By Choquet-Bruhat Y., Dewitt-Morette C., M.Dillard-Bleick

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**Sample text**

Lesson 6b _2. A runner’s position function p is graphed below: Graphing Solutions to Differential Equations 45 Runner’s starting position? ___ _3. 130 Given the velocity function graphed below, graph the runner’s position function on the grid below: the runner’s starting position is the 20-yard line. (Note: this exercise demonstrates the basic technique for graphing solutions to DEs) The runner finish position? ___ Velocity for interval 0-2? ___ p(0) = 20 Velocity for interval 2-4? ___ p(2) = ___ Velocity for interval 4-6?

So, we continue the graph of pA with a straight line having slope – 50, as shown to the left. pA is the trajectory of our approximate apple. It is an approximate solution to the DE p’(t) = -10*t. We’ll see how to improve it in the next lesson. We’ll start by dividing the time of interest, 0 to 10 seconds, into two 5-second intervals. The apple’s initial position is 500 m, and its velocity at the start of the 1st interval is 0. pA will start at 500, and its velocity will be 0 for the entire interval.

It is an approximate solution to the DE p’(t) = -10*t. We’ll see how to improve it in the next lesson. We’ll start by dividing the time of interest, 0 to 10 seconds, into two 5-second intervals. The apple’s initial position is 500 m, and its velocity at the start of the 1st interval is 0. pA will start at 500, and its velocity will be 0 for the entire interval. Since the approximate apple’s velocity on the interval is constant, its graph is a straight line as shown below: We know the apple’s true velocity function, v(t) = -10 * t.

### Analysis, manifolds and physics by Choquet-Bruhat Y., Dewitt-Morette C., M.Dillard-Bleick

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