Mathematically analyzing change

Teacher Cadet College DayMarch 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

Maximum

Minimum

Inflection

Calculus

Let’s “walk a few curves”

Maximum

Minimum

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

http://salempress.com

The rate of change in a body’s temperature is proportional to the difference between its current temperature and that of the surroundings.

Newton’s law of cooling

)]([ tTRkdt

dTRate of

temperature change

Cooling constant Constant

room temperature

Unknown temperature function

Time of death, Exercise 1The 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

]70[' TkTdt

dT

How do we determine this cooling constant?

Natural logarithms...

Yikes! Can you do the math?

Finding the cooling constant

7080

7090ln2

1

)(

)(ln

1

1

2

12 RtT

RtT

ttk

Let’s use those computers!

Log on as “visitor”Password is “winthrop”Go to Dr. Kull’s webpage http://faculty.winthrop.edu/kullt/Open Mathematica file: “Cooling Constant”

The formula

Enter all values from the investigation

The cooling constant (k)

We’ll find a function and follow it “back in time”Back to Dr. Kull’s webpageClick on Direction field linkClick on “DFIELD 2005.10”

Time to analyze!

Click OK

We’ll enter information

in this window

We’ll see the cool stuff in this window

Enter “T” Enter “t”

Enter “.3463574(70-T)”

-5

10

65

105

Click when ready

Close to (0,90)

Time = 0

Temp = 90

Temp = 98.66

Time = -1.042

Exercise 1: Solution

Time of death, Exercise 2Just 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 = 98.66

k = .291823

Time = -3.2227

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 skier’s 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 skier’s temperature drop to 86 ̊ F?

k = .00157484

Temp = 98.66Time = -62.471

Temp = 86.026Time = 61.882Exercise 3: Solution

What if the “room” is not a constant temperature?

Newton’s law of cooling is modified.If proportionality constant is known, we need a single data point.

)]()([ tTtRkdt

dT

A new differential equation

Suppose k=.221343Investigators record T(0)=58R(t) is periodic

)](50)(*20[ tTtSinkdt

dT

Temp = 58Time = 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 2005.10