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Cogito

AP Calculus AB · Chapter 7 · Lesson 3

Practice Worksheet

Exponential Models

When the rate is proportional to the amount.

12 problems · about 21 minutes · FUN-7.F

NameDate

Figure — use these to answer the problems

This field is dy/dx = ky. Make k negative and every solution curve decays instead of growing.
  • The equationdy/dx = a·y
  • Through(0, 1)
k, the constant1
Starting x0
Starting y1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A sample of 100 g halves every hour. How many grams remain after 2 hours?

    Answer

  2. What does dy/dt = ky say?

    • The rate of change is proportional to the amount present.
    • The rate of change is constant.

  3. dy/dt = ky with k = 0.03. Growing or decaying? 1 growing, 2 decaying.

    Answer

  4. dy/dt = ky with k = −0.2. Growing or decaying? 1 or 2?

    Answer

  5. A sample of 160 g halves every hour. Grams after 4 hours?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A population doubles every 5 years. By what factor in 15 years?

    Answer

  2. Does half-life depend on the starting amount? 1 yes, 0 no.

    Answer

  3. y = y₀e^(kt) at t = 0. What is y?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Which situations follow dy/dt = ky?

    • Radioactive decay
    • Continuously compounded interest
    • Unrestricted population growth
    • A tank filling at a constant litres per minute
    • Tick every box that applies.

  2. A sample of 64 g halves every hour. Grams after 3 hours?

    Answer

  3. The Constant

    dP/dt = 0.05P with P(0) = 200. What is P₀?

    Answer

  4. The Decay

    A sample of 80 g halves every hour. How many grams remain after 3 hours?

    Answerg