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**Question 1: The value of the function f(x) is given at n distinct values of x and its value is to be interpolated at the point x*, using all the n points. The estimate is obtained first by the Lagrange polynomial, denoted by I _{L} and then by the Newton polynomial, denoted by I_{N}. Which one of the following statements is correct?**

(a) I_{L} is always greater than I_{N}

(c) I

(d) I

Given

f'(x) = 4x - 3

By Newton-Rapshon

f(u , f) = 3t

u0 = 0

t0 = 0

Δt = 2

By Euler’s method

After first iteration u = 2 when t = 2

Absolute error = Exact value - approx value

= 10 -2

= 8

or, f' (x) = 3 - e

⇒

∴ X

f{x) = 0.2 + 25x - 200x

By Simpson’s 1/3 Rule

0 = y(0) = 0.2

y

y

= 1.367

f(x) = -2 + 6x - 4X

The correction, Δx, to be added to x

[2015 : 1 Mark, Set-II]

f (x) = - 2 + 6x - 4x

f '(x) = 6 - 8x + 1 .5x

x

By Newton Raphson Method,

⇒

∴

f(x) = x

f'(x) = 2x - 4

x

f (3) = 1, f'(3) = 2

By Newton Rapshon method,

f(5/2) = 25/4 -10 + 4 = 1/4

By Secant method,**Question 8: The integral is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If I is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship? ****(a) J > I ****(b) J < I ****(c) J = I ****(d) insufficient data to determine the relationship ****[2015 : 1 Mark, Set-I]****Answer: (a)****Solution:**

Exact value is computed by integration which follows thee exact shape of graph while computing the area.

Whereas, in **Trapezoidal rule**, the lines joining each points are considered straight lines which is not the exact variation of graph all the time like as shown in figure.

∴ J > I

OR

Here, f(x) = x^{2}

or, f'(x) = 2x

or, f''(x) = 2 > 0

Since f''(x) is positive, the error is negative.

Since error = exact - approximate.

= I - J

and since error is negative in this case J > I is true.**Question 9: The magnitude as the error (correct to two decimal places) in the estimation of following integral using simpson 1/3 rule. Take the step length as 1.**** [2013 : 2 Marks]****Solution: **Using Simpson’s Rule,

= 245.33

The value of integral,

∴Magnitude of error = 245.33 - 244.8 = 0.53**Question 10: The error in for a continuous function estimated with h = 0.03 using the central difference formula The values of x _{0} and f(x_{0}) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately. **

This means,

error ∝ h

If error for h = 0.03 is 2 x 10

Error for h = 0.02 is approximately

**Question 11: The estimate of Obtained using Simpson's rule with three-point function evaluation exceeds the exact value by ****(a) 0.235 ****(b) 0.068 ****(c) 0.024 ****(d) 0.012**** [2011 : 2 Marks]****Answer: (d)****Solution: **

Approximate value by Simpson’s rule with 3 point is,

(n_{pt }is the number of pts and n_{i} is the number of intervals)

Hero

The tabic is

So the estimate exceeds the exact value by,

Approximate value - Exact value

= 0.012499

≈ 0.012**Question 12: The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x ^{2} - N = 0. If i denotes the iteration index, the correct iterative scheme will be**

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