Cogito
AP Calculus AB · Chapter 7 · Lesson 3
Exponential Models
When the rate is proportional to the amount.
12 problems · about 21 minutes · FUN-7.F
What this lesson teaches
The student solves and applies exponential growth and decay models.
- dy/dt = ky has solution y = y₀e^(kt).
- Positive k grows and negative k decays.
- Half-life and doubling time depend only on k, never on the starting amount.
Warm Up
Straightforward practice. Get the method working first.
5 problemsA sample of 100 g halves every hour. How many grams remain after 2 hours?
Answer 25
Why 25 g.
What does dy/dt = ky say?
Answer The rate of change is proportional to the amount present.
Why Rate proportional to amount.
dy/dt = ky with k = 0.03. Growing or decaying? 1 growing, 2 decaying.
Answer 1
Why A positive k.
dy/dt = ky with k = −0.2. Growing or decaying? 1 or 2?
Answer 2
Why A negative k.
A sample of 160 g halves every hour. Grams after 4 hours?
Answer 10
Why 80, 40, 20, 10.
Build It Up
The same ideas with more to keep track of.
3 problemsA population doubles every 5 years. By what factor in 15 years?
Answer 8
Why Three doublings.
Does half-life depend on the starting amount? 1 yes, 0 no.
Answer 0
Why It depends only on k.
y = y₀e^(kt) at t = 0. What is y?
Answer 1
Why Enter 1 for y₀ times e to the zero, which is y₀ times one.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich situations follow dy/dt = ky?
Answer Radioactive decay; Continuously compounded interest; Unrestricted population growth
Why One of these has a rate that does not depend on the amount present.
A sample of 64 g halves every hour. Grams after 3 hours?
Answer 8
Why 32, 16, 8.
The Constant: dP/dt = 0.05P with P(0) = 200. What is P₀?
Answer 200
Why 200.
The Decay: A sample of 80 g halves every hour. How many grams remain after 3 hours?
Answer 10 g
Why 10 g.