Section 1.4 « Difference Equations to Differential Equations

Section 1.4

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1. (a) $x_0 = 10$ , $x_1 = 14$ , $x_2 = 18$ , $x_3 = 22$ , $x_4 = 26$ , $x_5 = 30$

(e) $x_0 = 2$ , $x_1 = 3$ , $x_2 = 5$ , $x_3 = 8$ , $x_4 = 13$ , $x_5 = 21$ $x_6 = 34$

2. (a) $x_n = 5 \cdot 2^n$ , $n = 0, 1, 2, \ldots$ ; $x_{10} = 5120$

(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

9. (a) $\displaystyle{T_{n+1} = \frac{17}{19}T_n + \frac{140}{19}}$ , $n = 0, 1, 2, \ldots$

(b) $\displaystyle{T_n = 95\left(\frac{17}{19}\right)^n + 70}$ , $n = 0, 1, 2, \ldots$

(d) $\displaystyle{\lim_{n \to \infty}T_n = 70^\circ}$ F

(f) No

11. $103.8^\circ$ F, rounded to one decimla place

15. $x_n = 4n + 10$ , $n = 0, 1, 2, \ldots, 10$

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Difference Equations to Differential Equations

Section 1.4

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