blob: 4c03b10083eebbd408300cefd077fffee425ae50 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
|
\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}
|
