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Due: Thu Feb 28 2013 MTH 437 Homework 1 Problem 1 (10 points) Consider the n-th Taylor polynomial Tn of a function f given by Tn(x) =Xn k=0 f(k)(a) k! (x ?? a)k 1. Compute the 4-th Taylor polynomial for ex sin(x) cos(x). 2. Use matlab to nd the maximum error kf(x) ?? T4(x)k1 for the three functions above and their 4-th order Taylor polynomial in [??=6; =6]. Print out the code that you used. 3. Verify at this order the formula eix = cos(x) + i sin(x) Problem 2 (10 points) Consider the following dierential equation _N= ??N; N(0) = N0 1. Find the analytical solution of the equation. 2. Find the time th such that N(th) = N0=2. 3. Write a matlab code using the Euler-scheme that solves the above equation for N0 = 10 = 0:1 dt = 0:1. Print out the code itself and a graph of the solution of your code in [0; 10]. What is the maximum error of the numerical solution and the analytical solution in this interval? 4. Repeat the above steps using the mid-point method. Problem 3 (10 points) Repeat the steps in problem 2 for the logistic model _N= rN 1 ?? NKYou can use the analytical solution discussed in class. Find appropriate parameters r and K.

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