Developing Exponential Functions through Recursion

Recursion Example In order to alleviate a headache, Joan decides to take a 200-mg ibuprofen tablet. Every four hours, her body filters out approximately 67% of the ibuprofen that was in her body at the beginning of that four-hour period. Taken from Contemporary Precalculus through Applications, NCSSM.

Solution: n = number of 4-hour periods

Simple Recursive System: New value = old value + change

Geometric Growth A quantity exhibits geometric growth if the value at time n is a fixed multiple of the value at time n-1.

Using Explicit Model When will the amount of ibuprofen left in Joan’s body be 10 mg?

Using Explicit Model When will the amount of ibuprofen left in Joan’s body be 10 mg?

Modifying Explicit Model Let t = number of hours since initial dose nt

General Geometric Growth Recursive Equations: for some constants A, k Explicit Equation: Exponential function with a base of 1+k

Compound Interest Example $1000 is deposited into an account that earns 5% interest compounded quarterly. Write recursive and explicit functions to model the amount of money in the account.

Recursive Solution Quarterly interest rate = Let n = number of quarters

Two Explicit Solutions Let t = number of years since initial deposit

Continuous Compounding As n gets larger and larger,