Page:Calculus Made Easy.pdf/280

(1) ${\displaystyle ab(\epsilon ^{ax}+\epsilon ^{-ax})}$.

(2) ${\displaystyle 2at+{\frac {2}{t}}}$.

(3) ${\displaystyle \log _{\epsilon }n}$.

(5) ${\displaystyle npv^{n-1}}$.

(6) ${\displaystyle {\frac {n}{x}}}$.

(7) ${\displaystyle {\frac {3\epsilon ^{-{\frac {x}{x-1}}}}{(x-1)^{2}}}}$.

(8) ${\displaystyle 6x\epsilon ^{-5x}-5(3x^{2}+1)\epsilon ^{-5x}}$.

(9) ${\displaystyle {\frac {ax^{a-1}}{x^{a}+a}}}$.

(10) ${\displaystyle \left({\frac {6x}{3x^{2}-1}}+{\frac {1}{2\left({\sqrt {x}}+x\right)}}\right)\left(3x^{2}-1\right)\left({\sqrt {x}}+1\right)}$.

(11) ${\displaystyle {\frac {1-\log _{\epsilon }\left(x+3\right)}{\left(x+3\right)^{2}}}}$.

(12) ${\displaystyle a^{x}\left(ax^{a-1}+x^{a}\log _{\epsilon }a\right)}$.

(14) Min.: ${\displaystyle y=0.7}$ for ${\displaystyle x=0.694}$.

(15) ${\displaystyle {\frac {1+x}{x}}}$.

(16) ${\displaystyle {\frac {3}{x}}(\log _{\epsilon }ax)^{2}}$.

(1) Let ${\displaystyle {\frac {t}{T}}=x}$ (∴ ${\displaystyle t=8x}$), and use the Table on page 159.

(2) ${\displaystyle T=34.627}$; ${\displaystyle 159.46}$ minutes.

(3) Take ${\displaystyle 2t=x}$; and use the Table on page 159.

(5) (a) ${\displaystyle x^{x}\left(1+\log _{\epsilon }x\right)}$; (b) ${\displaystyle 2x(\epsilon ^{x})^{x}}$; (c) ${\displaystyle \epsilon ^{x^{x}}\times x^{x}\left(1+\log _{\epsilon }x\right)}$.

(6) ${\displaystyle 0.14}$ second.

(7) (a) ${\displaystyle 1.642}$; (b) ${\displaystyle 15.58}$.

(8) ${\displaystyle \mu =0.00037}$, ${\displaystyle 31^{m}{\frac {1}{4}}}$.

(9) ${\displaystyle i}$ is ${\displaystyle 63.4\%}$ of ${\displaystyle i_{0}}$, ${\displaystyle 220}$ kilometres.

(10) ${\displaystyle 0.133}$, ${\displaystyle 0.145}$ ${\displaystyle 0.155}$, mean ${\displaystyle 0.144}$; ${\displaystyle -{10.2}\%}$, ${\displaystyle -{0.9}\%}$, ${\displaystyle +{77.2}\%}$.

(11) Min. for ${\displaystyle x={\frac {1}{\epsilon }}}$.

(12) Max. for ${\displaystyle x=\epsilon }$.

(13) Min. for ${\displaystyle x=\log _{\epsilon }a}$.