Any population of living creatures increases at a rate that is proportional to the number present (at least for a while). Other things that increase or decrease at a rate proportional to the amount present include radioactive material and money in an interest-bearing account. If the rate of change is proportional to the amount present, the change can be modeled by: 6.4 Exponential Growth and Decay

Rate of change is proportional to the amount present. Divide both sides by y. Integrate both sides.

Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication.

Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication. Since is a constant, let.

At,. This is the solution to our original initial value problem.

Exponential Change: If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay.

Continuously Compounded Interest If money is invested in a fixed-interest account where the interest is added to the account k times per year, the amount present after t years is: If the money is added back more frequently, you will make a little more money. The best you can do is if the interest is added continuously.

Of course, the bank does not employ some clerk to continuously calculate your interest with an adding machine. We could calculate: but we won’t learn how to find this limit until chapter 8. Since the interest is proportional to the amount present, the equation becomes: Continuously Compounded Interest: You may also use: which is the same thing.

Radioactive Decay The equation for the amount of a radioactive element left after time t is: This allows the decay constant, k, to be positive. The half-life is the time required for half the material to decay.

Half-life Half-life:

Newton’s Law of Cooling Espresso left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air. (It is assumed that the air temperature is constant.) If we solve the differential equation: we get: Newton’s Law of Cooling where is the temperature of the surrounding medium, which is a constant.