Math 101 Exam 2

Exponential Functions

• $Q(t)=ab^t$, where $a$ is the initial value, $b$ is the growth factor, and $b-1$ is the growth rate
• Exponential Growth occurs when $b>1$ and exponential decay occurs when $0<b<1$.

Compound Growth

• If a bank has a nominal interest rate of $r$ and compounds $n$ times a year, the amount of money in an account $t$ years after the principal $P$ is deposited $$A(t)=P(1+\frac{r}n)^{nt}.$$
• Effective growth rate: $(1+\frac{r}n)^n-1$

Continuous growth

• If a bank has a nominal interest rate of $r$ and compounds continuously, the amount of money in an account $t$ years after the principal $P$ is deposited is $$A(t)=Pe^{rt}.$$
• Effective growth rate: $e^r-1$

Overview of all exponential formulas: (In tex version)

Inverses

• If $y=f(x)$ is a function, the inverse is $x=f^{-1}(y)$ with inputs $y$ and outputs $x$.
• A function is invertible if its inverse is also a function.
  • Horizontal Line Test: If a horizontal line intersects a function at more then one spot, it's not invertible. (When the inputs and outputs swap roles, this horizontal line becomes a vertical line intersecting the graph at more then one spot...)

Logarithms

• The logarithmic function $t=\log_b(P)$ is the inverse of $P=b^t$.
  • So, solving for $t$ in $t=\log_3(9)$ is the same as finding $t$ so that it satisfies $3^t=9$. So $t=2$ in this case.
• $\log(t)=\log_{10}(t)$, so if $t=\log(P)$, $P=10^t$.
• $\ln(t)=\ln_e(t)$, so if $t=\ln(P)$, $P=e^t$.
• Properties:
  • $\log_b(x)+\log_b(y)=\log_b(xy)$
  • $\log_b(x)-\log_b(y)=\log_b(x/y)$
  • $\log_b(x^y)=y\log_b(x)$
    • So... $\log_b(b^y)=y\log_b(b)=y$
  • $b^{\log_b(x)}=x$
• The half-life of a substance is the time it takes for half of the substance to decay.
• The doubling-time of a substance is the time it takes for a substance to double.

Compositions

• If $f(x)$ and $g(x)$ are functions, then $f(g(x))$ is called a composition of functions.
• Work with the inside function first:
  • If $f(x)=x+1$, and $g(x)=2x^2+3$,
    • $g(f(3))=g(3+1)=g(4)=2(4^2)+3=35$
    • $g(f(x))=g(x+1)=2(x+1)^2+3$
• When decomposing functions, there may be more then one way to do it:
  • Here are two ways to find $f(x)$ and $g(x)$ so that $f(g(x))=\frac{1}{\sqrt{x+3}}$:
    • $f(x)=\frac{1}{x}$, $g(x)=\sqrt{x+3}$
    • $f(x)=\frac{1}{\sqrt{x}}$, $g(x)=x+3$.

Transformations

• Horizontal transformations will effects the input values of points. Vertical transformations effects the output values of points.
• Shifts: If $k>0$,
  • $f(x)=g(x)+k$ is the graph of $g(x)$ shifted up $k$ units
  • $f(x)=g(x)-k$ is the graph of $g(x)$ shifted down $k$ units
  • $f(x)=g(x+k)$ is the graph of $g(x)$ shifted left $k$ units
  • $f(x)=g(x-k)$ is the graph of $g(x)$ shifted right $k$ units
• Refelctions:
  • $f(x)=-g(x)$ is the graph of $g(x)$ reflected across the $x$-axis
  • $f(x)=g(-x)$ is the graph of $g(x)$ reflected across the $y$-axis
• A function is even if $f(x)=f(-x)$, that is, $f(x)$ is even if you get the same graph after reflecting across the $y$-axis.
• A function is odd if $f(x)=-f(-x)$, that is, $f(x)$ is odd if you get the same graph after reflecting it across the $x$ and $y$-axis