Project ForgeEngineering Mathematics
Numerical Methods
GATE 2008
Hard
The minimum number of equal length subintervals needed to approximate ∫[1 to 2] x·eˣ dx to an accuracy of at least (1/3)×10⁻⁶ using the trapezoidal rule is:
- 1000e
- B1000
- C100e
- D100
Explanation
The trapezoidal rule's error bound is E ≤ (b−a)³/(12n²) · max|f''(x)| on [a,b]. Here f(x)=x·eˣ, f'(x)=eˣ(1+x), f''(x)=eˣ(2+x), which is maximized on [1,2] at x=2: f''(2)=4e². With (b−a)=1, E ≤ 4e²/(12n²) = e²/(3n²). Requiring e²/(3n²) ≤ (1/3)×10⁻⁶ gives n² ≥ e²×10⁶, i.e. n ≥ 1000e ≈ 2718.3.
Source: GATE CS 2008 (2008)
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