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 — de2de:chapter-1 [2012/06/25 11:10] (current)dcs created 2012/06/25 11:10 dcs created 2012/06/25 11:10 dcs created Line 1: Line 1: + ====Differential Equations to Difference Equations==== + ===Answers for selected problems=== + + **Chapter 1** + + ---- + + **Section 1.1** + + ​4. ​    ​(a) ​ $y = 4(x - 1) + 2$ + + ​(b) ​ $y = 2(x - 1) + 2$ + + ​%%(c)%% ​  $y = 3(x - 1) + 1$ + + ​(d) ​ $y = 4(x - 2) + 4$ + + ​5. ​ For  $6$ digits, ​ $n = 5624$. + + ​6. ​   $n \ge 201$ + + ​7. ​ (a)  $p = 101$ + + ​(b) ​ $|m_n - 2| < 0.01$ when  $n > p$  ​ + + ​8. ​ (a)  $a_{20} = 0.9996 ​$ to four decimal places. ​ + + ​%%(c)%% ​ $a_{20} = 2.6533 ​$ to four decimal places. ​ + + ​(d) ​ $a_{20} = 41.1032 ​$ to four decimal places. + + ---- + + **Section 1.2** + + 1.    (a) $\displaystyle{a_n = \frac{1}{3^n},​ n=0,​1,​2,​\ldots}$ + + (b) $\displaystyle{a_n = \frac{1}{n},​ n=1,2,3, \ldots}$ + + %%(c)%% $\displaystyle{a_n = \frac{2n+1}{n+1},​ n=0,​1,​2,​\ldots}$ + + (d) $\displaystyle{a_n = \frac{(-1)^n}{2n+1},​ n=1,​2,​3,​\ldots}$ + + 2. (a) Converges: $\displaystyle{\lim_{n \to \infty}\frac{1}{3^n} = 0}$ + + (b) Diverges: $\displaystyle{\lim_{n \to \infty}\pi^n = \infty}$ + + %%(c)%% Converges: $\displaystyle{\lim_{n \to \infty}\frac{3n-1}{2n+6} = \frac{3}{2}}$ + + (d) Diverges + + (e) Diverges: $\displaystyle{\lim_{n \to \infty}\frac{3n^4-6n^3+1}{5n^3+n^2+2}=\infty}$ + + (f) Converges: $\displaystyle{\lim_{n \to \infty}\frac{2n^5-3n^2+23}{7n^5+13n^4-12} = \frac{2}{7}}$ + + (g) Converges: $\displaystyle{\lim_{n \to \infty}\frac{45-16n^2}{13+5n+6n^3} = 0}$ + + (h) Converges: $\displaystyle{\lim_{n \to \infty}\frac{3n+1}{\sqrt{4n^2+1}}=\frac{3}{2}}$ + + (i) Diverges: $\displaystyle{\lim_{n \to \infty}(-2)^{2n+1} = -\infty}$ + + (j) Divergs: $\displaystyle{\lim_{n \to \infty}\frac{10-16n^3}{1+n^2}=-\infty}$ + + (k) Converges: $\displaystyle{\sqrt{\frac{3n^2+n-6}{5n^2+16}}=\sqrt{\frac{3}{5}}}$ + + (l) Converges: $\displaystyle{\lim_{n \to \infty}\frac{(-1)^n}{5^n}=0}$ + + 3. $\displaystyle{\lim_{n \to \infty}\frac{\sin(n)}{n}=0}$ + + 4.  (a) $a_1 = 2$, $a_2 = 2.25$, $a_3 = 2.3704$, $a_4 = 2.4414$, and $a_5 = 2.4883$, where the last three have been rounded to four decimal places. + + (e) $a_{74} > 2.7$, but $a_n < 2.7$ for $n < 74$. + + (f) $n = 135$ + + 5. (e) $n = 13$ + + (f) $n = 41$ + + 7. %%(c)%% $\displaystyle{a_9 = a_{10} = \frac{1562500}{567}}$ + + 11. (a) $\displaystyle{N = \left\lfloor \frac{1}{\epsilon}\right\rfloor}$; for $\epsilon = 0.001$, $N = 1000$ + + (b) $\displaystyle{N = \left\lfloor \frac{\log(\epsilon)}{\log(0.98)}\right\rfloor}$; for $\epsilon = 0.001$, $N = 341$ + + %%(c)%% $\displaystyle{N = \left\lfloor \frac{1}{\sqrt{\epsilon}}\right\rfloor}$; for $\epsilon = 0.001$, $N = 31$ + + (d) $\displaystyle{N = \left\lfloor \frac{1}{\sqrt[3]{\epsilon}}\right\rfloor}$; for $\epsilon = 0.001$, $N = 10$ + + ---- + + **Section 1.3** + + 1. (a) $\displaystyle{\sum_{n=1}^\infty \left(\frac{1}{3}\right)^{n-1} = \frac{3}{2}}$ + + %%(c)%% $\displaystyle{\sum_{n=1}^\infty \left(\frac{2}{5^n}\right) = \frac{1}{2}}$ + + (e) $\displaystyle{\sum_{n=1}^\infty 7\left(\frac{1}{3}\right)^n\frac{2}{5}^{n-1} = \frac{35}{13}}$ + + (g) Does not converge + + (i) Does not converge + + (k) Does not converge + + (m) $\displaystyle{\sum_{n=1}^\infty \sin(n\pi) = 0}$ + + (n) Does not converge + + 3. (a) $\pi$ + + (b) $99$ + + 5. (a) $10$ billion dollars + + %%(c)%% $\displaystyle{\frac{A}{1 - r}}$ dollars + + 7. $70$ meters; $\displaystyle{frac{50}{3}}$ meters ​ + + 9. $\displaystyle{\frac{4}{9}}$ + + 11. $83$ terms for a partial sum larger than $5$; $12,367$ terms for a partial sum larger than $10$ + + ---- + + **Section 1.4** + + 1. (a) $x_0 = 10$, $x_1 = 14$, $x_2 = 18$, $x_3 = 22$, $x_4 = 26$, $x_5 = 30$ + + %%(c)%% $x_0 = 40$, $x_1 = 60$, $x_2 = 100$, $x_3 = 180$, $x_4 = 340$, $x_5 = 660$ + + (e) $x_0 = 2$, $x_1 = 3$, $x_2 = 5$, $x_3 = 8$, $x_4 = 13$, $x_5 = 21$ $x_6 = 34$ + + 2. (a) $x_n = 5 \cdot 2^n$, $n = 0, 1, 2, \ldots$; $x_{10} = 5120$ + + %%(c)%% $x_n = 32.5(1.8)^n - 12.5$, $n = 0, 1, 2, \ldots$; $x_{10} = 11,591.5$, rounded to one decimal place + + (e) $\displaystyle{x_n = \frac{10}{3} \cdot 4^n - \frac{4}{3}}$,​ $n = 0, 1, 2, \ldots$; $x_{10} = 3,495,252$ + + 3. (a) $w_{n+1} = 1.03w_n$, $n = 0, 1, 2, \ldots$ + + (b) $w_n = 350(1.03)^n$, $n = 0, 1, 2, \ldots$ + There will be $545$ weasels in $15$ years. + + (d) $24$ years + + (e) $\displaystyle{\lim_{n \to \infty}w_n = \infty}$ + This says the population will grow without any upper bound to its size, which is not possible due to the physical limitations of the habitat. + + 5. (a) $w_n = 1.03w_n - 6$, $n = 0, 1, 2, \ldots$ + + (b) $w_n = 150(1.03)^n + 200$, $n = 0, 1, 2, \ldots$ + There will be $434$ weasels in $15$ years. + + %%(c)%% $\displaystyle{\lim_{n \to \infty}w_n = \infty}$ + This says the population will grow without any upper bound to its size. + + (d) The population will double in $41$ years. + + 7. (a) $155.30$ grams, rounded to two decimal places + + (b) About $686$ years + + %%(c)%% About $2,279$ years + + 9. (a) $\displaystyle{T_{n+1} = \frac{17}{19}T_n + \frac{140}{19}}$,​ $n = 0, 1, 2, \ldots$ + + (b) $\displaystyle{T_n = 95\left(\frac{17}{19}\right)^n + 70}$, $n = 0, 1, 2, \ldots$ + + %%(c)%% $75.9^\circ$ F, rounded to one decimal place + + (d) $\displaystyle{\lim_{n \to \infty}T_n = 70^\circ}$ F + + (f) No + + 11. $103.8^\circ$ F, rounded to one decimla place + + 15. $x_n = 4n + 10$, $n = 0, 1, 2, \ldots, 10$ + + ---- + + **Section 1.5** + + 1. (a) $w_{n+1} = w_n + 0.00003w_n(1000 - w_n)$, $n = 0, 1, 2, \ldots$ + + %%(c)%% The population will double in $42$ years and triple in $102$ years. + + (d) From the plot, it appears that $\displaystyle{\lim_{n \to \infty}w_n = 1000}$ ​ + + 3. (a) $p_{n+1} = 1.045p_n$, $n = 0, 1, 2, \ldots$ + + %%(c)%% The population will double in $16$ years and triple in $25$ years. + + (d) $\displaystyle{\lim_{n \to \infty}p_n = \infty}$ + + (e) $52$ years + + 4. (a) It appears that $\displaystyle{\lim_{n \to \infty}r_n = 10,000}$ + + %%(c)%% There appears to be a limiting cycle of period $2$, with values alternating between $6,403$ and $11,930$. + + (e) There appears to be a limiting cycle of period $8$, with values cycling thorough $4851$, $11245$, $7661$, $12248$, $5198$, $11588$, $6876$, and $12375$. + + 7. (a) $\displaystyle{\lim_{x \to \infty}x_n = \infty}$ + + %%(c)%% It appears that $\displaystyle{\lim_{n \to \infty}x_n = 0.7391}$ to four decimal places. + + (e) It appears that $\displaystyle{\lim_{n \to \infty}x_n =\sqrt{2}}$. + + (g) It appears that $\displaystyle{\lim_{n \to \infty}x_n = \infty}$. + + (i) There appears to be a limiting cycle of order $2$. + + (k) The limiting behavior, if any, is not clear. + + ----