\sethandouttitle{The Exponential and Logarithm Functions} \begin{enumerate} \item Evaluate $f(x) = x^2$ and $g(x) = 2^x$ at $x=-3,-2,-1,0,1,2,3,4,5$. Graph both functions on the same graph. \item Find the domain of $\displaystyle f(x) = \frac{ 1 - e^{x^2} }{ 1 - e^{1-x^2} }$ and $\displaystyle g(x) = \frac{ 1+x }{ e^{\cos x} }$. \item Find the limit. \begin{enumerate} \item $\displaystyle\lim_{ x \rightarrow \infty } (1.0001)^x $. \item $\displaystyle\lim_{x \rightarrow \infty} \frac{ e^{3x} - e^{-3x} }{ e^{3x} + e^{-3x} }$. \item $\displaystyle \lim_{x \rightarrow \infty} ( e^{-2x} \cos x)$. \end{enumerate} \item Is $g(x) = 1/x$ one-to-one? \item If $g(x) = 3 +x + e^x$, find $g^{-1}(4)$. \newpage \item The formula $C = \frac{5}{9}(F-32)$ expresses Celsius $C$ as a function of Fahrenheit $F$. Find a formula for the inverse function and interpret it. \item Find the inverse of the function $f(x) = 1 + \sqrt{ 2 + 3x }$. \item Find the inverse of the function $f(x) = e^{2x-1}$. \item Find $ (f^{-1})'(a)$ using the inverse derivative formula. \begin{enumerate} \item $f(x) = x^3$. \item $f(x) = 9-x^2$. \end{enumerate} Then, find the inverse function first and differentiate it. Do your answers agree? \item Find the exact value of each expression (without a calculator). \begin{enumerate} \item $\log_5(125)$. \item $\displaystyle \log_3\left( \frac{1}{27} \right)$. \item $\log_2 6 - \log_2 15 + \log_2 20$. \end{enumerate} \item Solve for $x$: \begin{enumerate} \item $e^{7-4x}=6$. \item $\ln(3x-10) = 2$. \item $\ln x + \ln (x-1) = 1$. \end{enumerate} \end{enumerate}