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GATE CSE 1997 | Question: 4.10

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The trapezoidal method to numerically obtain $\int_a^b f(x) dx$ has an error E bounded by $\frac{b-a}{12} h^2 \max f’’(x), x \in [a, b]$ where $h$ is the width of the trapezoids. The minimum number of trapezoids guaranteed to ensure $E \leq 10^{-4}$ in computing $\ln 7$ using $f=\frac{1}{x}$ is

  1. 60
  2. 100
  3. 600
  4. 10000
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Interval [a b] = [1 7]

f(x) = 1/x

Max f"(x) in [a b] is 2 only ..  // f"(x) = 2/x^3

Error = (b -a)*h*h*max f"(x)/12.   // given

         = (7-1)*h*h*2/12

         = h*h

Error <= 10^(-4)         // given

h*h <= 10^(-4)

h <= 10^(-2)

 

(b - a)/n = h                // n is no of trapezoid

(7-1)/n >=10^(-4)

n <= 600

MinimumMinimum value of n is 600..
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