91Ó°ÊÓ

Finding an Inverse Function In Exercises \(35-46\) , (a) find the inverse function of \(f,\) (b) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs, and (d) state the domains and ranges of \(f\) and \(f^{-1}\) . $$f(x)=\frac{x}{\sqrt{x^{2}+7}}$$

In Exercises \(43-46\) , find the general solution of the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. $$\frac{d y}{d x}=\frac{2 x}{x^{2}-9}, \quad(0,4)$$

Finding a Derivative In Exercises \(43-66,\) find the derivative of the function. $$f(x)=\ln \left(x^{2}+3\right)$$

In Exercises \(33-54,\) find the derivative of the function. $$y=\ln \left(2-e^{5 x}\right)$$

In Exercises \(41-56,\) find the derivative of the function. $$f(x)=\arctan e^{x}$$

Finding a Derivative In Exercises \(39-60,\) find the derivative of the function. $$f(t)=\frac{-2 t^{2}}{8^{t}}$$

Finding an Inverse Function In Exercises \(35-46\) , (a) find the inverse function of \(f,\) (b) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs, and (d) state the domains and ranges of \(f\) and \(f^{-1}\) . $$f(x)=\frac{x+2}{x}$$

Evaluating a limit In Exercises \(43-62\) , (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hopital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). $$\lim _{x \rightarrow \infty} x \tan \frac{1}{x}$$

In Exercises 43-46, use the specified substitution to find or evaluate the integral. $$\begin{array}{l}{\int_{0}^{1} \frac{d x}{2 \sqrt{3-x} \sqrt{x+1}}} \\\ {u=\sqrt{x+1}}\end{array}$$

Finding a Derivative In Exercises \(39-60,\) find the derivative of the function. $$f(t)=\frac{3^{2 t}}{t}$$