Consider the definite integral, from 0 to 4, e^(cosx) dx
a) Compute the estimate for this integral using the trapezoid rule with n = 4.
b) Find a bound for the error in part a.
c) How large should n be to guarantee that the size of the error in using Tn is less than 0.0001
3. The Attempt at a Solution
delta x = b-a/n = 4-0/4 = 1.
1 * [f(0) + 2f(1) + 2f(2) + 2f(3) + f(4)] =
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e^(cos(0)) + 2 * e^(cos(1)) + 2 * e^(cos(2)) + 2 * e^(cos(3)) + e^(cos(4))
= 2.718 + 3.433 + 1.319 + 0.743 + 0.5220 = 8.733
Problem here: when I calculate this integral with my TI, I get a vastly different number (4.335). Anyway, onto B
First derivative: -sin(x) e^(cos(x))
Second derivative: sin^2 (x) e^ (cos(x)) - cos(x) * e^(cos(x))
From here, I'm confused. I don't know exactly how to find "K". I know that it is the absolute value of the largest number possible of the 2nd derivative of the original function (3rd derivative for Simpsons Rule), and that it would be: k(b-a)^3 /12(n)^2.
I have read that you may want to make sin/cos functions equal to 1, since that is the highest value they can have, however doing that would yield:
1 * e^1 - 1 * e^1 =
e - e = 0
C) Haven't gotten to that point.