Homework 6-due March 12th
Homework problems.
1. Determine whether the following integral is convergent or divergent.
Z ∞
sin2 (x)
dx
1 + x2
0
2. (a) (You may use a calculator for the final calculation) Find the approximation M10 (The MidR2 1
point Rule approximation) for 1 e x dx.
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(b) If f (x) = e x prove that | f (2) (x) |≤ 3e for every x in [1, 2],where f (2) (x) is the second derivative
of f (x)
(c) (You may use a calculator for the final calculation) How large do we have to choose n so
that the approximation Mn to the integral in part (a) is accurate to within 0.0001? Hint : Use
part(b) and the bound for the error in the Midpoint Rule.
x
3. (a) If f (x) = ee ,prove that | f (4) (x) |≤ 15e for every x in [−1, 0], where f (4) (x) is the fourth derivative
of f (x). Hint : The fact that ex is an increasing function may be useful.
(b) (You may use a calculator for the final calculation) How large should we take n in order
R0 x
to guarantee that the Simpson’s Rule approximation for −1 ee dx is accurate to within 0.001.
Hint : Use part(a) and the bound for the error in the Simpson’s Rule.
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