Exponential Functions Notes

Definition

• Exponential function f is defined by the equation $f(x)=b^{x},$ where the base $b>0$ and $b\neq 1.$
• Domain: all real numbers.
• Range: all positive real numbers.
• Major contrast with power functions: in $f(x)=2^x$ the exponent is the variable; in $g(x)=x^2$ the base is the variable.
• Key takeaway: base determines how the function behaves as x varies; not all bases produce the same shape.

Graphs and Basic Properties

• Theorem 1: Basic Properties of the Graph of $f(x)=b^{x},\ b>0,\ b\neq 1$

  1. The graph contains the point $(0,1)$ since $b^{0}=1.$
  2. The graph is a continuous curve (no holes or jumps).
  3. The x-axis is a horizontal asymptote (as $x\to -\infty$, (f(x)\to 0^+)).
  4. If $b>1$, then $b^{x}$ increases as $x$ increases.
  5. If $0<b<1$, then $b^{x}$ decreases as $x$ increases.

• Graph behavior examples:

  • For $f(x)=2^{x}$, the graph increases to the right and approaches 0 on the left; as $x\to\infty$, $f(x)\to\infty$; as $x\to -\infty$, $f(x)\to 0^+.$
  • For $f(x)=2^{-x}$, the function is the same as $f(-x)=2^{x}$ and thus the left-right behavior is reversed correspondingly.

• Theorem 2: Properties of Exponential Functions (positive bases)

  • For positive constants $a>0$ and $b>0$ with $a\neq 1,\ b\neq 1$, the properties described apply to functions like $y=a^{x}$ and $y=b^{x}$ for real $x,y$.
  • Note: The statement in the slides mentions (−2)² = 22 does not contradict the property because Theorem 2's standard domain/range assumptions require positive bases; negative bases with real exponents are not well-defined in general. In short, the standard, clean exponential behavior is stated for bases $>0$.

• Base e and the natural exponential:

  • The base $e$ is special in calculus and modeling; it appears naturally in growth/decay and calculus limits.
  • Calculators commonly feature keys for $10^x$ and for the natural exponential $e^x$.
  • The base $e$ is used because many processes in calculus simplify when the base is $e$.
  • The function with base e is $y=e^{x}$, often simply called the exponential function.

• Base e details:

  • The base $e$ is irrational and cannot be represented exactly by any finite decimal or fraction.
  • The common decimal approximation is $e\approx 2.718281828459\ldots$

• Exponential functions with base e and base 1/e:

  • Defined by $y=e^{x}$ and $y=e^{-x}$, respectively.
  • Domain: $(-\infty,\infty)$; Range: $(0,\infty)$ for both.

Exponential Growth and Decay Models

• General form: $y=c\cdot e^{k t}$ where

  • $c$ is the initial quantity (the value at time $t=0$).
  • $k$ is the relative growth rate (a decimal); per unit of time. If $k>0$, growth; if $k<0$, decay.
  • The independent variable $t$ represents time.

• Cholera bacteria growth example

  • Model: $N=N<em>0\,e^{1.386 \;t}$ where $N</em>0$ is the initial number of bacteria and $1.386$ is the relative growth rate per hour (approximately ln(4)).
  • Given $N_0=25$:
  • (A) After $t=0.8$ hours: $N=25\,e^{1.386\cdot 0.8}\approx 76$ bacteria.
  • (B) After $t=2.5$ hours: $N=25\,e^{1.386\cdot 2.5}\approx 799$ bacteria.

• Carbon-14 decay (radioactive decay):

  • Decay model: $A=A_0\,e^{-0.000124\,t}$ where
  • $A_0$ is the amount at time $t=0$,
  • $t$ is time in years.
  • Example: If $A_0=500$ mg, after $t=15{,}000$ years:
  • $A=500\,e^{-0.000124\cdot 15000}\approx 77.84\text{ mg}.$

• Half-life concept (for carbon-14):

  • The half-life $t<em>{1/2}$ is the time when the amount is half of the initial: $A=A</em>0/2.$
  • Using the decay model, solving $A<em>0/2=A</em>0 e^{-0.000124 t}$ gives $t = \frac{\ln 2}{0.000124}\approx 5590\text{ years}$ (approximate value; graphing calculator refinement yields about 5590–6000 years depending on method).

• Graphical method for half-life refinement (calculator approach):

  • Solve $250=500 e^{-0.000124 t}$ for $t$ using intersection of $y<em>1=500 e^{-0.000124 x}$ and $y</em>2=250$ within a window such as [0, 50{,}000] by [0, 500]. By this method the intersection occurs at about $t\approx 5{,}590$ years.

Exponential Regression

• When data suggests exponential behavior, fit a model of the form $y=a\cdot b^{x}$ where $a,b>0$.

• Example regression: The regression yields

  • $y = 0.4833\cdot 1.1343^{x}$ (values rounded to four decimal places).
  • The associated $r^2$ value indicates the fit is good for the data.

• Use for prediction: If the independent variable is time since a baseline year (e.g., $x$ years after 2000), and $x=24$ for year 2024, then

  • $y = 0.4833 \cdot 1.1343^{24} \approx 9.95\text{ (billion internet users).}$

• Practical notes:

  • The shape of the regression curve should resemble the scatterplot if the exponential model is appropriate.
  • Regression outputs include the regression equation and an $r^2$ indicating goodness of fit.

Compound Interest

• Compound interest formula: if principal $P$ is invested at annual rate $r$ (as decimal) compounded $m$ times per year, for $t$ years, amount $A$ is:
  $A = P\left(1+\frac{r}{m}\right)^{m t}.$
• Example: ${P=10{,}000}, \ {r=0.075}, \ {m=12}, \ {t=20}$ yields an amount approximately $A\approx 44{,}608.17.$

Continuous Compound Interest

• Concept: with continuous compounding, interest is added infinitely often; the base of the exponential is $e$.

• Formula:
  $A = P e^{r t}.$

  • Here, $e\approx 2.71828\ldots$ is the base of the natural exponential.

• Example: Continuous compounding with $P=10{,}000$, $r=0.075$, and $t=20$:
  $A = 10000 e^{0.075\cdot 20} = 10000 e^{1.5} \approx 44816.89.$

Quick Reference of Key Facts

• Exponential function form: $f(x)=b^{x}$ with base $b>0, b\neq 1.$
• Domain and range:
  • Domain: $(-\infty,\infty)$
  • Range: $(0,\infty)$
• Fundamental points: $f(0)=1$; continuous; horizontal asymptote at the x-axis.
• Growth/decay depends on base:
  • If $b>1$, increasing function.
  • If $0<b<1$, decreasing function.
• Special base: $e$ is the natural base; $y=e^{x}$ and $y=e^{-x}$ are canonical forms used in calculus due to simplifications in differentiation and integration.
• Useful decay model: $A=A_0 e^{-\lambda t}$ with decay constant $\lambda>0$; here for Carbon-14, $\lambda = 0.000124$ per year.
• Useful growth model: $y=c e^{k t}$ with relative growth rate $k$ per unit time.
• Regression form for data: $y=a\cdot b^{x}$; regression can produce predictions like $y=a\cdot b^{x}$ with an $r^{2}$ value indicating fit quality.

Important Equations (LaTeX)

• Exponential function: $f(x)=b^{x},\quad b>0,\ b\neq 1.$
• Basic properties (Theorem 1):
  • $f(0)=1$, continuous, horizontal asymptote at the x-axis, increasing if $b>1$, decreasing if $0<b<1.$
• Growth/decay model: $y=c\cdot e^{k t}.$
• Cholera growth: $N=N_0\,e^{1.386\,t}.$
• Nuclear decay (Carbon-14): $A=A_0\,e^{-0.000124\,t}.$
• Half-life relation (example): $t_{1/2}=\dfrac{\ln 2}{0.000124}\approx 5590\text{ years}.$
• Exponential regression: $y=a\cdot b^{x},\quad y=0.4833\cdot 1.1343^{x}.$
• Compound interest: $A = P\left(1+\frac{r}{m}\right)^{m t}.$
• Continuous compound interest: $A = P e^{r t}.$