The trapezoidal method for numerically computing integral_a^b f(x) dx has an…
1997
The trapezoidal method for numerically computing integral_a^b f(x) dx has an error E bounded by ((b - a)/12) h^2 max |f''(x)|, x in [a, b], where h is the width of each trapezoid. The minimum number of trapezoids guaranteed to ensure E <= 10^-4 while computing ln 7 using f(x) = 1/x is
- A.
60
- B.
100
- C.
600
- D.
10,000
Show answer & explanation
Correct answer: C
To compute ln 7 using f(x) = 1/x, use ln 7 = integral_1^7 (1/x) dx. Thus a = 1 and b = 7, so b - a = 6.
For f(x) = 1/x,
f'(x) = -1/x^2,
f''(x) = 2/x^3.
On [1, 7], the maximum value of |f''(x)| is 2, attained at x = 1.
The given error bound is
E <= ((b - a)/12) h^2 max |f''(x)|
= (6/12) h^2 * 2
= h^2.
We need E <= 10^-4, so h^2 <= 10^-4 and h <= 10^-2 = 0.01.
If n trapezoids are used, h = (b - a)/n = 6/n. Therefore,
6/n <= 0.01, so n >= 600.
The minimum number of trapezoids guaranteed by the bound is 600.