Suppose that T (A) is the trapezoidal approximation of J f(x) dx·It is known that for every h > 0...
Given the following table of data: 0.00 0.250.500.751.00 f(x) 0.39890.38670.35210.30110.2420 Estimate f(x) dx Estimate Jo f (Q) dx (i) by composite trapezoidal rule (ii) by Romberg integration of 0(h6), R33 Given the following table of data: 0.00 0.250.500.751.00 f(x) 0.39890.38670.35210.30110.2420 Estimate f(x) dx Estimate Jo f (Q) dx (i) by composite trapezoidal rule (ii) by Romberg integration of 0(h6), R33
please clear handwriting (Richardson Extrapolation Applied to Differentiation). (a) Suppose that N(h) is an approximation to M for every h > 0 and that M = N(h) + Kih? + Kh? + K3h3 +... ), and N ) for some constants K1, K2, K3, .... Use the values N(h), N to produce an O(h) approximation to M. (b) Recall that f(xo+h)-f(20) df (20) dc f(x) Use the formula you constructed in part (a) to construct an imation to df 20)...
10. Trapezoidal Rule is used to approximate the integral f(a) dx using 1- (yo +2y1 + 2y2 + x-na b-a + 2yn-1 +%),where Use this approximation technique to estimate the area under the curve y = sinx over。 a. π with n 4 partitions. x A 0 B: @ Δy B-A b. The error formula for the trapezoidal rule is RSL (12ba)1 where cischosen on the interval [a, b] to maximize lf" (c)l. Use this to compute the error bound...
4. Consider an underlying function f(x) and nodes at the locatons of oo+ h, for some ћ 〉 0, Assuming ћ is small enough so that f"(x) f"(y), for any x,y E f"(t) zo - h, + h], use the result: fo,1,t write down an approximation formula for f"( f f(xo - h), f(zo), f(xoh). 2, to write down an approximation Co) aS a linear combination o 4. Consider an underlying function f(x) and nodes at the locatons of oo+...
Exercise 6: Given the table of the function f(x)-2" 2 X 0 3 2 f(x) 1 2 4 8 a) Write down the Newton polynomials P1(x), P2(x), Pa(x). b) Evaluate f(2.5) by using Pa(x). c) Obtain a bound for the errors E1(x), E2(x), Es(x) Exercise 7: Consider f(x)- In(x) use the following formula to answer the given questions '(x) +16-30f+16f,- 12h a) Derive the numerical differentiation formula using Taylor Series and find the truncation error b) Approximate f'(1.2) with h-0.05...
4. Compute the k-th order derivative for f(0) = arctan x for every k = 1,2,3,... Use the result to derive the Taylor formula for f(x) = arctans.
7. (a) Use the well known Maclaurin series expansion for the cosine function: f (x ) = cos x = 1 x? 2! + 4! х 6! + (-1)" (2n)! . * 8! 0 and a substitution to obtain the Maclaurin series expansion for g(x) = cos (x²). Express your formula using sigma notation. (b) Use the Term-by-Term Integration Theorem to obtain an infinite series which converges to: cos(x) dx . y = cos(x²) (c) Use the remainder theorem associated...
Section A Q1 0 Using the following Taylor series expansion: f(x+h) = f(x)+hf'(x)+22 h 3! f"(x)+ (+0) (1.1) 4! show that the central finite difference formula for the first derivative can be written as: f'(x)= f(x+h)-f(x-1) + ch" +0(hº) (1.2) 2h Determine cp and of the derived equation. [4 marks] Consider the function: f(x) = sin +COS (1.3) 2 2 Let x =ih with n=0.25, give your answer in 3 decimals for (ii) to (vi): (ii) Evaluate f(x) for i...
Need help on this thermodynamics question. Thanks Data given from function Cp=22.64 + 6.28 x (10^-3)*T [Jmol-1 K-1 ] Cp(J/mol.K) T(K) 24.524 300 24.838 350 400 25.152 25.466 450 500 25.78 550 26.408 26.722 650 27.036 700 750 27.35 27.664 800 27.978 28.292 28.606 950 1000 28.92 1050 29.234 1100 29.548 1150 29.862 30.176 1200 1250 30.49 1300 30.804 1350 31.118 358 31.16824 The specific heat capacity of solid copper above 300 K is given by Cp-22.64+6.28 x 103TJmol K1]...
(a) Suppose the equation defines a differentiable function y-f(z) (0) Find the derivatint ()-(e,l) (ii) Write the linear approximation for f(x) around a = e and use this to approx- dy Hence, e T,y 5 marks imate f(3). markS (b) Evaluate the following limits. Simplify your results if possible. 5 marks 5 marks] lim cot 5x sin 6x cos 7a (i) (ii) limIn (a) Suppose the equation defines a differentiable function y-f(z) (0) Find the derivatint ()-(e,l) (ii) Write the...