Chapter 10

In Exercises, find \(d y / d x\) implicitly. $$ x^{2}-3 \ln y+y^{2}=10 $$

In Exercises, solve for \(x\) or \(t\). $$ \ln x=0 $$

Use Example 6 as a model to show that the graph of the normal probability density function with \(\mu=0\) \(f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-x^{2} / 2 \sigma^{2}}\) has points of inflection at \(x=\pm \sigma .\) What is the maximum value of the function? Use a graphing utility to verify your answer by graphing the function for several values of \(\sigma\).

In Exercises, find \(d y / d x\) implicitly. $$ \ln x y+5 x=30 $$

In Exercises, solve for \(x\) or \(t\). $$ 2 \ln x=4 $$

In Exercises, find \(d y / d x\) implicitly. $$ 4 x^{3}+\ln y^{2}+2 y=2 x $$

In Exercises, solve for \(x\) or \(t\). $$ \ln 2 x=1.2 $$

In Exercises, find \(d y / d x\) implicitly. $$ 4 x y+\ln \left(x^{2} y\right)=7 $$

In Exercises, solve for \(x\) or \(t\). $$ \ln 5 x=1 $$

In Exercises, use implicit differentiation to find an equation of the tangent line to the graph at the given point. $$ x+y-1=\ln \left(x^{2}+y^{2}\right), \quad(1,0) $$

In Exercises, solve for \(x\) or \(t\). $$ 3 \ln 5 x=8 $$

In Exercises, use implicit differentiation to find an equation of the tangent line to the graph at the given point. $$ y^{2}+\ln (x y)=2, \quad(e, 1) $$

In Exercises, solve for \(x\) or \(t\). $$ 2 \ln 4 x=7 $$

In Exercises, solve for \(x\) or \(t\). $$ e^{x+1}=4 $$

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

In Exercises, solve for \(x\) or \(t\). $$ e^{-0.5 x}=0.075 $$

In Exercises, find the second derivative of the function. $$ f(x)=3+2 \ln x $$

In Exercises, solve for \(x\) or \(t\). $$ 300 e^{-0.2 t}=700 $$

In Exercises, find the second derivative of the function. $$ f(x)=2+x \ln x $$

In Exercises, solve for \(x\) or \(t\). $$ 400 e^{-0.0174 t}=1000 $$

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

In Exercises, solve for \(x\) or \(t\). $$ 4 e^{2 x-1}-1=5 $$

In Exercises, find the second derivative of the function. $$ f(x)=5^{x} $$

In Exercises, solve for \(x\) or \(t\). $$ 2 e^{-x+1}-5=9 $$

In Exercises, find the second derivative of the function. $$ f(x)=\log _{10} x $$

In Exercises, solve for \(x\) or \(t\). $$ \frac{10}{1+4 e^{-0.01 x}}=2.5 $$

In Exercises, solve for \(x\) or \(t\). $$ \frac{50}{1+12 e^{-0.02 x}}=10.5 $$

The temperatures \(T\left({ }^{\circ} \mathrm{F}\right)\) at which water boils at selected pressures \(p\) (pounds per square inch) can be modeled by \(T=87.97+34.96 \ln p+7.91 \sqrt{p}\) Find the rate of change of the temperature when the pressure is 60 pounds per square inch.

In Exercises, solve for \(x\) or \(t\). $$ 5^{2 x}=15 $$

In Exercises, find the slope of the graph at the indicated point. Then write an equation of the tangent line to the graph of the function at the given point. $$ f(x)=1+2 x \ln x, \quad(1,1) $$

In Exercises, solve for \(x\) or \(t\). $$ 2^{1-x}=6 $$

In Exercises, find the slope of the graph at the indicated point. Then write an equation of the tangent line to the graph of the function at the given point. $$ f(x)=2 \ln x^{3}, \quad(e, 6) $$

In Exercises, solve for \(x\) or \(t\). $$ 500(1.07)^{t}=1000 $$

In Exercises, find the slope of the graph at the indicated point. Then write an equation of the tangent line to the graph of the function at the given point. $$ f(x)=\ln \frac{5(x+2)}{x}, \quad(-2.5,0) $$

In Exercises, find the slope of the graph at the indicated point. Then write an equation of the tangent line to the graph of the function at the given point. $$ f(x)=\ln (x \sqrt{x+3}), \quad(1.2,0.9) $$

In Exercises, find the slope of the graph at the indicated point. Then write an equation of the tangent line to the graph of the function at the given point. $$ f(x)=x \log _{2} x, \quad(1,0) $$

In Exercises, solve for \(x\) or \(t\). $$ \left(1+\frac{0.07}{12}\right)^{12 t}=3 $$

In Exercises, find the slope of the graph at the indicated point. Then write an equation of the tangent line to the graph of the function at the given point. $$ f(x)=x^{2} \log _{3} x, \quad(1,0) $$

In Exercises, solve for \(x\) or \(t\). $$ \left(1+\frac{0.06}{12}\right)^{12 t}=5 $$

In Exercises, graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=x-\ln x $$

In Exercises, graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=\frac{x}{\ln x} $$

In Exercises, graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=\frac{\ln x}{x} $$

In Exercises, \$ 3000\( is invested in an account at interest rate \)r$, compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$ r=0.085 $$

In Exercises, graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=x \ln x $$

In Exercises, \$ 3000\( is invested in an account at interest rate \)r$, compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$ r=0.12 $$

In Exercises, graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=x^{2} \ln \frac{x}{4} $$

A deposit of \(\$ 1000\) is made in an account that earns interest at an annual rate of \(5 \%\). How long will it take for the balance to double if the interest is compounded (a) annually, (b) monthly, (c) daily, and (d) continuously?

In Exercises, graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=(\ln x)^{2} $$

Use a spreadsheet to complete the table, which shows the time \(t\) necessary for \(P\) dollars to triple if the interest is compounded continuously at the rate of \(r\). $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% & 14 \% \\ \hline t & & & & & & & \\ \hline \end{array} $$

In Exercises, find \(d x / d p\) for the demand function. Interpret this rate of change when the price is \(\$ 10\). $$ x=\ln \frac{1000}{p} $$