Section 4.6 « Difference Equations to Differential Equations

Section 4.6

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1. (a) $\displaystyle{\int_1^\infty \frac{1}{x^3} \ dx = \frac{1}{2}}$

(b) $\displaystyle{\int_4^\infty \frac{3}{x^7} \ dx = \frac{1}{8192}}$

(d) This integral diverges.

(e) $\displaystyle{\int_0^\infty \frac{3}{2x + 3)^2} \ dx = \frac{1}{2}}$

(f) This integral diverges.

2. (a) $\displaystyle{\int_{-\infty}^{-2} \frac{3}{x^2} \ dx = \frac{3}{2}}$

(b) $\displaystyle{\int_{-\infty}^\infty \frac{x}{(x^2 + 4)^4} \ dx = 0}$

(d) This integral diverges.

3. (a) Converges

(b) Converges

(d) Converges

(e) Converges

(f) Converges

4. (a) $\displaystyle{\int_0^8 \frac{1}{x^{\frac{1}{3}}} \ dx = 6}$

(b) This integral diverges.

(d) $\displaystyle{\int_0^5 \frac{5}{(t - 2)^{\frac{2}{5}}} \ dt = \frac{25}{3}\left(2^{\frac{3}{5}} + 3^{\frac{3}{5}}\right)}$

(e) This integral diverges.

(f) $\displaystyle{\int_{-1}^2 \frac{3}{x^\frac{1}{3}} \ dx = \frac{9}{2}\left(2^\frac{2}{3} - 1\right)}$

5. (a) $\displaystyle{\int_1^\infty \frac{1}{x^p} \ dx = \frac{1}{p - 1}}$

6. (a) $\displaystyle{\int_0^1 \frac{1}{x^p} \ dx = \frac{1}{1 - p}}$

8. (a) $\displaystyle{P(x) = \left(\frac{\sigma}{x}\right)^\alpha}$

(b) $\displaystyle{A = \frac{\alpha}{\alpha - 1}\sigma}$

(d) $A = 60,000$ , $P(A) = 0.1165$ , $P(2A) = 0.0507$
These values tell us that only $11.65$ % of the population have incomes exceeding the average income of $60,000$ and only $5.07$ % of the population have incomes exceeding $120,000$ .

(e) $\displaystyle{P(A) = \left(\frac{\alpha - 1}{\alpha}\right)^\alpha}$ , $\alpha > 1$

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Difference Equations to Differential Equations

Section 4.6

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