1. What do all of these have to do with Calculus?!?!?

2. Application of Differential Equations
AP Calculus
Mrs. Mongold

3. Common Problems: Exponential Growth and Decay Compound Interest
Radiation and half-life
Newton’s law of cooling
Other fun topics

4. Suppose we are interested in a quantity that increases at a rate proportional to the amount present. If we also know the amount at a certain time, we can find y as a function of time by solving the differential equation:

5. When t = o, y = y0.
Solving for C,
C = y0
Initial Amount

6. The rate of change of a population of rabbits is proportional to the number of rabbits present at any given time. If 10 rabbits are present initially, and 195 rabbits are present in 6 months, how many rabbits will there be in 2 years?

7. Suppose you deposit $800 in an account that pays 6. 3% annual interest
Suppose you deposit $800 in an account that pays 6.3% annual interest. How much will you have 8 years later if interest is:
A. compounded continuously
B. compounded quarterly

8. The rate at which the population of a group of organisms grows is directly proportional to the number of organisms in the population. If the population at time zero is 3,500 and the population after one year is 5,250, what will the population be after 3 years?

9. Half-Life After t years, half of the original amount is left
t is the half-life. K does not depend on the initial amount!

10. Example ½ life of carbon-14 is 5700 years.
Find the age of a sample in which 10% of the original radioactivity is depleted.
90% is still present

11. Newton’s Law of Cooling
Rate of change of object’s temp with respect to time
Difference between temp of surroundings and temp of object is
Proportional to

12. The crime was committed at 1:23 pm
When a murder is committed, the body, originally at 37C, cools according to Newton’s Law of Cooling. Suppose on the day of a very difficult calc quiz, Mrs. Mongold’s body is found at 2pm with a body temperature of 35C in a room with a constant temperature of 20C. 2 hours later the body temperature is 30C. Being the bright students you are, you should be able to tell Mr. Miller when the heinous crime was committed.
The crime was committed at 1:23 pm