Thread: Finding "K" - Approximating the Error with Trapezoid and Midpoint Rule

Hello everyone. I'm trying to approximate the integral of cos(x^2) over the interval 0 to 1. I'm using the trapezoid and midpoint rule with 8 subintervals which is not a problem. However, I am to find the error bounds using the formulas given in the book and I am having trouble finding what "K" is. I need to find the second derivative of cos(x^2) and find the maximum value over the interval. I get the second derivative to be [(-4)*cos(x^2)*(x^2)] - [2sin(x^2)].
Over the interval 0 to 1, the maximum value of this equation I believe is 0, which would give me K = 0, but that can't be right because then the error bound would also be 0. Can someone please help me and tell me what I'm doing wrong to find K?

Originally Posted by thejabronisayz

Hello everyone. I'm trying to approximate the integral of cos(x^2) over the interval 0 to 1. I'm using the trapezoid and midpoint rule with 8 subintervals which is not a problem. However, I am to find the error bounds using the formulas given in the book and I am having trouble finding what "K" is. I need to find the second derivative of cos(x^2) and find the maximum value over the interval. I get the second derivative to be [(-4)*cos(x^2)*(x^2)] - [2sin(x^2)].
Over the interval 0 to 1, the maximum value of this equation I believe is 0, which would give me K = 0, but that can't be right because then the error bound would also be 0. Can someone please help me and tell me what I'm doing wrong to find K?

You need the maximum of |f''(x)| x in [0,1], not the maximum of f''(x).

RonL

thank you, that would make sense. It's just that my professor made it seem as though we needed to find the maximum value that the second derivative would produce, not the maximum value after taking the absolute value of the numbers in the interval. Thanks again