Question

The error in the trapezoidal rule is
1. of the order h³
2. of the order h²
3. of the order h⁴
4. of the order h

Answer 1

Correct Answer - Option 2 : of the order h²

Concept:

Trapezoidal rule states that for a function y = f(x)

For the trapezoidal method with only a single subinterval, we have

\(\mathop \smallint \nolimits_{{x_0}}^{{x_0} + nh} f\left( x \right)dx - \frac{h}{2}\left[ {f\left( \alpha \right) + f\left( {\alpha + h} \right)} \right] = - \frac{{{h^2}}}{{12}}{f^{''}}\left( c \right)\)

for some c in the interval [α, α + h].

The general trapezoidal rule Tn(f) was obtained by applying the simple trapezoidal rule to a subdivision of the original interval of integration.

Then the error, En(f) = \(\mathop \smallint \limits_b^a f\left( x \right)dx - Tn\left( f \right)\)

Then combining these errors, we obtain

∴ \(En\left( f \right) = - \frac{{{h^2}n}}{{12}}\left[ {\frac{{{f^{''}}\left( {{y_1}} \right) + ... + {f^{''}}\left( {{y_n}} \right)}}{n}} \right]\)

∴ The error in the trapezoidal rule is of the order h²