Homework 5.5: Exponential and Compounding Models

Foundations of Exponential Functions and the Natural Base

• Definition of Exponential Functions:

  • An exponential function is a mathematical function of the form $f(x) = a \cdot b^x$, where $a$ represents the initial value (or vertical stretch/compression factor, with $a \neq 0$) and $b$ represents the positive base ($b > 0$ and $b \neq 1$).

  • When $b > 1$, the function models exponential growth, and the function is strictly increasing across its entire domain.

  • When $0 < b < 1$, the function models exponential decay, and the function is strictly decreasing across its entire domain.

  • Every exponential function passes the Horizontal Line Test, proving that all exponential functions are strictly one-to-one and invertible.

• The Natural Base $e$:

  • The number $e$, known as the natural base, is an irrational mathematical constant approximately equal to $2.718281828459...$.

  • Historical origins trace back to an appendix within John Napier's work on logarithms; however, Leonhard Euler solidified the notation $e$.

  • Limit Definition: The constant $e$ is formally defined as the limiting value of the expression $\left(1 + \frac{1}{n}\right)^n$ as $n$ approaches infinity ($n \rightarrow \infty$):

  • For $n = 10$: $\left(1 + \frac{1}{10}\right)^{10} = (1.1)^{10} \approx 2.59374$

  • For $n = 100$: $\left(1 + \frac{1}{100}\right)^{100} = (1.01)^{100} \approx 2.70481$

  • For $n = 1000$: $\left(1 + \frac{1}{1000}\right)^{1000} = (1.001)^{1000} \approx 2.71692$

  • For $n = 10000$: $\left(1 + \frac{1}{10000}\right)^{10000} = (1.0001)^{10000} \approx 2.71815$

  • For $n = 100000$: $\left(1 + \frac{1}{100000}\right)^{100000} = (1.00001)^{100000} \approx 2.71827$

  • Infinite Series Definition: The constant $e$ can also be computed using the Maclaurin series expansion:     $e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots = \sum_{k=0}^{\infty} \frac{1}{k!}$

Characteristics, Graphs, and Transformations of Exponential Functions

• Parent Function $f(x) = 2^x$:

  • Table of Values:

  • $x = -2 \implies f(-2) = 2^{-2} = \frac{1}{4} = 0.25$

  • $x = -1 \implies f(-1) = 2^{-1} = \frac{1}{2} = 0.5$

  • $x = 0 \implies f(0) = 2^0 = 1$

  • $x = 1 \implies f(1) = 2^1 = 2$

  • $x = 2 \implies f(2) = 2^2 = 4$

  • Key Graphical Characteristics:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • Horizontal Asymptote: $y = 0$

  • $y$-intercept: $(0, 1)$

  • $x$-intercept: None

  • One-to-one status: Yes

  • Interval of increase: $(-\infty, \infty)$

  • Interval of decrease: None

• Parent Function $f(x) = \left(\frac{1}{2}\right)^x$:

  • Table of Values:

  • $x = -2 \implies f(-2) = \left(\frac{1}{2}\right)^{-2} = 4$

  • $x = -1 \implies f(-1) = \left(\frac{1}{2}\right)^{-1} = 2$

  • $x = 0 \implies f(0) = \left(\frac{1}{2}\right)^0 = 1$

  • $x = 1 \implies f(1) = \left(\frac{1}{2}\right)^1 = \frac{1}{2} = 0.5$

  • $x = 2 \implies f(2) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} = 0.25$

  • Key Graphical Characteristics:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • Horizontal Asymptote: $y = 0$

  • $y$-intercept: $(0, 1)$

  • $x$-intercept: None

  • One-to-one status: Yes

  • Interval of increase: None

  • Interval of decrease: $(-\infty, \infty)$

• Parent Natural Exponential Function $f(x) = e^x$:

  • Table of Values:

  • $x = -2 \implies f(-2) = e^{-2} \approx 0.135$

  • $x = -1 \implies f(-1) = e^{-1} \approx 0.368$

  • $x = 0 \implies f(0) = e^0 = 1$

  • $x = 1 \implies f(1) = e^1 \approx 2.718$

  • $x = 2 \implies f(2) = e^2 \approx 7.389$

  • Key Graphical Characteristics:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • Horizontal Asymptote: $y = 0$

  • $y$-intercept: $(0, 1)$

  • $x$-intercept: None

  • Interval of increase: $(-\infty, \infty)$

  • Interval of decrease: None

• **Universal Commonalities of Untransformed Exponential Functions ($y = b^x$ with