13.1.1: Resources and Key Concepts

Prerequisite Topics

The following is a list of prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Arithmetic
  • Percents: Understanding how to convert percentages to decimals (e.g., 9% to 0.09) is crucial for working with growth/decay rates in exponential functions and compound interest formulas.
• Systems
  • Solving 2x2 Systems of Linear Equations: This skill is needed to find equations of exponential functions going through two given points.

Videos

• Exponential functions
  • Intermediate Algebra - Exponential Functions: The Concepts
  • Exponential Functions - A Brief Review of Exponential Functions
• Exponential growth and decay
  • Exponential Functions - Types of Exponential Growth and Decay
• Finding equation of exponential function
  • Exponential Functions - Finding Equations of Exponential Functions
• Compound interest formula
  • Exponential Functions - Applying the Compound Interest Formula
• Natural exponential base \( e \)
  • Exponential Functions - Investigating Continuous Growth
  • Exponential Functions - Calculus and the Natural Base
• Applications of exponential functions
  • Exponential Functions - Applications Involving Exponential Functions
  • Intermediate Algebra - Exponential Functions: Creating from Applications

Definitions

• Percent Change: A change based on a percent of the original amount.
• Exponential Growth: An increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.
• Exponential Decay: A decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.
• Exponential Function: A function with the form \(f(x) = ab^x\), where \(a\) is a non-zero real number called the initial value and \(b\) is any positive real number, called the base, such that \(b \neq 1\).
• Initial Value (of an exponential function): In the form \(f(x) = ab^x\), \(a\) is the initial value (the value of the function when \(x=0\)).
• Base (of an exponential function): In the form \(f(x) = ab^x\), \(b\) is the base. It must be a positive real number not equal to 1.
• Exponential Growth/Decay Function (alternative form): A function that models exponential growth or decay can be written as \(P(t) = P_0 b^t\), where \(P_0\) is the initial value, and \(b\) is the growth/decay factor. If \(b>1\), it's a growth factor. If \(0 < b < 1\), it's a decay factor.
• Compounding: The process of earning interest not only on the initial investment (principal) but also on the accumulated interest itself.
• Annual Percentage Rate (APR) / Nominal Rate: The yearly interest rate earned by an investment account before considering the effect of compounding frequency.
• Effective Rate / Annual Percentage Yield (APY): The percentage of interest on a loan or financial product if compound interest accumulates in periods different than a year; it reflects the actual annual rate earned after compounding.
• The number \(e\) (Euler's number / natural exponential base): An irrational number approximately equal to 2.71828, which arises as the limit of \((1 + \frac{1}{n})^n\) as \(n\) increases without bound. It is used as the base for continuous growth/decay models.
• Continuous Growth/Decay: Exponential growth or decay represented by the formula \(P(t) = P_0 e^{rt}\), where \(P_0\) is the initial value, \(r\) is the continuous growth rate (if \(r>0\)) or continuous decay rate (if \(r<0\)) per unit time, and \(t\) is time.
• Continuous Growth/Decay Rate: The parameter \(r\) in the continuous growth/decay formula \(P(t) = P_0 e^{rt}\).

Theorems

• Slope Addition Property: For a linear function with slope \(m\), every increase of 1 in input results in adding \(m\) to the function value.
• Base Multiplier Property: For an exponential function with base \(b\), every increase of 1 in input results in multiplying the previous function value by \(b\).
• Properties of the Exponential Function \(f(x)=ab^x\), \(a>0\):
  • Domain: All real numbers.
  • Range: All positive real numbers \((0, \infty)\).
  • \(y\)-intercept: \((0,a)\).
  • Includes point \((1,ab)\).
  • Horizontal asymptote: \(y=0\).
  • If \(b>1\), \(f\) is increasing; as \(x \rightarrow -\infty, f(x) \rightarrow 0\); as \(x \rightarrow \infty, f(x) \rightarrow \infty\).
  • If \(0 < b < 1\), \(f\) is decreasing; as \(x \rightarrow -\infty, f(x) \rightarrow \infty\); as \(x \rightarrow \infty, f(x) \rightarrow 0\).
• Compound Interest Formula: The amount \(A(t)\) an investment of \(P_0\) dollars is worth after \(t\) years earning an annual percentage rate of \(r \times 100\%\) compounding \(n\) times per year is given by \(A(t) = P_0 \left(1 + \frac{r}{n}\right)^{nt}\).

Common Mistakes

• Confusing Exponential Growth with Linear Growth: Mistaking a situation with a constant additive change for exponential growth, or vice-versa. Exponential growth involves a constant multiplicative factor (percent change).
• Incorrect Base for Exponential Functions: Using a base \(b \le 0\) or \(b=1\) for an exponential function \(f(x)=ab^x\). The base must be positive and not equal to 1.
• Order of Operations Errors in Evaluation: When evaluating exponential functions like \(f(x)=ab^{x+c}+d\), performing operations in the wrong order (e.g., multiplying by \(a\) before evaluating the exponent).
• Incorrectly Using the Continuous Compounding Formula: Applying \(A=P_0e^{rt}\) when interest is compounded discretely (e.g., monthly, quarterly), or using the discrete compounding formula for continuous compounding.
• Sign Errors with Decay Rates: For continuous decay \(P(t)=P_0e^{rt}\), forgetting that the rate \(r\) must be negative.