\infty \frac{1}{1 + x^2} \ dx = \pi}$+ + + 9. (a) For$ -1 \le x \le 1 $, + $+ \tan^{-1}(x) = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2n + 1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots . +$ + + (b)$ \displaystyle{\pi = 4\tan^{-1}(1) \approx P_{3999}(1) = 3.1411}$, rounded to 4 decimal places + + ---- + + **Section 6.6** + + 1. (a)$ \displaystyle{\int \frac{3}{\sqrt{16-x^2}} \ dx = 3\sin^{-1}\left(\frac{x}{4}\right) + c}$+ + %%(c)%%$ \displaystyle{\int \sqrt{5-z^2} \ dz = \frac{z}{2}\sqrt{5-z^2} + \frac{5}{2}\sin^{-1}\left(\frac{z}{\sqrt{5}}\right) + c}$+ + (e)$ \displaystyle{\int \frac{1}{\sqrt{4+x^2}} \ dx = \log\left|\sqrt{4+x^2} + x\right| + c}$+ + + 2. (a)$ \displaystyle{\int z\sqrt{1-z^2} \ dz = -\frac{1}{3}(1 - z^2)^{\frac{3}{2}} + c}$+ + %%(c)%%$ \displaystyle{\int \frac{1}{\sqrt{x^2 - 4}} \ dx = \log\left|x + \sqrt{x^2-4}\right| + c}$+ + (e)$ \displaystyle{\int \frac{4}{\sqrt{3-2x^2}} ​ \ dx = 2\sqrt{2}\sin^{-1}\left(\sqrt{\frac{2}{3}}x\right) + c}$+ + (g)$ \displaystyle{\int \sqrt{4-t^2} \ dt = \frac{t}{2}\sqrt{4-t^2} + 2\sin^{-1}\left(\frac{t}{2}\right)}$​ + + + 3. (a)$ \displaystyle{\int_0^3 \sqrt{9-t^2} \ dt = \frac{9\pi}{4}}$​ + + %%(c)%%$ \displaystyle{\int_0^1 \frac{1}{(1+x^2)^{\frac{3}{2}}} \ dx = \frac{1}{\sqrt{2}}}$​ + + (e)$ \displaystyle{\int_{5\sqrt{2}}^{10} \frac{1}{\sqrt{x^2-25}} \ dx = \log(2+\sqrt{3}) - \log(1 + \sqrt{2})}$​ + + + 5.$ \displaystyle{\int \frac{1}{\sqrt{1-x^2}} \ dx = -\cos^{-1}(x) + c}$+ + ---- + + **Section 6.7** + + 1. (a)$ f'(x) = 3\cosh(3x) $+ + %%(c)%%$ f'(t) = 6t\sinh(t)\sinh(2t) + 3t\cosh(t)\cosh(2t) + 3\sinh(t)\cosh(t) $+ + (e)$ y'(t) = 40t^2\cosh(4t)\sinh(4t) + 10t\cosh^2(4t) $+ + + 2. (a)$ \displaystyle{\int \sinh(3x)dx = \frac{1}{3}\cosh(3x) + c}$+ + %%(c)%%$ \displaystyle{\int \sinh(z)\cosh(z)dz = \frac{1}{2}\sinh^2(z) + c}$+ + (e)$ \displaystyle{\int e^{-2t}\cosh(2t)dt = \frac{1}{2}t - \frac{1}{8}e^{-4t} + c}$+ + (g)$ \displaystyle{\int 5t^2\cosh(2t)dt = \frac{5}{2}t^2\sinh(2t) - \frac{5}{2}t\cosh(2t) + \frac{5}{4}\sinh(2t) + c}$+ + + 4. (a)$ \displaystyle{\int \frac{1}{\sqrt{4+x^2}} \ dx = \sinh^{-1}\left(\frac{x}{2}\right) + c}$+ + %%(c)%%$ \displaystyle{\int \frac{3}{\sqrt{9+3t^2}} \ dt = \sqrt{3}\sinh^{-1}\left(\frac{t}{\sqrt{3}}\right) + c}$+ + (d) For$ x < -1 $,$ \displaystyle{\int \frac{1}{\sqrt{x^2-1}} \ dx = -\cosh^{-1}|x| + c}$. + + + 6. (a)$ \displaystyle{f'​(x) = 12x\mathrm{sech}^2(4x) + 3\tanh(4x)}$​ + + %%(c)%%$ \displaystyle{h'​(\theta) = 8\tanh(\theta)\mathrm{sech}^3(\theta) - 4\tanh^3(\theta)\mathrm{sech}(\theta)}$​ + + + 7. (a)$ \displaystyle{\int \tanh(x)dx = \log(\cosh(x)) + c}$+ + %%(c)%%$ \displaystyle{\int \frac{1}{4 - x^2} \ dx = \frac{1}{2}\tanh^{-1}\left(\frac{x}{2}\right) + c}\$ + + ----