Clicker Question 1 The position of an object (in feet) is

given by s (t ) = 4 + ln(t 2) where t is in seconds. What is the object’s velocity at time t = 3 sec? A. 4 + ln(9) feet/sec B. ln(9) feet/sec C. 2/3 feet/sec D. 1/3 feet/sec E. 1/9 feet/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

Exponential Growth and Decay(2/9/11) 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? What is the rate of growth (in bacteria/day)

at 10 days?

Example of Decay

My savings account started with $1000 bit seems to be dwindling at a continuous rate of 5% per year. How much is in the account after 5

years? At what rate is it dwindling (in $/year)

after 5 years?

Assignment for Friday

Read Section 3.8. Do Exercises 1, 3, and 5b,. Hand-in #1 will be given out Friday

and will be due next Tuesday. Test #1 is Monday, Feb 21.