Help with math
265 Project 1: Koch-ing up Mackerel Name: 1. (15 points) The Koch Snowflake has an infinite perimeter but finite area. The first step is to draw an equilateral triangle. Then on each side, draw another equilateral triangle one-third of the size. Continue. Here is a picture after two steps with the first piece of the third step included. Finish the third step and add the fourth step. Determine the area of the shape you have drawn if the length of the original triangle is 3 inches. Include units. The Koch Snowflake results after an infinite number of iterations. Construct a geometric series that represents the area of the Koch Snowflake. Compute the sum of the geometric series you constructed. 2. (15 points ) Suppose the P (t) models a population of mackerel measured in thousands of kilograms of biomass at time t in years. P (t) satisfies dP = 2P − P 2 where P (0) = 3. dt Determine P 00 (t) by taking the derivative of the differential equation using implicit differentiation. That means dtd P 2 = 2P P 0 . Get P 000 (t) in the same way. Use the above derivatives and the equation, itself, to construct a third-degree Taylor polynomial for P (t). Approximate P (0.1) using your Taylor polynomial. 2t The explicit form of P (t) is 3e6e2t −1 . Ascertain the error between the Taylor polynomial and the explicit form at t = 0.1. 2
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