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s01 - Ma 449 Numerical Applied Mathematics Model Solutions...

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Ma 449: Numerical Applied Mathematics Model Solutions to Homework Assignment 1 Prof. Wickerhauser Exercise 9(a*) of section 1.1, p.12 Let g(x) be the integral from 0 to x of t^2 cos(t). Then by the Second Fundamental Theorem of Calculus (2FT), we have g'(x)= x^2 cos(x). Exercise 9(b) of section 1.1, p.12 Let g(x) be the integral from 1 to x of exp(t^2). Then by the Second Fundamental Theorem of Calculus (2FT), we have g'(x)=exp(x^2). But the function we need to differentiate is f(x)=g(x^3), so we use the chain rule: (d/dx)f(x) = (d/dx)g(x^3) = 3 x^2 g'(x^3) = 3 x^2 exp(x^6). Exercise 11(b*) of section 1.1, p.12 Factor out 2/3 and apply the geometric sum formula to get 1/(1-1/3)=3/2 for the sum of 1/3^n from n=0 to infinity. The result is 1. Exercise 11(c) of section 1.1, p.12 Write 3/(n(n+1)) as 3*[ 1/n - 1/(n+1) ] to get a telescoping series: each term -1/(n+1) is cancelled by the 1/n term for the next n. Since 1/n tends to 0 as n tends to infinity, the only surviving term is 1/n for the first value of n, namely n=1. Thus the sum is 3*(1/1) = 3.
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