Suppose we have a rate of change equation and initial condition for the populati

Question

Suppose we have a rate of change equation and initial condition for the population of raccoons in Lake County. Below is a graph of an exact solution. Merry, Pippin, and Sam used the "tip to tail" Euler method to predict what the population of raccoons would be at time t = 2, with time increments one unit. However, they arrived at different graphs for their predictions. Their predictions are given below, and are shown with the exact solution. For each prediction, give reasons as to whether or not each person illustrated the correct relationship between Euler's method and the exact solution.

Explanation / Answer

In Euler method, P(1) is evaluated by approximating P(t) as its tangent at t=0. Then P(2) is evaluated by approximating P(t) as its tangent at t=1 and so on.

a) Merry's Prediction:

Since P(t) is concave down, the tangents to P(t) must be above the curve of P(t). In Merry's prediction, the tangents are drawn below the curve of P(t). Hence Merry's illustration of Euler's method is incorrect.

b) Pippin's Prediction:

Pippin's illustration is correct, because the tangents in Pippin's graph are drawn above the curve of P(t) and they correctly represent the slope of P(t) at t=0 and t=1.

c) Sam's Prediction:

Similar to Merry's graph, the tangents in Sam's graph are drawn below the curve of P(t). Hence Sam's illustration of Euler's method is incorrect.

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