Differential Equations Math meets the real world!

Why is it that the more Math I learn the harder it gets? One learning theory claims that the more a person knows about a task the slower he or she will learn the rest of the task. How can we “model” this mathematically and develop a mathematical way of determining how fast learning will occur?

Setting up the model… Assign variables and parameters: –Let ‘y’ = the percent of the task that has been learned How can we represent the statement: –When will a person learn fastest according to this model? –How do we represent the rate at which a person learns? more a person knows about a task the slower he or she will learn the rest of the task.

From “words to action”… Rate a person learns = Percentage of task not yet learned Where t is the time in weeks

A simple First Order Differential Equation Re-arrange this by separating all “y” terms from “t” terms… and then integrate both sides

Initial Conditions… In order to solve this you will need to supply the appropriate values for y and t at the start - IC’s: y(0)=0

First Order Differential Equation Basic form Initial Condition: y(a) = b

Sometimes it’s the little things in life that count! Suppose that you were having tea with the queen. The tea was poured into identical cups and at the same time but you added your cream immediately. The queen said that she always waits 5 minutes before adding her cream and asked her butler to put the cream back in the cooler. It would be improper to start drinking before the queen so you wait. Five minutes later the queen has her butler add the same amount of cream to her tea as you put in. Just as you reach for your first sip the queen gives you a mischievous smile and says: “I always do this to test whether young diplomats can think on their feet – tell me, which of us is drinking the hottest cup of tea?” A Matter of Protocol (or why diplomats should know calculus)

Other Simple Examples Newton’s Law of Cooling –“the rate at which a body cools is directly proportional to the difference between its temperature and the ambient or background temperature” Exponential Growth and Decay –Application: Radioactive Dating Try… 11.1 #7,13,19, 23 Practice: 8.1: 6, 8, 16