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Section A Q1 0 Using the following Taylor series expansion: f(x+h) = f(x)+hf'(x)+22 h 3! f"(x)+...
5. High accuracy Differentiation formulas. Using Taylor series.
5. High accuracy Differentiation formulas. Using Taylor series.(a) Prove the following centered finite difference formula that is \(\mathrm{O}\left(\mathrm{h}^{4}\right)\) for the first derivative$$ f^{\prime}\left(x_{i}\right)=\frac{-f\left(x_{i+2}\right)+8 f\left(x_{i+1}\right)-8 f\left(x_{i}-1\right)+f\left(x_{i-2}\right)}{12 h}+O\left(h^{4}\right) $$(b) Compute the centered finite difference approximation of \(\mathrm{O}\left(\mathrm{h}^{2}\right)\) and \(\mathrm{O}\left(\mathrm{h}^{4}\right)\) for the first derivative of \(y=\sin x\) at \(x=\pi / 4\) using the value of \(h=\pi / 12\). Calculate the true percent relative error in both cases.
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the finite difference formula defined by the second row of Table 5.2. Table 5.2. Weights for forward finite difference formulas (p 0 in (5.4.2). The values given here are for approximating the derivative at zero. See the text about the analogous backward differences where q=0. The term order of accuracy is explained in Section 5.5. Order of Node location 2h 3h 4h accuracy 1 2...
Given that f(x) = 3x + 1 g(x) = 5x - 8 and h(x) = 2x...
Given that f(x) = 3x + 1 g(x) = 5x - 8 and h(x) = 2x – 1 3 Find:- i) f(-4) = ii) g[h(5)] = iii) f[g(3)] = iv) g[h(x)] = vi) h-1(7) =
Approximating derivatives f(z +h) - f(z) f(x)-f( -h) f(x + h) - f(x - h) Forward difference Backw...
Added the formulas, thank you!
Approximating derivatives f(z +h) - f(z) f(x)-f( -h) f(x + h) - f(x - h) Forward difference Backward difference Centered difference for 1st derivative s(a) (3) 2h t)-2e-bCentered diference for 2nd derivative (4) 2 2. Write a short program that uses formulas (1), (3) and (4) to approximate f(1) and f"(1) for f(x)e with h 1, 2-1, 2-2,.., 2-60. Format your output in columns as follows: h (6+f)() error (öf(1 error f error Indicate the...
Question 1 The functions f(x) g(x) and h(x) are defined as follows: f(x) = e* XER...
Question 1 The functions f(x) g(x) and h(x) are defined as follows: f(x) = e* XER g(x) = x x 20 h(x) = 2x + 1 XER [3 marks] Find fg(x) and state its domain and range Find hf (x) and state its domain and range Find hº(x) and state its domain and range __[3 marks] (1111) [3 marks] WA (b) The figure below shows the graph of y=-x-shifted to four new positions. Write an equation for each new graph....
5. Let f(x)- arctan(x) (a) (3 marks) Find the Taylor series about a 0 for f(x). Hint: - arctan(x)...
5. Let f(x)- arctan(x) (a) (3 marks) Find the Taylor series about a 0 for f(x). Hint: - arctan(x) - dx You may assume that the Taylor series for f(x) converges to f (x) for values of x in the interval of convergence (b) (3 marks) What is the radius of convergence of the Taylor series for f(x)? Show that the Taylor series converges at x-1. (c) (3 marks) Hence, write T as a series (d) (3 marks) Go to...
3. Consider the function f(x) = -0.1.24 – 0.15x3 – 0.522 – 0.25x + 1.2. (a)...
3. Consider the function f(x) = -0.1.24 – 0.15x3 – 0.522 – 0.25x + 1.2. (a) Obtain the analytical expression (i.e. True or Exact Solution) for the first derivative, Eval- uate its value at 1 =0.5. Box your answer and label it as fexact- (b) Now assume the function is discretized on a grid with uniform spacing of h. Evaluate your finite difference approximation at x = 0.5 using central differencing with step sizes starting at 1 and re- duced...
3h Q6. Consider f'(X) f2 -f- hf"C) 2 1. Derive the formula using Taylor expansion. 2....
3h Q6. Consider f'(X) f2 -f- hf"C) 2 1. Derive the formula using Taylor expansion. 2. Derive the formula using Lagrange interpolation. 3. Find the optimal h.
Differentiate the following function: f(x) = ex -2x +1
MatlabMECE 2350 Numerical Methods Lab 8.1. Differentiate the following function: f(x) = ex -2x +1 and solve its first derivative atx = 8 2. Numerically evaluate the approximated first derivative from the above function at x = 8 and h = 0.15 by the following: (a) Forward finite difference method (b) Backward finite difference method (c) Centered finite difference method 3. Calculate the error of each method by comparing the numerical derivative with the result from problem 1.
h 5. In Homework 8 we looked at the finite difference approximation of f'(x). f(x +...
h 5. In Homework 8 we looked at the finite difference approximation of f'(x). f(x + 1) = f(x) + f'(z)h + }}" (-x)+2 +0(h2). Rearranging terms, and dividing by h leads to D(h) f(x+h)-f(x) != f'(x) +}s"(x)h + O(n?). Find constants a and 8 such that A(h) = aD(h/2) + BD(h) = f'(x) + (h) (similar to convergence acceleration in Chapter 2, we are using our knowledge of the be- haviour of the error to get a better approximation!)...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the finite difference formula defined by the second row of Table 5.2. Table 5.2. Weights for forward finite difference formulas (p 0 in (5.4.2). The values given here are for approximating the derivative at zero. See the text about the analogous backward differences where q=0. The term order of accuracy is explained in Section 5.5. Order of Node location 2h 3h 4h accuracy 1 2...
Added the formulas, thank you!
Approximating derivatives f(z +h) - f(z) f(x)-f( -h) f(x + h) - f(x - h) Forward difference Backward difference Centered difference for 1st derivative s(a) (3) 2h t)-2e-bCentered diference for 2nd derivative (4) 2 2. Write a short program that uses formulas (1), (3) and (4) to approximate f(1) and f"(1) for f(x)e with h 1, 2-1, 2-2,.., 2-60. Format your output in columns as follows: h (6+f)() error (öf(1 error f error Indicate the...
Question 1 The functions f(x) g(x) and h(x) are defined as follows: f(x) = e* XER g(x) = x x 20 h(x) = 2x + 1 XER [3 marks] Find fg(x) and state its domain and range Find hf (x) and state its domain and range Find hº(x) and state its domain and range __[3 marks] (1111) [3 marks] WA (b) The figure below shows the graph of y=-x-shifted to four new positions. Write an equation for each new graph....
5. Let f(x)- arctan(x) (a) (3 marks) Find the Taylor series about a 0 for f(x). Hint: - arctan(x) - dx You may assume that the Taylor series for f(x) converges to f (x) for values of x in the interval of convergence (b) (3 marks) What is the radius of convergence of the Taylor series for f(x)? Show that the Taylor series converges at x-1. (c) (3 marks) Hence, write T as a series (d) (3 marks) Go to...
3. Consider the function f(x) = -0.1.24 – 0.15x3 – 0.522 – 0.25x + 1.2. (a) Obtain the analytical expression (i.e. True or Exact Solution) for the first derivative, Eval- uate its value at 1 =0.5. Box your answer and label it as fexact- (b) Now assume the function is discretized on a grid with uniform spacing of h. Evaluate your finite difference approximation at x = 0.5 using central differencing with step sizes starting at 1 and re- duced...
3h Q6. Consider f'(X) f2 -f- hf"C) 2 1. Derive the formula using Taylor expansion. 2. Derive the formula using Lagrange interpolation. 3. Find the optimal h.
h 5. In Homework 8 we looked at the finite difference approximation of f'(x). f(x + 1) = f(x) + f'(z)h + }}" (-x)+2 +0(h2). Rearranging terms, and dividing by h leads to D(h) f(x+h)-f(x) != f'(x) +}s"(x)h + O(n?). Find constants a and 8 such that A(h) = aD(h/2) + BD(h) = f'(x) + (h) (similar to convergence acceleration in Chapter 2, we are using our knowledge of the be- haviour of the error to get a better approximation!)...
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