The trapezoidal method is used to evaluate the numerical value of ∫₀¹ eˣ dx.…
2007
The trapezoidal method is used to evaluate the numerical value of ∫₀¹ eˣ dx. Consider the following values for the step size h:
(i) 10⁻²
(ii) 10⁻³
(iii) 10⁻⁴
(iv) 10⁻⁵
For which of these values of h is the computed value guaranteed to be correct to seven decimal places? Assume that there are no round-off errors in the computation.
Answer: B. (iii) and (iv) only — For the composite trapezoidal rule, the error bound is |E_T| ≤ ((b - a)/12)h² max |f''(x)| on [a, b]. Here f(x) = eˣ, so f''(x) = eˣ. On [0, 1], max |f''(x)|…
- A.
(iv) only
- B.
(iii) and (iv) only
- C.
(ii), (iii) and (iv) only
- D.
(i), (ii), (iii) and (iv)
Attempted by 2 students.
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Correct answer: B
For the composite trapezoidal rule, the error bound is
|E_T| ≤ ((b - a)/12)h² max |f''(x)| on [a, b].
Here f(x) = eˣ, so f''(x) = eˣ. On [0, 1], max |f''(x)| = e. Also b - a = 1.
Thus,
|E_T| ≤ (e/12)h².
To be correct to seven decimal places, the error must be less than 0.5 × 10⁻⁷.
So,
(e/12)h² ≤ 0.5 × 10⁻⁷
h² ≤ (12 × 0.5 × 10⁻⁷)/e
h ≤ approximately 4.7 × 10⁻⁴.
Among the given step sizes, 10⁻⁴ and 10⁻⁵ satisfy this condition, while 10⁻² and 10⁻³ do not.
Therefore, the correct choice is (iii) and (iv) only.
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