Section 3.4 « Difference Equations to Differential Equations

Section 3.4

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1. (a) $f'(x) =16(4x + 5)^3$

(c) $\displaystyle{h'(t) = -\frac{18}{(6t - 3)^3}}$

(e) $g'(z) = 2(3z+4)^3(2z^2 + z)(4z + 1) + 9(3z + 4)^2(2z^2 + z)^2$

2. (a) $\displaystyle{\frac{ds}{dt} = 16t^3(t^2 - 1) + 8t(t^2 - 1)^2}$

(c) $\displaystyle{\frac{dq}{dt} = \frac{9t^2 - 4}{2\sqrt{3t^3 - 4t}}}$

(e) $\displaystyle{\frac{dx}{dt} = \frac{40 - 32t}{(4t + 5)^2}}$

(g) $\displaystyle{\frac{dy}{dx} = \frac{3}{5(3x - 1)^{\frac{4}{5}}}}$

3. $\displaystyle{T(x) = -\frac{33}{125}(x - 2) + \frac{6}{25}}$

4. (a) $T(x) = hx + 1$

(c) $\displaystyle{\sqrt[3]{1.06} \approx \frac{1}{3}(0.06) + 1 = 1.02}$
Rounded to four decimal places, $\sqrt[3]{1.06} = 1.0196$ .

5. (a) $\displaystyle{y = -\frac{1}{2}(x - 2) + 3}$

(c) $\displaystyle{y = -\frac{7}{8}(x - 2) + 1}$

(e) $y = 2 - x$

6. (a) $k'(0) = 9$

(c) $k'(0) = -6$

(e) $k'(0) = 8$

8. $900 \text{ cm}^3\text{/sec}$

9. $400\pi \text{ cm}^2\text{/sec}$

11. $\displaystyle{\frac{dK}{dt}\bigg|_{v=10} = 98m}$

13. $1.6 \text{ in}^2\text{/sec}$

15. (a) $0$ units per month

(b) $-100$ units per month

17. $\dfrac{5}{2\pi} \text{ ft/sec}$

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