Page:Calculus Made Easy.pdf/276

(2) ${\displaystyle -b}$, ${\displaystyle 0}$.

(3) ${\displaystyle 2}$, ${\displaystyle 0}$.

(4) ${\displaystyle 56440x^{3}-196212x^{2}-4488x+8192}$.
${\displaystyle \quad \quad \quad 169320x^{2}-392424x-4488}$.

(5) ${\displaystyle 2}$, ${\displaystyle 0}$.

(6) ${\displaystyle 371.80453x}$, ${\displaystyle 371.80453}$.

(7) ${\displaystyle {\dfrac {30}{(3x+2)^{3}}}}$, ${\displaystyle -{\dfrac {270}{(3x+2)^{4}}}}$.

(Examples, p. 41):

(1) ${\displaystyle {\dfrac {6a}{b^{2}}}x}$, ${\displaystyle {\dfrac {6a}{b^{2}}}}$.

(2) ${\displaystyle {\dfrac {3a{\sqrt {b}}}{2{\sqrt {x}}}}-{\dfrac {6b{\sqrt[{3}]{a}}}{x^{3}}}}$, ${\displaystyle {\dfrac {18b{\sqrt[{3}]{a}}}{x^{4}}}-{\dfrac {3a{\sqrt {b}}}{4{\sqrt {x^{3}}}}}}$.

(3) ${\displaystyle {\dfrac {2}{\sqrt[{3}]{\theta ^{8}}}}-{\dfrac {1.056}{\sqrt[{5}]{\theta ^{11}}}}}$, ${\displaystyle {\dfrac {2.3232}{\sqrt[{5}]{\theta ^{16}}}}-{\dfrac {16}{3{\sqrt[{3}]{\theta ^{11}}}}}}$.

(4) ${\displaystyle 810t^{4}-648t^{3}+479.52t^{2}-139.968t+26.64}$.
${\displaystyle \quad \quad \quad 3240t^{3}-1944t^{2}+959.04t-139.968}$.

(5) ${\displaystyle 12x+2}$, ${\displaystyle 12}$.

(6) ${\displaystyle 6x^{2}-9x}$, ${\displaystyle 12x-9}$.

(7) ${\displaystyle {\begin{aligned}&{\dfrac {3}{4}}\left({\dfrac {1}{\sqrt {\theta }}}+{\dfrac {1}{\sqrt {\theta ^{5}}}}\right)+{\dfrac {1}{4}}\left({\dfrac {15}{\sqrt {\theta ^{7}}}}-{\dfrac {1}{\sqrt {\theta ^{3}}}}\right).\\&{\dfrac {3}{8}}\left({\dfrac {1}{\sqrt {\theta ^{5}}}}-{\dfrac {1}{\sqrt {\theta ^{3}}}}\right)-{\dfrac {15}{8}}\left({\dfrac {7}{\sqrt {\theta ^{9}}}}+{\dfrac {1}{\sqrt {\theta ^{7}}}}\right).\end{aligned}}}$

(2) ${\displaystyle 64}$; ${\displaystyle 147.2}$; and ${\displaystyle 0.32}$ feet per second.

(3) ${\displaystyle {\dot {x}}=a-gt}$; ${\displaystyle {\ddot {x}}=-g}$.

(4) ${\displaystyle 45.1}$ feet per second.

(5) ${\displaystyle 12.4}$ feet per second per second. Yes.

(6) Angular velocity ${\displaystyle =11.2}$ radians per second; angular acceleration ${\displaystyle =9.6}$ radians per second per second.

(7) ${\displaystyle v=20.4t^{2}-10.8}$. ${\displaystyle a=40.8t}$. ${\displaystyle 172.8in./sec.}$, ${\displaystyle 122.4in./sec^{2}}$.

(8) ${\displaystyle v={\dfrac {1}{30{\sqrt[{3}]{(t-125)^{2}}}}}}$, ${\displaystyle a=-{\dfrac {1}{45{\sqrt[{3}]{(t-125)^{5}}}}}}$.

(9) ${\displaystyle v=0.8-{\dfrac {8t}{(4+t^{2})^{2}}}}$, ${\displaystyle a={\dfrac {24t^{2}-32}{(4+t^{2})^{3}}}}$, ${\displaystyle 0.7926}$ and ${\displaystyle 0.00211}$.

(10) ${\displaystyle n=2}$, ${\displaystyle n=11}$.