Exponential Growth & Decay – AP Calculus AB | ShowMeClass

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Exponential Growth & Decay

AP Calculus AB · Unit 7: Differential Equations

Learning Objectives

Solve the DE: dy/dt = ky with initial condition y(0) = y₀

Distinguish growth (k>0) from decay (k<0)

Calculate half-life and doubling time

Interpret slope fields for exponential models

Apply to real-world: population, radioactive decay, Newton's cooling

Key Equations

Graph Legend

y(t) = y₀eᵏᵗ — solution curve

Tangent line at t-probe

Half-life / Doubling time

Slope field segments

Key Insight

The solution y(t) = y₀eᵏᵗ is the only function whose rate of change is proportional to itself. When k>0, doubling time = ln(2)/k. When k<0, half-life = ln(2)/|k|.

Diff. Equations Exponential AP Calculus AB Unit 7

y₀100

k0.30

t-probe2.00

y(t)—

y(t) =—

dy/dt =—

Half-life =—

Mode:Growth

SLOPE FIELD: dy/dt = ky

Initial Value y₀ 100

Rate Constant k 0.30

t-Probe (tangent point) 2.00

Time Range 10

Presets

Display

Solution Details

📝 Quick Quiz▼

1. The general solution to dy/dt = ky is:

y = kt + C y = y₀eᵏᵗ y = k·y₀·t y = ln(kt)

2. A substance has a half-life of 5 years. Its decay constant k is:

−0.5 −ln(2)/5 ≈ −0.1386 −5/ln(2) ln(5)/2

3. If a bacteria population doubles every 3 hours, starting at 500, the population at t=9 hours is:

1500 2000 4000 8000