Chapter 6 – Exponential and Logarithmic Functions

# Activity 6a – Compound Interest

This activity develops the standard formula for compounding interest in discrete time steps. This formula is connected to the continuously compounding interest formula in the next activity. We begin here with interest compounded annually, then proceed to shorter time intervals. The recursive nature of the process is stressed.

Activity 6a – Compound Interest

# Activity 6b – Continuously Compounded Interest

This activity begins with the discrete compounding interest formula $\displaystyle{A = P \left(1 + \frac{r}{n} \right)^{nt}}$ and looks at what happens as $n$ gets larger. It begins with a review of the discrete compounding process, and then gives a brief intuitive discussion of $e$, which here is defined as $\displaystyle{e = \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x}$. The final step is to derive the continuously compounded interest formula $A = Pe^{rt}$.

Activity 6b – Continuously Compounded Interest

# Activity 6c – Susceptible, Infected, Recovered (S-I-R) model

This activity begins with exploring exponential growth models that represent the rapid spread of bacteria growing in a culture when there are plenty of resources. The S–I–R model, which in its simplest form is logistic growth, is a better t once further growth is inhibited. This is the case in the spread of infectious diseases; once a significant percentage of the population is immune, the disease will no longer spread exponentially. The second part of this activity goes through a discrete approximation of the S–I–R model based on recursive formulas. The calculations can be performed with a calculator or a spreadsheet.

Activity 6c – SIR model