Page:Calculus Made Easy.pdf/280

Exercises XII. (p. 153.)

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

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

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

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

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

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

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

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

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

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

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

(14) Min.: $y=0.7$ for $x=0.694$.

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

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

 

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Exercises XIII. (p. 162.)

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

(2) $T=34.627$; $159.46$ minutes.

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

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

(6) $0.14$ second.

(7) (a) $1.642$; (b) $15.58$.

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

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

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

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

(12) Max. for $x=\epsilon$.

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

 

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