One must be careful to distinguish between the inverse of a function and the reciprocal of the function. If [math]f(x) = 3x + 4[/math], write

• [math]f^{-1}(x)[/math], the value of the inverse function [math]f^{-1}[/math] at [math]x[/math].
• [math][f(x)]^{-1}[/math], the reciprocal of [math]f(x)[/math].

If [math]f[/math] has an inverse [math]f^{-1}[/math], what relations hold between the domains and ranges of [math]f[/math] and [math]f^{-1}[/math]?

Find the derivative with respect to [math]x[/math] of each of the following functions.

• [math]e^{7x}[/math]
• [math]\frac13e^{3x+2}[/math]
• [math]xe^x[/math]
• [math]e^{-x^2}[/math]
• [math]e^x \ln x[/math]
• [math]\frac{e^x+e^-x}2[/math]
• [math]\frac{e^x-e^-x}2[/math]
• [math]\frac3{2e^x}[/math]
• [math]\frac{e^x}x[/math]
• [math]\frac{5\ln x}x[/math]
• [math]e^{3x^2 - 4x + 5}[/math]
• [math]e^{ax+b}[/math].

Solve each of the following integrals.

• [math]\int e^{3x} \;dx[/math]
• [math]\int 2xe^{x^2} \;dx[/math]
• [math]\int \frac1x e^{\ln x} \;dx[/math]
• [math]\int \frac{e^x + e^{-x}}{e^x - e^{-x}} \;dx[/math]
• [math]\int \frac{xe^{x^2} \;dx}{4e^{x^2} + 5}[/math]
• [math]\int \frac {3\;dx}{2e^{4x}}[/math]
• [math]\int \frac {x^2\;dx}{e^{x^3-2}}[/math]
• [math]\int (x+1) e^{x^2+2x} \; dx[/math]
• [math]\int e^\pi \;dx[/math]
• [math]\int e^{ax+b} \;dx[/math].

Evaluate each of the following integrals.

• [math]\int_1^2 \frac{dx}x[/math]
• [math]\int_0^3 e^{2x} \;dx[/math]
• [math]\int_x^{x^2} e^t \; dt[/math]
• [math]\int_4^8 \frac{dx}{e^x}[/math].

Sketch the graph of each of the following equations. Label all extreme points and point of inflection, and give the values of [math]x[/math] at which these occur. Classify each extreme point as a local maximum or minimum.

• [math]y=e^{3x}[/math]
• [math]y=x \ln x[/math]
• [math]y = xe^{-x}[/math]
• [math]y = x^2e^{-x}[/math]
• [math]y = e^{-x^2}[/math].

In each of the following examples, find the area of the region above the [math]x[/math]-axis, below the graph of the function [math]f[/math], and between two vertical lines whose equations are given.

• [math]f(x) = 2e^{4x}[/math], [math]x=0[/math] and [math]x=2[/math].
• [math]f(x) = xe^{x^2}[/math], [math]x=2[/math] and [math]x=4[/math].
• [math]f(x) = \frac1{e^x}[/math], [math]x=5[/math] and [math]x=7[/math].

Suppose [math]f[/math] is a function which has the property that it is equal to its own derivative; i.e., [math]f^\prime = f[/math].

• lab{5.2.8a} Compute the derivative of the quotient [math]\frac{f(x)}{e^x}[/math].
• Using the result of \ref{ex5.2.8a}, prove that [math]f(x) = ke^x[/math] for some constant [math]k[/math].

Let [math]f[/math] be a function with domain [math][0,1][/math] and defined by

Draw the graph of [math]f[/math] and the graph of the inverse function [math]f^{-1}[/math].

Compute each of the following limits using L'H\^opital's Rule or some other method if you prefer.

• [math]\lim_{x\goesto0} \frac{e^x-1}x[/math]
• [math]\lim_{x\goesto0} \frac{e^x-1-x}{x^2}[/math]
• [math]\lim_{x\goesto1} \frac{e^x-e}{x-1}[/math]
• [math]\lim_{x\goesto0} \frac{x^2e^x}{1-e^{x^2}}[/math]
• [math]\lim_{x\goesto\infty} \frac{x}{e^x}[/math]
• [math]\lim_{x\goesto\infty} \frac{e^x}{x^6}[/math]
• [math]\lim_{x\goesto0+} \frac{\ln x}{e^\frac1x}[/math]
• [math]\lim_{x\goesto0} \frac{e^x-e^{-x}}x[/math].