The Main Idea

• Compound Interest: Interest earned on both the principal and previously accumulated interest.
• Annual Percentage Rate (APR): The nominal yearly interest rate.
• Compounding Frequency: How often interest is calculated and added to the principal.
• Annual Percentage Yield (APY): The effective annual rate of return, taking compounding into account.

Key Formulas

1. Compound Interest Formula: [latex]A(t) = P(1 + \frac{r}{n})^{nt}[/latex] Where:
   • [latex]A(t)[/latex] = Final amount
   • [latex]P[/latex] = Principal (initial investment)
   • [latex]r[/latex] = Annual interest rate (as a decimal)
   • [latex]n[/latex] = Number of times interest is compounded per year
   • [latex]t[/latex] = Number of years
2. Annual Percentage Yield (APY) Formula: [latex]APY = (1 + \frac{r}{n})^n - 1[/latex]

The Main Idea

• The Number [latex]e[/latex]: An irrational number approximately equal to [latex]2.718282[/latex].
• Definition of [latex]e[/latex]: [latex]e = \lim_{n \to \infty} (1 + \frac{1}{n})^n[/latex]
• Natural Exponential Function: [latex]f(x) = e^x[/latex]
• Applications: Widely used in modeling natural phenomena and financial calculations.

Key Properties of [latex]f(x) = e^x[/latex]

1. Domain: All real numbers [latex](-\infty, \infty)[/latex]
2. Range: All positive real numbers [latex](0, \infty)[/latex]
3. [latex]y[/latex]-intercept: ([latex]0, 1)[/latex]
4. Horizontal asymptote: [latex]y = 0[/latex]
5. Always increasing
6. [latex]e^0 = 1[/latex]

The Main Idea

• Continuous Growth/Decay Formula: [latex]A(t) = Pe^{rt}[/latex]
• Components:
  • [latex]P[/latex]: Initial value or principal
  • [latex]r[/latex]: Growth or decay rate per unit time
  • [latex]t[/latex]: Time period
  • [latex]e[/latex]: Mathematical constant ([latex]≈ 2.71828[/latex])
• Types:
  • Continuous Growth: [latex]r > 0[/latex]
  • Continuous Decay: [latex]r < 0[/latex]

Key Properties

1. Exponential nature: Growth/decay occurs continuously
2. Compound interest limit: As compounding frequency approaches infinity
3. Time independence: Rate of change proportional to current value

Problem-Solving Strategy

1. Identify initial value ([latex]P[/latex])
2. Determine growth/decay rate ([latex]r[/latex])
   • Ensure [latex]r[/latex] is expressed as a decimal
   • Use negative [latex]r[/latex] for decay
3. Identify time period ([latex]t[/latex])
4. Substitute values into [latex]A(t) = Pe^{rt}[/latex]
5. Calculate final value using a calculator