Chapter 3

Find the limit. $$\lim _{x \rightarrow 2^{-}} e^{3 /(2-x)}$$

Differentiate the function. $$ y=\ln \left|2-x-5 x^{2}\right| $$

Find the derivative. Simplify where possible. $$ h(x)=\ln (\cosh x) $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{e^{x}-1}\right)$$

Find the derivative of the function. Simplify where possible. \(y=\arccos \left(\frac{b+a \cos x}{a+b \cos x}\right), \quad 0 \leqslant x \leqslant \pi, a>b>0\)

Find the limit. $$\lim _{x \rightarrow \infty}\left(e^{-2 x} \cos x\right)$$

Differentiate the function. $$ y=\sqrt{1+x e^{-2 x}} $$

Find the derivative. Simplify where possible. $$ y=x \operatorname{coth}\left(1+x^{2}\right) $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0}(\csc x-\cot x)$$

Find the derivative of the function. Find the domains of the function and its derivative. \(f(x)=\arcsin \left(e^{x}\right)\)

Find the limit. $$\lim _{x \rightarrow(\pi / 2)^{+}} e^{\tan x}$$

Differentiate the function. $$ f(t)=\tan \left(e^{t}\right)+e^{\tan t} $$

Find the derivative of the function. Find the domains of the function and its derivative. \(g(x)=\cos ^{-1}(3-2 x)\)

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow \infty}(x-\ln x)$$

(a) Show that \(f\) is one-to-one. (b) Use Theorem 7 to find \(\left(f^{-1}\right)^{\prime}(a)\) . (c) Calculate \(f^{-1}(x)\) and state the domain and range of \(f^{-1}\) (d) Calculate \(f^{-1}(x)\) and state the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of \(f\) and \(f^{-1}\) on the same axes. $$ f(x)=x^{3}, \quad a=8 $$

If you graph the function \(f(x)=\frac{1-e^{1 / x}}{1+e^{1 / x}}\) you'll see that \(f\) appears to be an odd function. Prove it.

Differentiate the function. $$ y=e^{k \tan \sqrt{x}} $$

$$y^{\prime} \text { if } \tan ^{-1}(x y)=1+x^{2} y$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1^{+}}\left[\ln \left(x^{7}-1\right)-\ln \left(x^{5}-1\right)\right]$$

Find \(y^{\prime}\) if tan \(^{-1}(x y)=1+x^{2} y.\)

Graph several members of the family of functions \(f(x)=\frac{1}{1+a e^{b x}}\) where \(a>0 .\) How does the graph change when \(b\) changes? How does it change when \(a\) changes?

Differentiate the function. $$ y=\ln \left(e^{-x}+x e^{-x}\right) $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0^{+}} x^{\sqrt{x}}$$

If \(g(x)=x \sin ^{-1}(x / 4)+\sqrt{16-x^{2}},\) find \(g^{\prime}(2).\)

Differentiate the function. $$ y=\left[\ln \left(1+e^{x}\right)\right]^{2} $$

Find an equation of the tangent line to the curve \(y=3 \arccos (x / 2)\) at the point \((1, \pi) .\)

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0^{+}}(\tan 2 x)^{x}$$

(a) Show that \(f\) is one-to-one. (b) Use Theorem 7 to find \(\left(f^{-1}\right)^{\prime}(a)\) . (c) Calculate \(f^{-1}(x)\) and state the domain and range of \(f^{-1}\) (d) Calculate \(f^{-1}(x)\) and state the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of \(f\) and \(f^{-1}\) on the same axes. $$ f(x)=1 /(x-1), \quad x>1, \quad a=2 $$

Differentiate the function. $$ y=2 x \log _{10} \sqrt{x} $$

Find the limit. $$\lim _{x \rightarrow-1^{+}} \sin ^{-1} x$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0}(1-2 x)^{1 / x}$$

Find $$\left(f^{-1}\right)^{\prime}(a)$$ $$ f(x)=2 x^{3}+3 x^{2}+7 x+4, \quad a=4 $$

Differentiate the function. $$ y=x^{2} e^{-1 / x} $$

Find the limit. $$\lim _{x \rightarrow \infty} \arccos \left(\frac{1+x^{2}}{1+2 x^{2}}\right)$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{b x}$$

Find $$\left(f^{-1}\right)^{\prime}(a)$$ $$ f(x)=x^{3}+3 \sin x+2 \cos x, \quad a=2 $$

Differentiate the function. $$ f(t)=\sin ^{2}\left(e^{\sin ^{2} t}\right) $$

Find the derivative. Simplify where possible. \(y=\sinh ^{-1}(\tan x)\)

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1^{+}} x^{1 /(1-x)}$$

Find the limit. $$\lim _{x \rightarrow \infty} \arctan \left(e^{x}\right)$$

Find $$\left(f^{-1}\right)^{\prime}(a)$$ $$ f(x)=3+x^{2}+\tan (\pi x / 2),-1 < x < 1, \quad a=3 $$

Differentiate the function. $$ y=\log _{2}\left(e^{-x} \cos \pi x\right) $$

Find the derivative. Simplify where possible. \(y=x \tanh ^{-1} x+\ln \sqrt{1-x^{2}}\)

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow \infty}\left(e^{x}+x\right)^{1 / x}$$

Find the limit. $$\lim _{x \rightarrow 0^{+}} \tan ^{-1}(\ln x)$$

Find the derivative. Simplify where possible. $$ y=x \sinh ^{-1}(x / 3)-\sqrt{9+x^{2}} $$

\(39-40=\) Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value. $$\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x}$$

A ladder 10 ft long leans against a vertical wall. If the bottom of the ladder slides away from the base of the wall at a speed of 2 \(\mathrm{ft} / \mathrm{s}\) , how fast is the angle between the ladder and the wall changing when the bottom of the ladder is 6 \(\mathrm{ft}\) from the base of the wall?

Suppose \(f^{-1}\) is the inverse function of a differentiable function \(f\) and \(f(4)=5, f^{\prime}(4)=\frac{2}{3} .\) Find \(\left(f^{-1}\right)^{\prime}(5)\)

Differentiate the function. $$ y=2^{3^{x^{2}}} $$