Exponential Growth and Decay

In many natural phenomena (such as population growth, radioactive decay, etc.), quantities grow or decay at a rate proportional to their size. In other words, they satisfy the following differential equation


If we call it the law of natural growth. If we call it the law of natural decay.

THEOREM: The only solution of the differential equation (1) are the exponential functions


REMARK: It is easy to check that (2) satisfies (1). In fact,

Population Growth

Radioactive Decay

Newton's Law of Cooling

Continuous Compounded Interest