91Ó°ÊÓ

\(39-40=\) Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value. $$\lim _{x \rightarrow 0} \frac{5^{x}-4^{x}}{3^{x}-2^{x}}$$

A lighthouse is located on a small island, 3 \(\mathrm{km}\) away from the nearest point \(P\) on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 \(\mathrm{km}\) from \(P\) ?

Suppose \(f^{-1}\) is the inverse function of a differentiable function \(f\) and let \(G(x)=1 / f^{-1}(x)\) . If \(f(3)=2\) and \(f^{\prime}(3)=\frac{1}{y}\) , find \(G(2)\) .

Find the derivative. Simplify where possible. $$ y=\operatorname{coth}^{-1}(\sec x) $$

Prove that $$\quad \lim _{x \rightarrow \infty} \frac{e^{x}}{x^{n}}=\infty$$ for any positive integer \(n .\) This shows that the exponential function approaches infinity faster than any power of \(x .\)

(a) How is the logarithmic function \(y=\log _{a} x\) defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function y \(=\log _{a} x\) if \(a>1\)

$$41-44=\( Find \)y^{\prime}\( and \)y^{\prime \prime}$$ $$y=e^{\alpha x} \sin \beta x$$

Prove that $$\lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}}=0$$ for any number \(p>0 .\) This shows that the logarithmic function approaches \(\infty\) more slowly than any power of \(x .\)

(a) Sketch the graph of the function \(f(x)=\sin \left(\sin ^{-1} x\right)\) (b) Sketch the graph of the function \(g(x)=\sin ^{-1}(\sin x)\) \(x \in \mathbb{R} .\) (c) Show that \(g^{\prime}(x)=\frac{\cos x}{|\cos x|}\) (d) Sketch the graph of \(h(x)=\cos ^{-1}(\sin x), x \in \mathbb{R},\) and find its derivative.

$$41-44=\( Find \)y^{\prime}\( and \)y^{\prime \prime}$$ $$y=\frac{\ln x}{r^{2}}$$