Math 1082 Midterm Exam I
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Math 1082 Midterm Exam I (continued)
5. Suppose that a certain numerical approximation of an integral with N-subdivisions had an error of 0.0375, then find the accuracy of this numerical approximation if five times the subdivisions were used for each of the following methods: (find Error(5N)) (10pts)
Left/Right Rule: Error(5N) ≈
Midpoint/Trapezoid Rule: Error(5N) ≈
Simpson’s Rule: Error(5N) ≈
6. The following questions refer to estimating the area under the function over the interval 1 ≤ x ≤ 4. Find an error bound for estimating this area using Left/Right rule and using Trapezoid/Midpoint rule with N = 40,000 subdivisions. (10pts) (do NOT use the programs/apps)
Left/Right Error Bound:
Trapezoid/Midpoint Error Bound:
7. The following are the left, right, trapezoid, and midpoint approximations (not necessarily in that order) to the definite integral 1 0 ( ) f x dx ∫, where () 0 f x′ < and () 0 f x ′′ > : (10pts)
I) 3.753
II) 3.974
III) 3.669
IV) 3.532
a) Which is which? How do you know?
Left Rule:
Right Rule:
Trapezoid Rule:
Midpoint Rule:
b) Find the Simpson’s rule estimate for this definite integral.
8. Given an increasing concave down function f (x) on the interval [1, 4] with f ′′′(1) 2 = − and (4) 1 2 f ′′′ = − , find the number of subdivisions N needed for using Simpson’s rule to estimate the definite integral 4 1 ( ) f x dx ∫ within an accuracy of 0.0001.
9. Evaluate the following improper integrals: (20pts
i)
ii)
iii)
iv)
10. Use the comparison test to determine whether or not the following improper integrals converge or diverge.
Explain why.
i)
ii)
iii)
2023-09-25