1. 7-4 Exponential Growth and Decay

2. Ex 1) Solve for y if and y = 1 when x = 1. Initial conditions:
Separable Differential Equation … basically you have a derivative with x’s and y’s
You put the dy and dx on either side of an equation and then antidifferentiate both sides
Ex 1) Solve for y if and y = 1 when x = 1.
Initial conditions:
*only 1 constant needed

3. The Law of Exponential Change
k > 0 growth constant
k < 0 decay constant
y0 = t = 0
Interest
Compounded Continuously Interest
r = continuous interest rate

4. Compounded continuously:
Ex 2) Suppose you deposit $800 in an account that pays 6.3% annual interest. How much will you have 8 years later if the interest is (a) compounded continuously? (b) compounded quarterly?
Compounded continuously:
Compounded quarterly:

5. Radioactivity:
Half-life =
(Other bases – does not have to be e)
related to
h will be the reciprocal of the time period required for pop to grow (or decay) by a factor of b

6. Since dy/dt = ky, growth is exponential
Ex 4) At the beginning of the summer, the population of a hive of bald-faced hornets (which are actually wasps) is growing at a rate proportional to the population. From a population of 10 on May 1, the number of hornets grows to 50 in thirty days. If the growth continues to follow the same model, how many days after May 1 will the population reach 100?
May 1 = days = 50 when 100?
Since dy/dt = ky, growth is exponential
Pop grows by a factor of 5 in 30 days
Use y = y0bht y0 = 10 b = h = 1/30
Solve for y = 100

7. Ex 5) Scientists who use carbon-14 dating use 5700 years as its half-life. Find the age of a sample in which 10% of the radioactive nuclei originally present have decayed.

8. Newton’s Law of Cooling: T – TS = (T0 – TS)e–kt
t = time TS = surrounding temp T0 = temp at t = 0
Ex 6) A hard-boiled egg at 98°C is put in a pan under running 18°C water to cool. After 5 minutes, the egg’s temperature is found to be 38°C. How much longer will it take the egg to reach 20°C?
Egg’s temp is:
Graph this eqtn and y = 20 to see where they intersect
13.3
13.3 – 5 = 8.3 more min

9. homework
Pg. 361 Quick Review # 1-10; # 3-30 (mult of 3)