Page:Calculus Made Easy.pdf/280

 (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) $$\frac{3}{x} (\log_\epsilon ax)^2$$.

(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 $$x=\log_\epsilon a$$.