7-4 Exponential Growth and Decay

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

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

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:

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

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 = 10 30 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 = 5 h = 1/30 Solve for y = 100

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.

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 Intersect @ 13.3 13.3 – 5 = 8.3 more min

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