Short Questions and Answers

SHORT QUESTIONS AND ANSWERS

1. State Newton's formula to find f'(x) and f'(x) using the forward differences.

[M.U. Oct., 1995,A.U N/D 2009, Trichy A/M 2010] [A.U CBT N/D 2011, CBT A/M 2011]

Solution:

Let, y = f (x) be a function taking the values y₀, y₁, … y_n corresponding to x₀, x₁, … y_n of the independent variable x. Let the values of x be at equidistant intervals of size h.

2. Find dy / dx at x = 1 from the following table : [MU April 96]

Solution:

The forward difference table is as follows.

3. Fill in the blank :

Numerical differentiation can be used only when the difference of some order …..

Solution: Numerical differentiation can be used only, if it is clear from the tabulated values that differences of some order are constant.

4. If f(x) = a× (a ≠ 0), is given for x = 0, 0.5, 1 show by numerical differentiation that f'(0) = 4√a-a-3

Solution: For x = 0, 0.5, 1, the values of f (x) = ax are respectively ao,

a^0.5, a¹ (i.e.,) 1, √a, a

The formal difference table is as follows:

5. Using forward differences, the formula for f' (a).

6. Write the formula for dy / dx at x = xo using forward difference operator.

Solution :

7. Using Newton's backward difference formula, write the formulae for the first and second order derivatives at the end values x = xn upto the fourth order difference term. [A.U. A/M 2004] [A.U Tvli M/J 2010, CBT N/D 2010] [A.U CBT N/D 2011] [A.U N/D 2006, N/D 2010, M/J 2014] [A.U A/M 2015 R-8]

Solution :