SECTION 1.7: Exponential Growth and Decay

Many functions can be defined in the form of P = P0 (e ^kt) where P0 is still defined as the initial quantity and k is the continuous growth or decay rate. When k is negative it represents a decay. When it is positive it refers to exponential growth. "e", as mentioned in the previous post, is defined as a natural base.

Two new concepts defined in this sections were doubling time and half-life:

Doubling time of an increasing exponential function is the time required for the quantity to double itself.

Half- life of a decaying exponential function is the time required for the quantity to be reduced to half its current amount.

All growing functions have a doubling time, and all decaying functions have a half-life.

Compound interest on an amount saved in a bank can be defined exponentially. If an initial
amount of P0 is deposited in a bank that has an interest rate of r/ year, assuming P is the balance in the bank over t years, it can be said that
- interest compounded annually implies P = P0 (1+r)^t
- interest conpounded continuously implies P= P0(e^rt) where e is again the natural base.

In business terminology, there are two important terms to remember:

The future value of an amount P is the amount to which P will grow in an interest-bearing bank account.

The present value, P, of a future payment is an amount that will have to be deposited in a bank now to produce the amount of future payment at the relevant time.

In this reading, I was confused by the term 'interest compounded continuously' because I could not easily understand how this could happen when the interest rate itself was defined per year.

It is clearly seen that this section of exponential functions is most related to Economics. Interest rates are crucial to bankers, borrowers and lenders. Understanding how one's savings grow in a bank is equally important and necessary in managing one's own finances.