1. Differential Equations: Growth and Decay Calculus 5.6

2. 2 Differential equations so far.. Calculus 5.6

3. 3 In general Separate variables Each variable is only on one side of the equation Then integrate Calculus 5.6

4. 4 Examples Calculus 5.6

5. 5 Exponential growth and decay If y is a differentiable function of t such that y > 0 and y´ = ky, for some constant k, then C is the initial value of y k is the proportionality constant k > 0 means growth, k < 0 means decay See proof on page 363 Calculus 5.6

6. 6 Example Write and solve the differential equation. The rate of change of P with respect to t is proportional to 10 – t. Calculus 5.6

7. 7 Example Find the exponential function that passes through the points (0, 4) and (5, 2). Calculus 5.6

8. 8 Radioactive decay Uses half-life Exponential decay Use the initial amount to find C Use the half-life to find k Use the equation to find whatever else you need. Calculus 5.6

9. 9 Examples Carbon-14 has a half-life of 5,730 years. If the initial quantity is 3.0 g, how much is left after 10,000 years? Plutonium-239 has a half-life of 24, 360 years. After 10, 000 years, 0.4 g remains. What was the initial amount? Calculus 5.6