h 5. In Homework 8 we looked at the finite difference approximation of f'(x). f(x +...
Question 1 15 Points) It is always desirable to have/ use the finite difference approximation with error term. Please using the Taylor Series: higher order of truncation sw(x) h" +R 2! 3! (I) Derive the following forward difference approximation of the 2nd orde 2) What is the order of error for this case? derivative of f(x). f" derivative off(x) h2 Question 1 15 Points) It is always desirable to have/ use the finite difference approximation with error term. Please using...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
ME/AE 342 - Numerical Methods in Engineering with Applications Homework 2 1. Use finite difference approximation to compute f'(2.36), /'(2.37) and /"(2.37) from the data 2.36 2.37 2.38 2 .39 0.85866 0.86289 0.86710 0.87129 1() 2. Estimate f'(1) from the following data using either forward or backward dif- ference 1 f(0) 0 .97 0.85040 1.00 1.05 0.84147 0.82612 3. Use polynomial interpolation to compute f' and l" at r = 0, using the data be- low. Hint: Find the Lagrange...
7. (5 points) Find the linear approximation for f(x) = tan(2x) at a = 0 and use it to approximate the value of tan(0.002). Hint: The linear approximation is just the tangent line to the curve at a = 2. 8. (5 points) Use the Mean Value Theorem for derivatives to find the value of x = c for f(x) = Vx on the interval (1,9). 9. (5 points) The acceleration of an object moving along the number line at...
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...
2. Show that the approximation 2 1 h) - 5(x- h)- S(x + 2h) - f(x- 2h) (x) can be obtained by a Richardson extrapolation. [Hint: Consider (h) f(z+2h)-f(x-2h) 2h 2. Show that the approximation 2 1 h) - 5(x- h)- S(x + 2h) - f(x- 2h) (x) can be obtained by a Richardson extrapolation. [Hint: Consider (h) f(z+2h)-f(x-2h) 2h
12 Our treatment of the three-spring problem was incomplete because we looked only at the cosine parts of the solutions, ignoring the sines. (a) Show that the following equations are valid solutions to Equations 6.1.1 and 6.1.2 for any constants A and A2 x (t) = Al cos (V/2 t) + A2 sin (v 2 ) (b) Show that the initial conditions h(0) = (0) = 0 lead to A2-0 and therefore to the solution we used in the Explanation...
Problem 5. Consider least squares polynomial approximation to f(x) = cos (nx) on x E [-1,1] using the inner product 1. In finding coefficients you will need to compute the integral By symmetry, an 0 for odd n, so we need only consider even n. (a) Make a change of variables and use appropriate identities to transform the integral for a to cos (Bcos 8)cos (ne) de (b) The Bessel function of even order, (x), can be defined by the...
3. At the beginning of 8.6, we investigated the graph of f(x) = ? and the graphs of several partial sums of its series 3x". You are now going to investigate the graphs of (-1)**(x - 2)", which is the series representation of the function f(x) = -centered at a = 2 a. Find the radius and interval of convergence (x - 2)". Show all your work. (3 points) b. Find the first five terms of Sn for Ž (-1)**(x-2)",...
Problem statement: Use forward and backward difference approximations of 0(h) and a centered difference approximation of 0(h') to estimate the first derivative of f(x)- 0.x-0.15x-0.5x-0.25x +12 Problem #2 Steady-state temperatures (K) at three nodal points of a long rectangular rod as shown. The rod experiences a uniform volumetric generation rate of 5 X 10 Wm and has a thermal conductivity of 20 W/m-K. Two of its sides are maintained at a constant temperature of 300K, while others are insulated. Problem...