Question

(Part B) The actual value of fº3cos(7tx)dx is 0. As you can see there is a discrepancy between the approximation with Trapezoid Rule and its actual value. The reason for this discrepancy is a) The trapezoidal rule samples only function values that are positive. b) Both values are equivalent. c) The trapezoidal rule samples only function values that are zero. d) The trapezoidal rule samples only function values that are negative. e) None of the above.

Answer 1

In the case of $\small \int_{0}^{20}3cos(7\pi x)dx$

For example If we choose nodes to be {0, 1,2,...20} f takes psitive values, negetive values, 0s.

In this case the value of the integration is 0.

Again if we take the partition to be {0 , 2 , 4 , ... , 20} then every value taken is 3. i.e positive. The integration value is 30.

OPTION A is the correct answer.

Since the perticular value of the step length is not given the problem can be visualized for different set of nodes by running the following MATLAB code

>> for j=1:20
x=0:20/j:20;
y=3*cos(7*pi*x);
i=1:length(x);
[i',x',y']
q=trapz(y)
endfor

trapz represents the formula for the trapezoidal rule.

In every case either trapizoidal rule gives value 0. Or it has a discrepency since it takes all positive values.

NOTE

The actual reason that the value of the integration obtained by trapezoidal rule has discrepancy is that Trapezoidal rule approximates the area of the curve by joining straight lines between the function values at the nodes.

AY x1 X2 x3 Х

The case that the trapezoidal rule samples only positive, negetive or zero values, entirely depends upon how we choose the nodes and what is the value of the function at that point.