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Math 112 Solutions for 2.7, p. 179

1.
(a) $ 5x^{4}$. (c) $ -7x^{-8}$. (f) 0. (h) $ 15x^{4}-5$. (i) $ 2x-2x^{-3}$. (j) $ 3x^{2}+3x^{-4}$. (l) $ 16x^{3}+9x^{2}$.
2.
(b) $ (10x^{4}-8x)(3x^{10}+x^{2})+(2x^{5}-4x^{2})(30x^{9}+2x)$.

(c) $ 4(2x+1)$. (g) $ 1/(x+1)^{2}$. (h) $ \dfrac{2x^{2}-6x+2}{(2x-3)^{2}}$.

3.
(b) $ v=-32t, a=-32, s(4)=144, v(4)=-128, a(4)=-32$.
4.
(b) $ 60x^{3}$. (f) $ -\dfrac{2abc}{(bx+c)^{3}}$
5.
(a) $ \dfrac{(-1)^{n}n!}{(x+a)^{n+1}}$. (c) $ 100\cdot99\cdots
(101-n) x^{100-n}$ if $ n\leq 100$; 0 if $ n>100$.
6.
(a) $ y=2x-2$. (b) $ y=2x+3$ (c) $ y=13x-11$. (d) $ y=4$.
7.
(a) $ y=-\frac{1}{2}x+\frac{1}{2}$. (c) $ y=-\frac{1}{13}x+\frac{27}{13}$.
8.
(a) $ \frac{d}{dr}(\pi r^{2})=2\pi r$.
9.
$ \left(\dfrac{1}{\sqrt 3},\dfrac{1}{3\sqrt 3}\right)$ and $ \left(\dfrac{-1}{\sqrt 3},\dfrac{-1}{3\sqrt 3}\right)$
16.
$ (fgh)'=[(fg)h]' =(fg)'h+(fg)h'=(f'g+fg')h+(fg)h'=f'gh+fg'h+fgh'$.
19.
$ (\frac{f}{g})'=(f\cdot\frac{1}{g})'=
f'\cdot(\frac{1}{g})+f(\frac{1}{g})' =\fr...
...\cdot(\frac{-g'}{g^{2}})
= \frac{f'}{g}-\frac{fg'}{g^{2}}=\frac{f'g-fg'}{g^{2}}$.
20.
$ (fg)'''= f'''g+3f''g'+3f'g''+fg'''$, $ (fg)^{(4)}=f^{(4)}g+4f'''g' +6f''g''+4f'g'''+fg^{(4)}$
24.
Differentiation lowers the degree of a polynomial by 1. If $ p(x)$ is a polynomial of degree $ n$, then $ p^{(n)}(x)$ has degree 0, so is a constant. $ p^{(n+1)}(x)=0$.





Jason Grout 2003-02-07