The idea of Richardson’s extrapolation is to combine two computed values of derivative of y using the same method but with ……… different step sizes.

• A) Three
• B) None of the given choices
• C) Two
• D) Four

The single definite integral of a function is called ………

• A) Area under the curve
• B) Length of the curve
• C) None of the given choices
• D) Volume of the curve

$$For\,the\,given\,data\,{\text{points}}\,(1,0.3),\,(3,1),\,and\,(5,1.2)\,\,the\,divide\,difference\,table\,will\,be\,given\,as$$

• A) $$\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.225} \\\ 6&{1.2}&{}&{} \end{array}$$
• B) $$\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.125} \\\ 6&{1.2}&{}&{} \end{array}$$
• C) $$\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.0625} \\\ 6&{1.2}&{}&{} \end{array}$$
• D) $$\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.525} \\\ 6&{1.2}&{}&{} \end{array}$$

Simpson’s 3/8 rule is based on fitting ……………… points by a cubic.

• A) Three
• B) Four
• C) None of the given choices
• D) Two

$$\begin{gathered} What\,will\,be\,the\,value\,of\,'a'\,in\,the\,given\,divide\,diffidence\,table? \hfill \\ \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D}&{3rdD.D} \\\ 1&{0.7}&{0.25}&{0.025}&{} \\\ 3&{1.2}&{0.35}&{ - 0.0625}&a \\\ 5&{1.9}&{0.1}&{}&{} \\\ 7&{2.1}&{}&{}&{} \end{array} \hfill \\\ \end{gathered}$$

• A) -0.0146
• B) -0.0245
• C) -0.0021
• D) -0.0387

Which of the following is the Richardson’s Extrapolation limit: F2(h/4) provided that F1(h/4) = F1(h/2) = 1 ?

• A) 1
• B) 16
• C) 15
• D) -1

$$The\,first\,divide\,difference\,y[{x_0},{x_1}]\,can\,be\,given\,as\,$$

• A) All
• B) $$\frac{{\Delta {y_0}}}{h}$$
• C) $$\frac{{{y_1} - {y_0}}}{{{x_1} - {x_0}}}$$
• D) $$\frac{{\nabla {y_1}}}{h}$$

In Newton-Cotes formula for finding the definite integral of a tabular function, which of the following is taken as an approximate function then find the desired integral?

• A) Exponential Function
• B) Logarithmic Function
• C) Polynomial Function
• D) Trigonometric Function

Given the following data x:1 3 8 y:2 4 9 f(3) can be found by using

• A) Newton’s backward difference interpolation formula
• B) None
• C) Lagrange’s interpolation formula
• D) Newton’s forward difference interpolation formula

If the area under ‘f(x) = x’ in interval [0,2] is subdivided into two equal sub-intervals of width ‘1’ with left end points, then which of the following will be the Truncation Error provided that I(definite integral) = 2 and approximate sum = 3 ?

• A) 1
• B) 3
• C) 0
• D) -1

Two-point formula for the first derivative is defined as

• A) y(xi+h)+y(xi-h)/h^2
• B) y(xi+h)-y(xi-h)/h^2
• C) y(xi+h)+y(xi-h)/2h
• D) y(xi+h)-y(xi-h)/2h

In Trapezoidal rule, we assume that f(x) is continuous on [a, b] and we divide [a, b] into n subintervals of equal length using the ………points.

• A) None of the given choices
• B) n
• C) n-1
• D) n+1

Three-point formula for the second derivative is defined as

• A) y(xi+h)-2y(xi)+y(xi-h)/h^2
• B) y(xi+h)+2y(xi)+y(xi-h)/h^2
• C) y(xi+h)-2y(xi)+y(xi-h)/2h
• D) y(xi+h)-2y(xi)-y(xi-h)/h^2

If the given tabular function f(x) is approximated by the polynomial ‘P1(x) = x+1’ then which of the following polynomial will approximate the derivative of f(x) ?

• A) x-1
• B) 1
• C) 2x
• D) x

Which of the following is the Richardson’s Extrapolation limit: F1(h/2) provided that F(h/2) = F(h) = 1 ?

• A) 3
• B) 1
• C) 4
• D) 0

The area of a trapezoid is obtained by adding the area of a …………and a triangle.

• A) Circle
• B) None of the given choices
• C) Rectangle
• D) Square

For a function ‘f(x) = x’, with a step size of ‘h=0.01’, which of the following gives the 1st derivative at x =1 by using two point formula?

• A) y’(x) = 1+Some Truncation Error
• B) y’(x) = 1.01 + Some Truncation Error
• C) y’(x) = -0.1 + Some Truncation Error
• D) y’(x) = 0.1 + Some Truncation Error

Differential operator in terms of forward difference operator is given by

• A) $$D = \frac{1}{h}(\Delta - \frac{{{\Delta ^2}}}{{2!}} + \frac{{{\Delta ^3}}}{{3!}} - \frac{{{\Delta ^4}}}{{4!}} + \frac{{{\Delta ^5}}}{{5!}} - ...)$$
• B) $$D = \frac{1}{h}(\Delta + \frac{{{\Delta ^2}}}{2} + \frac{{{\Delta ^3}}}{3} + \frac{{{\Delta ^4}}}{4} + \frac{{{\Delta ^5}}}{5} + ...)$$
• C) $$D = \frac{1}{h}(\Delta - \frac{{{\Delta ^2}}}{2} + \frac{{{\Delta ^3}}}{3} - \frac{{{\Delta ^4}}}{4} + \frac{{{\Delta ^5}}}{5} - ...)$$
• D) $$D = \frac{1}{h}(\Delta + \frac{{{\Delta ^2}}}{{2!}} + \frac{{{\Delta ^3}}}{{3!}} + \frac{{{\Delta ^4}}}{{4!}} + \frac{{{\Delta ^5}}}{{5!}} + ...)$$

In Richardson’s extrapolation method, we usually use two different step sizes ………and …… to yield a higher order method.

• A) h/2,h/4
• B) h,h/2
• C) h/2,h/3
• D) None of the given choices

Given the following data x:0 1 4 y:2 1 4 Value of first order divided difference y[0,1]is

• A) 2
• B) -1
• C) 1
• D) -2