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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) 

iv)