Exponential Functions and Compound Interest in Calculus

Exponential Functions and Compound Interest

Natural Exponential Function

The natural exponential function is a fundamental concept in calculus, widely used in modeling growth and decay processes. The function is defined as follows:

• Definition: The function $y = e^x$ is called the natural exponential function.

• Constant e: The number $e$ is defined by the limit $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$ and is approximately equal to 2.7.

• Application: The constant $e$ is the most convenient base to use in calculus, especially for continuous growth and decay models.

Applications of Exponential Functions

Exponential functions are commonly used to model real-world scenarios such as population growth, radioactive decay, and financial calculations involving compound interest.

• Revenue Modeling: If the price-demand function for a product is given by $p(x) = 200e^{-0.01x}$, where $x$ is the number of units produced and $p(x)$ is the price per unit, the revenue for selling $x$ units is $R(x) = x \cdot p(x)$.

• Example: To find the revenue when 100 units are produced: $R(100) = 100 \cdot 200e^{-0.01 \times 100}$ $= 100 \cdot 200e^{-1}$ $= 20,000e^{-1}$ Additional info: $e^{-1} \approx 0.3679$, so $R(100) \approx 20,000 \times 0.3679 = 7,358$ (rounded to the nearest dollar).

• Solving for Quantity: If the price per unit is set at $p(x) = 150$, solve for $x$: $150 = 200e^{-0.01x}$ $\frac{150}{200} = e^{-0.01x}$ $0.75 = e^{-0.01x}$ $\ln(0.75) = -0.01x$ $x = \frac{\ln(0.75)}{-0.01}$ Additional info: $\ln(0.75) \approx -0.2877$, so $x \approx 28.77$ units.

Compound Interest

Compound interest is a key application of exponential functions in finance. It describes how an initial principal grows over time when interest is added periodically.

• General Formula: If a principal $P$ is invested at an annual interest rate $r$ for $t$ years, and interest is compounded $m$ times per year, the future value $A$ is: $A = P \left(1 + \frac{r}{m}\right)^{mt}$

• Continuous Compounding: If interest is compounded continuously, the formula becomes: $A = Pe^{rt}$

• Example: What amount will be in an account after 5 years if $100$is invested at an annual nominal rate of $8\%$ compounded:

  • Semiannually ($m=2$): $A = 100 \left(1 + \frac{0.08}{2}\right)^{2 \times 5}$ $= 100 (1.04)^{10}$

  • Monthly ($m=12$): $A = 100 \left(1 + \frac{0.08}{12}\right)^{12 \times 5}$ $= 100 (1.0066667)^{60}$

  • Daily ($m=365$): $A = 100 \left(1 + \frac{0.08}{365}\right)^{365 \times 5}$

  • Continuously: $A = 100e^{0.08 \times 5}$ $= 100e^{0.4}$

  Additional info: These formulas allow comparison of how different compounding frequencies affect the final amount.

Comparison of Compounding Methods

The following table summarizes the formulas for different compounding frequencies:

Compounding Frequency	Formula
Semiannually ($m=2$)	$A = P \left(1 + \frac{r}{2}\right)^{2t}$
Monthly ($m=12$)	$A = P \left(1 + \frac{r}{12}\right)^{12t}$
Daily ($m=365$)	$A = P \left(1 + \frac{r}{365}\right)^{365t}$
Continuously	$A = Pe^{rt}$

Key Terms

• Exponential Function: A function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Natural Exponential Function: The exponential function with base $e$.

• Compound Interest: Interest calculated on both the initial principal and the accumulated interest from previous periods.

• Continuous Compounding: The process of calculating interest and adding it to the principal an infinite number of times per year.