Chapter 10

In Exercises, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$ N(t)=500 e^{-0.2 t} $$

In Exercises, find the derivative of the function. $$ y=x^{2} e^{x}-2 x e^{x}+2 e^{x} $$

In Exercises, sketch the graph of the function. $$ y=5+\ln x $$

In Exercises, find the derivative of the function. $$ y=\ln \frac{x}{x+1} $$

In Exercises, sketch the graph of the function. $$ y=3 \ln x $$

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=e^{-2 x+x^{2}}, \quad(2,1) $$

In Exercises, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$ g(x)=\frac{2}{1+e^{x^{2}}} $$

Find the half-life of a radioactive material if after 1 year \(99.57 \%\) of the initial amount remains.

In Exercises, find the derivative of the function. $$ y=\ln \frac{x^{2}}{x^{2}+1} $$

In Exercises, sketch the graph of the function. $$ y=\frac{1}{4} \ln x $$

In Exercises, determine an equation of the tangent line to the function at the given point. $$ g(x)=e^{x^{3}}, \quad\left(-1, \frac{1}{e}\right) $$

In Exercises, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$ g(x)=\frac{10}{1+e^{-x}} $$

\({ }^{14} \mathrm{C}\) dating assumes that the carbon dioxide on the Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a similar tree today. A piece of ancient charcoal contains only \(15 \%\) as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of \({ }^{14} \mathrm{C}\) is 5715 years.)

In Exercises, find the derivative of the function. $$ y=\ln \sqrt[3]{\frac{x-1}{x+1}} $$

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=x^{2} e^{-x}, \quad\left(2, \frac{4}{e^{2}}\right) $$

In Exercises, use a graphing utility to graph the function. Determine whether the function has any horizontal asymptotes and discuss the continuity of the function. $$ f(x)=\frac{e^{x}+e^{-x}}{2} $$

In Exercises, find the derivative of the function. $$ y=\ln \sqrt{\frac{x+1}{x-1}} $$

In Exercises, analytically show that the functions are inverse functions. Then use a graphing utility to show this graphically. $$ \begin{aligned} &f(x)=e^{x}-1 \\ &g(x)=\ln (x+1) \end{aligned} $$

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=\frac{x}{e^{2 x}}, \quad\left(1, \frac{1}{e^{2}}\right) $$

In Exercises, use a graphing utility to graph the function. Determine whether the function has any horizontal asymptotes and discuss the continuity of the function. $$ f(x)=\frac{e^{x}-e^{-x}}{2} $$

In Exercises, find the derivative of the function. $$ y=\ln \frac{\sqrt{4+x^{2}}}{x} $$

In Exercises, analytically show that the functions are inverse functions. Then use a graphing utility to show this graphically. $$ \begin{aligned} &f(x)=e^{2 x-1} \\ &g(x)=\frac{1}{2}+\ln \sqrt{x} \end{aligned} $$

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=\left(e^{2 x}+1\right)^{3}, \quad(0,8) $$

In Exercises, use a graphing utility to graph the function. Determine whether the function has any horizontal asymptotes and discuss the continuity of the function. $$ f(x)=\frac{2}{1+e^{1 / x}} $$

In Exercises, find exponential models \(y_{1}=C e^{k_{1} t}\) and \(y_{2}=C(2)^{k_{2} t}\) that pass through the points. Compare the values of \(k_{1}\) and \(k_{2} .\) Briefly explain your results. $$ (0,8),\left(20, \frac{1}{2}\right) $$

In Exercises, find the derivative of the function. $$ y=\ln \left(x \sqrt{4+x^{2}}\right) $$

In Exercises, determine an equation of the tangent line to the function at the given point. $$ y=\left(e^{4 x}-2\right)^{2}, \quad(0,1) $$

In Exercises, use a graphing utility to graph the function. Determine whether the function has any horizontal asymptotes and discuss the continuity of the function. $$ f(x)=\frac{2}{1+2 e^{-0.2 x}} $$

The number of a certain type of bacteria increases continuously at a rate proportional to the number present. There are 150 present at a given time and 450 present 5 hours later. (a) How many will there be 10 hours after the initial time? (b) How long will it take for the population to double? (c) Does the answer to part (b) depend on the starting time? Explain your reasoning.

In Exercises, find the derivative of the function. $$ g(x)=e^{-x} \ln x $$

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ \ln e^{x^{2}} $$

In Exercises, find \(d y / d x\) implicitly. $$ x e^{y}-10 x+3 y=0 $$

In Exercises, use a graphing utility to graph the function. $$ y=2^{x-1} $$

In Exercises, find the derivative of the function. $$ f(x)=x \ln e^{x^{2}} $$

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ \ln e^{2 x-1} $$

In Exercises, find \(d y / d x\) implicitly. $$ x^{2} y-e^{y}-4=0 $$

Use a graphing utility to graph the function. Describe the shape of the graph for very large and very small values of \(x\). (a) \(f(x)=\frac{8}{1+e^{-0.5 x}}\) (b) \(g(x)=\frac{8}{1+e^{-0.5 / x}}\)

In Exercises, use a graphing utility to graph the function. $$ y=4^{x}+3 $$

In Exercises, find the derivative of the function. $$ g(x)=\ln \frac{e^{x}+e^{-x}}{2} $$

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ e^{\ln (5 x+2)} $$

In Exercises, find \(d y / d x\) implicitly. $$ x^{2} e^{-x}+2 y^{2}-x y=0 $$

In Exercises, use a graphing utility to graph the function. $$ y=-2^{x} $$

In Exercises, find the derivative of the function. $$ f(x)=\ln \frac{1+e^{x}}{1-e^{x}} $$

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ e^{\ln \sqrt{x}} $$

In Exercises, find \(d y / d x\) implicitly. $$ e^{x y}+x^{2}-y^{2}=10 $$

In Exercises, use a graphing utility to graph the function. $$ y=-5^{x} $$

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ -1+\ln e^{2 x} $$

In Exercises, find the second derivative. $$ f(x)=2 e^{3 x}+3 e^{-2 x} $$

In Exercises, use a graphing utility to graph the function. $$ y=3^{-x^{2}} $$

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ -8+e^{\ln x^{3}} $$