3. Using the Newton's backward interpolation formula construct an interpolation polynomial of degree 3 for the data [5 Marks] f(-0.75) = -0.07181250, f(-0.5) = -0.024750, f(-0.25) = 0.33493750, f(0) = 1.1010. Hence, find f(-1/3). 4. Compute f'(0) and f''(4) from the data [5 Marks] x 0 1 2 3 4 y 1 2.718 7.389 20.086 54.598 5. Evaluate ?_{-1}^{1} (3x^2 - 5x^4)dx. [10 Marks] (i) Using Trapezoidal rule by taking h = 1, h = 0.5 & h = 0.2. [6] (ii) Using Romberg's method. [2] (iii) Using Gaussian two point formula [2]
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We have the data points: $(-0.75, 0.07181250)$, $(-0.5, 0.024750)$, $(-0.25, 0.33493750)$, and $(0, 1.1010)$. Show moreâŠ
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a) Use the trapezoidal rule to find the value of f(0.5) if f(x) is given, and h=0.25. Given that f(0)=0, f(1)=f(0.25)=1, and f(0.75)=0.25. b) Approximate the same integral using Simpson's rule.
Adi S.
1. Evaluate the integral of the data that is tabulated below, with: a) the trapezoidal rule b) and Simpson's rules: X | -2 | 0 | 2 | 4 | 6 | 8 | 10 f(x) | 35 | 5 | -10 | 2 | 5 | 3 | 20 2. The function f(x) = 2e^-1.5x can be used to generate the following table of unevenly spaced data: X | 0 | 0.05 | 0.15 | 0.25 | 0.35 | 0.475 | 0.6 f(x) | 2 | 1.8555 | 1.5970 | 1.3746 | 1.1831 | 0.9808 | 0.8131 3. Use Romberg integration of order h^8 to evaluate: â«[0 to 3] xe^x dx Compare Δ_a Δ_r Obtain an estimate of the integral of Problem 3, but use Gauss-Legendre formulas with two, three and four points. Calculate Δ_t for each case based on the solution analytics. 4. Use numerical integration to evaluate the following: a. â«[2 to â] dx / x(x+2) b. â«[0 to â] e^-y sin^2 y dy c. â«[0 to â] 1 / ((1+y^2)(1+y^2/2)) dy d. â«[-2 to â] ye^-y dy e. â«[0 to â] (1/â2) e^(-x^2/2) dx
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05. (a). Using Newton's Forward and Backward interpolating polynomials, estimate the population in 1895 and 1925 from the following statistics: Year (x) Population 1891 46 1901 66 1911 81 1921 93 1931 101 (b). Apply Lagrange's interpolation formula, determine f(5) for the following data: x 1 2 3 4 7 f(x) 2 4 8 16 128 (c). Evaluate the integral â«[1,2] 1/x dx, using (i) Trapezoidal rule, (ii) Simpson's one third rule. Compare these values with the exact value of the integral
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