Chapter 3

Use the definitions of the hyperbolic functions to find each of the following limits. (a) \(\lim _{x \rightarrow \infty} \tanh x\) (b) \(\lim _{x \rightarrow-\infty} \tanh x\) (c) \(\lim _{x \rightarrow \infty} \sinh x\) (d) \(\lim _{x \rightarrow-\infty} \sinh x\) (e) \(\lim _{x \rightarrow \infty} \operatorname{sech} x\) (f) \(\lim _{x \rightarrow \infty} \operatorname{coth} x\) (g) \(\lim _{x \rightarrow 0^{+}}\) coth \(x\) (h) \(\lim _{x \rightarrow 0^{-}} \operatorname{coth} x\) (i) \(\lim _{x \rightarrow-\infty} \operatorname{csch} x\)

Find the derivative of the function. Simplify where possible. \(y=\sin ^{-1}(2 x+1)\)

If \(\$ 3000\) is invested at 5\(\%\) interest, find the value of the investment at the end of 5 years if the interest is compounded (a) annually (b) semiannually (c) monthly (d) weekly (e) daily (f) continuously

The formula \(C=\frac{5}{9}(F-32),\) where \(F \geqslant-459.67\) expresses the Celsius temperature \(C\) as a function of the Fahresseit temperature \(F .\) Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

Differentiate the function. $$ g(x)=\sqrt{x} e^{x} $$

\(1-38=\) 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. $$\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}-2 x}{x-\sin x}$$

Find the derivative of the function. Simplify where possible. $$y=\tan ^{-1}\left(x-\sqrt{1+x^{2}}\right)$$

(a) How long will it take an investment to double in value if the interest rate is 6\(\%\) compounded continuously? (b) What is the equivalent annual interest rate?

In the theory of relativity, the mass of a particle with speed \(v\) is $$ m=f(v)=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}} $$ where \(m_{0}\) is the rest mass of the particle and \(c\) is the speed of light in a vacuum. Find the inverse function of \(f\) and explain its meaning.

Compare the rates of growth of the functions \(f(x)=x^{5}\) and \(g(x)=5^{x}\) by graphing both functions in several view- ing rectangles. Find all points of intersection of the graphs correct to one decimal place.

Differentiate the function. $$ y=\frac{x}{e^{x}} $$

\(1-38=\) 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. $$\lim _{x \rightarrow 0} \frac{\cos x-1+\frac{1}{2} x^{2}}{x^{4}}$$

Find the derivative of the function. Simplify where possible. $$G(x)=\sqrt{1-x^{2}} \text { arccos } x$$

Find a formula for the inverse of the function. $$ f(x)=1+\sqrt{2+3 x} $$

Compare the functions \(f(x)=x^{10}\) and \(g(x)=e^{x}\) by graph- ing both \(f\) and \(g\) in several viewing rectangles. When does the graph of \(g\) finally surpass the graph of \(f ?\)

Differentiate the function. $$ y=\frac{e^{x}}{1-e^{x}} $$

Find the derivative of the function. Simplify where possible. $$F(\theta)=\arcsin \sqrt{\sin \theta}$$

\(1-38=\) 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. $$\lim _{x \rightarrow a^{+}} \frac{\cos x \ln (x-a)}{\ln \left(e^{x}-e^{a}\right)}$$

Find a formula for the inverse of the function. $$ f(x)=\frac{4 x-1}{2 x+3} $$

Differentiate the function. $$ y=\sqrt{1+2 e^{3 x}} $$

Find the derivative of the function. Simplify where possible. $$h(t)=\cot ^{-1}(t)+\cot ^{-1}(1 / t)$$

\(1-38=\) 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. $$\lim _{x \rightarrow 0} \cot 2 x \sin 6 x$$

Find a formula for the inverse of the function. $$ f(x)=e^{2 x-1} $$

Find the limit. $$\lim _{x \rightarrow \infty}(1.001)^{x}$$

Differentiate the function. $$ y=e^{-2 t} \cos 4 t $$

Find the derivative of the function. Simplify where possible. $$y=\cos ^{-1}\left(\sin ^{-1} t\right)$$

\(1-38=\) 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. $$\lim _{x \rightarrow \infty} \sqrt{x} e^{-x / 2}$$

Find a formula for the inverse of the function. $$ y=x^{2}-x, \quad x \geqslant \frac{1}{2} $$

Find the limit. $$\lim _{x \rightarrow \infty} e^{-x^{2}}$$

Differentiate the function. $$ y=5^{-1 / x} $$

Find the derivative of the function. Simplify where possible. $$y=\arctan (\cos \theta)$$

\(1-38=\) 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. $$\lim _{x \rightarrow \infty} x^{3} e^{-x^{2}}$$

Find a formula for the inverse of the function. $$ y=\ln (x+3) $$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{e^{3 x}-e^{-3 x}}{e^{3 x}+e^{-3 x}}$$

Differentiate the function. $$ y=10^{1-x^{2}} $$

Find the derivative of the function. Simplify where possible. $$ f(x)=\tanh \left(1+e^{2 x}\right) $$

\(1-38=\) 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. $$\lim _{x \rightarrow 0^{+}} \sin x \ln x$$

Find a formula for the inverse of the function. $$ y=\frac{e^{x}}{1+2 e^{x}} $$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{2+10^{x}}{3-10^{x}}$$

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

Find the derivative. Simplify where possible. $$ f(x)=x \sinh x-\cosh x $$

\(1-38=\) 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. $$\lim _{x \rightarrow 1^{+}} \ln x \tan (\pi x / 2)$$

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

Find an explicit formula for \(f^{-1}\) and use it to graph \(f^{-1}, f,\) and the line \(y=x\) on the same screen. To check your work, see whether the graphs of \(f\) and \(f^{-1}\) are reflections about the line. $$ f(x)=x^{4}+1, \quad x \geqslant 0 $$

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

Differentiate the function. $$ y=\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}} $$

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

\(1-38=\) 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. $$\lim _{x \rightarrow \infty} x \tan (1 / x)$$

Find the derivative of the function. Simplify where possible. \(y=\arctan \sqrt{\frac{1-x}{1+x}}\)

Find an explicit formula for \(f^{-1}\) and use it to graph \(f^{-1}, f,\) and the line \(y=x\) on the same screen. To check your work, see whether the graphs of \(f\) and \(f^{-1}\) are reflections about the line. $$ f(x)=2-e^{x} $$