Clicker Question 1 The radius of a circle is growing at a constant rate of 2 inches/sec. How fast is the area of the circle growing when the radius is 5 inches? A. 5 sq. in./sec B. 10 sq. in./sec C. 20 sq. in./sec D. 25 sq. in./sec E. 50 sq. in./sec

Clicker Question 2 If the height (in feet) of a ball thrown upward is h (t ) = 64t – 16t 2 (t in seconds), what is its maximum height? A. 48 feet B. 54 feet C. 64 feet D. 72 feet E. There is no maximum height

Clicker Question 3 The profit function (in $) at production level x is P (x ) = 30x + 200 ln(x). What is the marginal profit at a production level of 100? A. $ 50 / item B. $ 32 / item C. $ 17 / item D. $ 76 / item E. $110 / item

Exponential Growth and Decay (3/25/09) It is very common for a population P to grow (or decay) at a rate proportional to its size. This is described by the differential equation dP /dt = k P for some constant number k. A differential equation is an equation containing one or more derivatives. A solution is a function which makes it true, i.e., which satisfies it.

Exponential functions solve that equation: Check that P (t) = C e k t satisfies the differential equation on the last slide. Note that C is just the initial population P (0). It can be shown that this is the only solution to the equation. k is called the relative (or continuous) growth rate.

Example of Growth A population P of bacteria is growing at a relative rate of 5% each day. It starts with 100 bacteria. Write an equation for P (t) , t in days. How many bacteria are there after 10 days? How long (from the start) will it take for the population to reach 1000?

Example: Radioactive Decay The half-life of a radioactive substance is the number of years for half the substance to decay. Here it makes sense to use 1/2 as the base instead of e. The half-life of radium-226 is 1590 years. If I start with 100 mg, how much is there after 200 years? How long before only 20 mg are left?

Newton’s Law of Cooling A body cools at a rate proportional to the difference between its current temperature and the (constant) surrounding temperature . A cup of coffee sitting in a 65° room cools from 200° to 180° in the first minute. How hot will it be after 10 (total) minutes?

Assignment for Friday Read Section 3.8. Do Exercises 1, 3, 5b, 9, 13, and 18a. Hand-in #2 is due at 4:45 tomorrow (Thursday).