Mathematically analyzing change Teacher Cadet College Day March 11, 2011 Dr. Trent Kull Ms. Wendy Belcher Mr. Matthew Neal

Algebra Slope of a line run rise

Calculus The slope of a curve rise run MaximumMinimum Inflection

Calculus Lets walk a few curves MaximumMinimum Inflection

Calculus Know the curve, find the slopes rise run

Differential Equations Know the slopes, find the curve

Rates of change Slopes are rates of change Curves are functions Knowing how an unknown function changes can help us determine the function

Example: Crime scene Body temperature changing at known rates Unknown temperature function can be found

The rate of change in a bodys temperature is proportional to the difference between its current temperature and that of the surroundings. Newtons law of cooling Rate of temperature change Cooling constant Constant room temperature Unknown temperature function

Time of death, Exercise 1 The body of an apparent homicide victim is found in a room that is kept at a constant temperature of 70 ̊ F. At time zero (0) the temperature of the body is 90 ̊ and at time two (2) it is 80 ̊. Estimate the time of death.

Three temperature measurements First: 90 at time 0 Second: 80 at time 2 Room: 70 Determine change Time of death construction How do we determine this cooling constant?

Natural logarithms... Yikes! Can you do the math? Finding the cooling constant

Lets use those computers! Log on as visitor Password is winthrop Go to Dr. Kulls webpage Open Mathematica file: Cooling Constant

The formula Enter all values from the investigation The cooling constant (k)

Well find a function and follow it back in time Back to Dr. Kulls webpage Click on Direction field link Click on DFIELD Time to analyze!

Click OK Well enter information in this window Well see the cool stuff in this window

Enter T Enter t Enter (70-T) Click when ready

Close to (0,90) Time = 0 Temp = 90

Temp = Time = Exercise 1: Solution

Time of death, Exercise 2 Just before midday, the body of an apparent homicide victim is found in a room that is kept at a constant temperature of 68 ̊ F. At 12 noon the temperature of the body is 80 ̊ and at 3p it is 73 ̊. Estimate the time of death.

Temp = k = Time = Exercise 2: Solution

Estimating time, Exercise 3 You are on a search and rescue team in the mountains of Colorado. Your crew has found a hypothermic avalanche victim whose initial temperature reading is 92 ̊ F. 10 minutes later, the skiers temperature is 91 ̊ F. Assuming the surrounding medium is 28 ̊ F, estimate the time of the avalanche to assist rescue & medical crews. Under current conditions, when will the skiers temperature drop to 86 ̊ F?

k = Temp = Time = Temp = Time = Exercise 3: Solution

What if the room is not a constant temperature? Newtons law of cooling is modified. If proportionality constant is known, we need a single data point.

A new differential equation Suppose k= Investigators record T(0)=58 R(t) is periodic

Temp = 58 Time = 0 Back in time

Where there is change… Temperature, Motion, Population, etc. …there are differential equations.

Where there are differential equations… …visual solutions may provide tremendous insight.

Thanks to… Our visiting students Ms. Belcher and Mr. Neal John Polking for the educational use of DFIELD