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=N0\,e^{1.386 \;t} where N0 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{1/2} is the time when the amount is half of the initial: A=A0/2.
  • Using the decay model, solving A0/2=A0 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 y1=500 e^{-0.000124 x} and y2=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}.