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

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

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Application of Differential Equations

• AP Calculus
• Mrs. Mongold

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Common Problems

• Exponential Growth and Decay
• Compound Interest
• Radiation and half-life
• Newtons law of cooling
• Other fun topics?

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• 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

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When t o, y y0. Solving for C, C y0
Initial Amount
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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?
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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
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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?
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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!
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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
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Newtons Law of Cooling
Rate of change of objects temp with respect to
time
Difference between temp of surroundings and temp
of object
is
Proportional to
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When a murder is committed, the body, originally
at 37C, cools according to Newtons Law of
Cooling. Suppose on the day of a very difficult
calc quiz, Mrs. Mongolds 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 123 pm