Approximation of derivatives using interpolation polynomials

APPROXIMATION OF DERIVATIVES USING INTERPOLATION POLYNOMIALS

(a) Newton's forward difference formula

Our Aim is to find the derivative of y = f(x) passing through the (n + 1) points, at a point nearer to the starting value x = x_o.

Newton's forward difference interpolation formula is

Equations (5), (6) and (7) give the values of first second and third derivatives at the starting value x = xo-mol

(b) Newton's backward difference formula

Newton's backward difference formula is

Equations (5), (6) and (7) give the values of first, second and third es of fir derivatives at x = x_n.

(c) Maxima and Minima of a tabulated function

Newton's forward interpolation formula is

For maxima or minima dy / du = 0. Hence equating the right hand side of (1) to zero and retaining only upto third differences, we obtain

Substituting the values of Ay₀, A² y₀, A³y₀ from the difference table, we solve this quadratic for u. Then the corresponding values of x are given by x = x0+ uh at which y is maximum or minimum.

Note: If the interval of differencing is not constant (i.e., x's are not equally spaced), we get Newton's divided difference formula (or) Lagrange's interpolation formula for general x, and then differentiating it w.r.to x, we can get the derivatives at any x in the range.

1. Find f'(3) and f'' (3) for the following data: [A.U. April/May 2005]

Solution: Since we require f' (3) and f'' (3) we use Newton's forward formula.

2. Find the first, second and third derivatives of f (x) at x = 1.5 if

[A.U N/D 2013, N/D 2014, M/J 2013, A.U.Trichy A/M 2010] [A.U. Tvli N/D 2011] [A.U A/M 2017 R-13] [A.U N/D 2021 R-17]

Solution: We have to find the derivative at the point x = 1.5 which is the starting point of the given data. Therefore, we use Newton's forward interpolation formula.

Forward difference table

3. Compute f' (0) and f'' (4) from the data.

Solution:

Since, we require f' (0.5) and f'' (3.5), we use Newton's forward formula and Newton's backward formula.

4. Find dy/dx and d²y / dx² at x = 51, from the following data [A.U CBT N/D 2010, A.U M/J 2012]

Solution:

Here h = 10. To find the derivatives of y at x Newton's forward difference formula taking the origin at a = 50.

5. The following data gives the velocity of a particle for 20 seconds at an interval of 5 seconds. Find the initial acceleration using the entire data. [A.U. A/M 2004, CBT M/J 2010]

Solution: v is dependent on time t

 (i.e.,) v = v(t). We require acceleration dv / dt at 0.

6. Find the maximum and minimum value of y tabulated below.

Solution: Netwon's forward difference formula is

7. Find the maximum and minimum value of y from the following table.

Solution: Here, h = 1

Newton's forward difference formula is

8. Find y' (x) given [A.U Tvli A/M 2011]

Hence, find y' (x) at x = 0.5

Solution:

Here, h = 1. We apply Newton's forward interpolation formula for derivative

9. Consider the following table of data:

Find f'(0.25) using Newton's forward difference approximation, and f'(0.95) using Newton's backward difference approximation.

Solution : Here, h = 0.2

Newton's forward interpolation formula for derivative

10. The population of a certain town is given below. Find the rate of growth of the population in 1931, 1941, 1961 and 1971.

11. Find the first and second derivatives at x = 1.6 for the function represented by the following tabular data:

Solution:

Since the arguments are not equally spaced, we will use Newton's divided difference formula.

Divided difference table

12. Given the following data, find y' (6) & y' (5) and the maximum value of y.

 [A.U M/J 2006, N/D 2006, M/J 2007, N/D 2007, Tvli M/J 2009] [A.U N/D 2011, N/D 2012] [A.U N/D 2020 R-17 NM] [A/M 2021 R-17 NM]

Solution:

Since the arguments are not equally spaced, we will use Newton's divided difference formula

Divided Difference Table

y (x) is maximum if y' (x) = 0. 3x² + 4x + 3 = 0. But the roots are imaginary. Therefore, there is no extremum value in the range. In fact, it is an increasing curve.

13. From the following table of values of x and y, obtain

y' (x) and y (x) for x = 16

[A.U N/D 2010]

Solution :

Since we require f' (16) and f" (16), we use Newton's forward formula h = 2

14. From the following table find the rate of change of pressure with respect to volume when v = 2.

Solution :

dp / dt at v = 2 is required.

We use forward formula of Newton.

15. The following table gives the velocity v of a particle at time t. Find its acceleration at t = 2.

Solution :

v is dependent on time t

v = v(t), we require acceleration = dv/dt

Therefore, we have to find v' (2).

16. Find the first two derivatives of (x)3 at x = 50 and x = 56, for the given table : [A.U N/D 2011]

Solution:

We use Newton's backward formula