1. Chapter 3 – Differentiation Rules
3.8 Exponential Growth and Decay
Section 3.8 Exponential Growth and Decay

2. Section 3.8 Exponential Growth and Decay
In many natural phenomena, quantities grow or decay at a rate proportional to their size. For instance, if y = f (t) is the number of individuals in a population of animals or bacteria at time t, then it seems reasonable to expect that the rate of growth f (t) is proportional to the population f (t); that is, f (t) = k f (t) for some constant k.
Indeed, under ideal conditions (unlimited environment, adequate nutrition, immunity to disease) the mathematical model given by the equation f (t) = k f (t) predicts what actually happens fairly accurately.
Section 3.8 Exponential Growth and Decay

3. Section 3.8 Exponential Growth and Decay
Another example occurs in nuclear physics where the mass of a radioactive substance decays at a rate proportional to the mass.
In chemistry, the rate of a unimolecular first-order reaction is proportional to the concentration of the substance.
In finance, the value of a savings account with continuously compounded interest increases at a rate proportional to that value.
Section 3.8 Exponential Growth and Decay

4. Law of Natural Growth or Decay
In general, if y (t) is the value of a quantity y at time t and if the rate of change of y with respect to t is proportional to its size y (t) at any time, then
where k is a constant.
Section 3.8 Exponential Growth and Decay

5. Law of Natural Growth or Decay
If k > 0 then the equation below is the law of natural growth. If k < 0, then the equation below is the law of natural decay. (k is a constant) The only solutions to the above differential equation are the exponential functions
Section 3.8 Exponential Growth and Decay

6. Section 3.8 Exponential Growth and Decay
Book Example 1 – pg. 242 #3
A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour, the population has increased to 420.
Find an expression for the number of bacteria after t hours.
Find the number of bacteria after 3 hours.
Find the rate of growth after 3 hours.
When will the population reach 10,000?
Section 3.8 Exponential Growth and Decay

7. Newton’s Law of Cooling
Newton’s Law of Cooling (or warming) states that the rate of cooling an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large.
Section 3.8 Exponential Growth and Decay

8. Newton’s Law of Cooling
If T(t) is the temperature of the object at time t and Ts is the temperature of the surroundings, then Newton’s Law as a differential equation is
This equation is not quite the same as Equation 1, so we make the change of variable y (t) = T (t) – Ts. Because Ts is constant, we have y  (t) = T  (t) and so the equation becomes
Section 3.8 Exponential Growth and Decay

9. Section 3.8 Exponential Growth and Decay
Book Example 2 – pg. 244 #15
When a cold drink is taken from a refrigerator, its temperature is 5oC. After 25 minutes in a 20oC room, its temperature has increased to 10oC.
What is the temperature of the drink after 50 minutes?
When will its temperature be15oC?
Section 3.8 Exponential Growth and Decay

10. Section 3.8 Exponential Growth and Decay
Book Example 3 – pg. 243 #10
A sample of tritium-3 decayed to 94.5% of its original amount after a year.
What is the half life of tritium-3?
How long would it take the sample to decay to 20% of its original amount?
Section 3.8 Exponential Growth and Decay

11. Section 3.8 Exponential Growth and Decay
Book Example 4 – pg. 244 # 20
How long will it take an investment to double in value if the interest rate is 6% compounded continuously?
What is the equivalent annual interest rate?
Section 3.8 Exponential Growth and Decay

12. Section 3.8 Exponential Growth and Decay
Try this problem
Bismuth-210 has a half life of 5.0 days.
A sample originally has a mass of 800mg. Find a formula for the mass remaining after t days.
Find the mass remaining after 30 days.
When is the mass reduced to 1mg?
Sketch the graph of the mass function.
Section 3.8 Exponential Growth and Decay

13. Section 3.8 Exponential Growth and Decay
Try this problem
A freshly brewed cup of coffee has a temperature of 95oC in a 20oC room. When the temperature is 70oC, it is cooling at a rate of 1oC per minute. When does this occur?
Section 3.8 Exponential Growth and Decay