1. 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

2. 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

3. Clicker Question 3
The profit function (in $) at production level x is P (x ) = 30x 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

4. 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.

5. 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.

6. 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?

7. 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?

8. 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?

9. 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).