Question

Determine whether the following statements are true and give an explanation or a counter example. a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of​ 3, the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of​ 4, the error is expected to decrease by a factor of 16.

Answer 1

statement is TRUE

statement is FALSE

statement is TRUE

Explanation:

(a)

By using the Trapezoidal Rule, the definite integral can be computed by applying linear interpolating formula on each sub interval, and then sum-up them, to get the value of the integral

So, in computing a definite integral of a linear function, the approximated value occurred by using Trapezoidal Rule is same as the area of the region. Thus, the value of the definite integral of a linear function is exact, by using the Trapezoidal Rule.

Therefore, the statement is TRUE

(b) Recollect that for each rule of both the midpoint and trapezoidal rules, the number of sub-internals, n increases by a factor of a. then the error decreases by a factor of a^2.

So, for the midpoint rule, the number of sub-intervals, n is increased by a factor of 3, then the error is decreased by a factor of 32 = 9, not 8. Therefore, the statement is FALSE

(c) Recollect that for each rule of both the midpoint and trapezoidal rules, the number of sub-internals, n increases by a factor of a. then the error decreases by a factor of a^2.

So, for the trapezoidal rule, the number of sub-internals, n is increased by a factor of 4. then the error is decreased by a factor of 42 = 16

Therefore, the statement is TRUE