1. AP CALCULUS AB Chapter 6:
Differential Equations and Mathematical Modeling
Section 6.4:
Exponential Growth and Decay

2. What you’ll learn about
Separable Differential Equations
Law of Exponential Change
Continuously Compounded Interest
Modeling Growth with Other Bases
Newton’s Law of Cooling
… and why
Understanding the differential equation
gives us new insight into exponential growth and decay.

3. Separable Differential Equation

4. Example Solving by Separation of Variables

5. Section 6.4 – Exponential Growth and Decay
Law of Exponential Change
If y changes at a rate proportional to the amount present
and y = y0 when t = 0,
then
where k>0 represents growth and k<0 represents decay.
The number k is the rate constant of the equation.

6. Section 6.4 – Exponential Growth and Decay
From Larson: Exponential Growth and Decay Model
If y is a differentiable function of t such that y>0 and y’=kt, for some constant k, then
where C = initial value of y, and
k = constant of proportionality
(see proof next slide)

7. Section 6.4 – Exponential Growth and Decay
Derivation of this formula:

8. Section 6.4 – Exponential Growth and Decay
This corresponds with the formula for Continuously Compounded Interest
This also corresponds to the formula for radioactive decay

9. Continuously Compounded Interest

10. Example Compounding Interest Continuously

11. Example Finding Half-Life
Hint: When will the quantity be half as much?

12. Section 6.4 – Exponential Growth and Decay
The formula for Derivation:
half-life of a
radioactive
substance is

13. Newton’s Law of Cooling

14. Section 6.4 – Exponential Growth and Decay
Another version of Newton’s Law of Cooling
(where H=temp of object
& T=temp of outside medium)

15. Example Using Newton’s Law of Cooling
A temperature probe is removed from a cup of coffee and placed in water that
has a temperature of T = 4.5 C.
Temperature readings T, as recorded in the table below, are taken
after 2 sec, 5 sec, and every 5 sec thereafter.
Estimate
the coffee's temperature at the time
the temperature probe was removed.
the time when the temperature
probe reading will be 8 C. o S

16. Example Using Newton’s Law of Cooling
Use time for L1 and T-Ts for L2 to fit an exponential regression equation to the data. This formula is T-Ts.

17. Section 6.4 – Exponential Growth and Decay
Resistance Proportional to Velocity
It is reasonable to assume that, other forces being absent, the resistance encountered by a moving object, such as a car coasting to a stop, is proportional to the object’s velocity.
The resisting force opposing the motion is
We can express that the resisting force is proportional to velocity by writing
This is a differential equation of exponential change,