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Math 112 Solutions for 4.1, p. 316

2.
(b) $ \frac{3}{2}x^{2}+C$ (d) $ 6x^{1/2}+C$ (f) $ x^{4}+\frac{5}{3}x^{3}-\frac{7}{2}x^{2}-2x+C$
3.
(a) $ \dfrac{2^{x}}{\ln 2}+C$. (f) $ \dfrac{x^{\pi+1}}{\pi+1}+\dfrac{\pi^{x}}{\ln
\pi}-\dfrac{\pi^{-x}}{\ln\pi}+C$. (g) $ -\cos x+\frac{3}{2}x^{2}-\sec x + C$. (h) $ 3\sin x+4\sinh x+C$.

4.
(b) $ D_{x}(\frac{1}{a}\sin ax)=\frac{1}{a}\cos ax\cdot a=\cos
ax$.

(d) $ D_{x}\left[\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}\right]=\dfrac{1}{a}\cdot\dfrac{1/a}{1+(x/a)^{2}}
=\dfrac{1}{a^{2}+x^{2}}$

(e) $ D_{x}(\sin^{-1}\frac{x}{a})=\dfrac{1}{\sqrt{1-(x/a)^{2}}}\cdot\dfrac{1}{a} =
\dfrac{1}{\sqrt{a^{2}-x^{2}}}$.

(g) $ D_{x}(\frac{1}{a}e^{ax})=\frac{1}{a}\cdot ae^{ax}=e^{ax}$.

(h) $ D_{x}[\frac{1}{2}x\sqrt{a^{2}-x^{2}}+\frac{1}{2}a^{2}\sin^{-1}\frac{x}{a}]
=\f...
...}\sqrt{a^{2}-x^{2}}+\dfrac{a^{2}-x^{2}}{2\sqrt{a^{2}-x^{2}}}=\sqrt{a^{2}-x^{2}}$

5.
(b) $ \frac{1}{a}\cosh ax$. (c) $ \frac{1}{a}\tan ax$. (g) $ \dfrac{a^{bx}}{b\ln a}$. (h) $ \frac{1}{a}\ln\vert ax+b\vert$.

6.
$ f$: ii $ f'$: i $ F$: iii $ \phi$: iv
8.
(b) $ s=t^{2}$ (d) $ s=\frac{3}{\pi}\sin \pi t+4$

9.
(b) $ v=t-\frac{1}{2}t^{2}-24,
s=\frac{1}{2}t^{2}-\frac{1}{6}t^{3}-24t+55$

(d) $ v=\frac{5}{2}\cos 2t, s=\frac{5}{4}\sin 2t$

12.
24,806.25 feet. Seems too high, so air resistance is probably substantial.

15.
400 ft/sec.

16.
$ \sqrt{2gh}$.

18.
240 ft.

20.
$ \dfrac{d^{2}s}{dt^{2}}=\dfrac{d}{dt}[\dfrac{d}{dt}(C\cos\omega
t+D\sin\omega t...
...-D\omega^{2}\sin \omega t=-\omega^{2}(C\cos\omega
t+D\sin\omega t)=-\omega^{2}s$.

23.
$ s=\dfrac{pv_{0}}{2\pi}\sin\left(\dfrac{2\pi t}{p}\right)$.

26.
(a) $ g(0)=g(0+0)=g(0)+g(0)-3\cdot 0\cdot 0=g(0)+g(0)\Rightarrow
0=g(0)$.

(b) $ g'(x)=\displaystyle{\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}=\lim_{h\to 0}
\frac{g(x)+g(h)-3xh-g(x)}{h} =\lim_{h\to 0} \left(3x+\frac{g(h)}{h}\right)} =
3x+5$.

(c) $ g(x)=\frac{3}{2}x^{2}+5x+C,g(0)=C=0\Rightarrow g(x)=\frac{3}{2}x^{2}+5x$.





Jason Grout 2003-03-19