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6 (In this problem, three decimal place approximations suffice.) Use a finite difference approximation, with a...
Problem statement: Use forward and backward difference approximations of 0(h) and a centered difference approximation of...
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...
1. Use the method of undetermined coefficients to compute the coefficients of a finite difference approximation...
1. Use the method of undetermined coefficients to compute the coefficients of a finite difference approximation for u'(E) using the values u(0),u (1) and u(2). Choose the coefficients such that the formula is exact for polynomials with degree less or equal to 2. Can you use these ecoefficients to get an approximation for a first derivative based on function values v(r),v(x+h)and v(x +2h)? At which point z and for which functions v(a) is this approximation equal to '()? Determine the...
need help with 28,29,30 Write the formula for Newton's method and use the given initial approximation...
need help with 28,29,30
Write the formula for Newton's method and use the given initial approximation to compute the approximations X1 and x2. Round to six decimal places. 28) f(x) = e-x-ixo = In 4 Use a calculator to compute the first 10 iterations of Newton's method when applied to the function with the given initial approximation. Make a table for the values. Round to six decimal places. 29) f(x) = 3x - cos x; x0 = 1 Use Newton's...
4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0)...
4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0) = y'(1) = 0 Let h = 1/n, X; = jh, where j = 0,1,..., n and u; y(x;). Consider two "exterior" mesh points 2-1 = -h and 2n+1 = 1+h. Write out an 0(ha) approximate linear tridiagonal system for {u}. Hint: Let u-1 = y(x-1) = y(-h) and Un+1 = y(2n+1) = y(1 + h). Then using f(a+h) – f(a – h). f'(a)...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0,...
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...
Use a step size of 0.1 and round your answers to five decimal places if needed. Use Euler's method to approx...
Use a step size of 0.1 and round your answers to five decimal places if needed. Use Euler's method to approximate the solution x1o for the IVP y' Ty, y(0)-1. The Euler approximation for xio is Find all equilibrium solutions of y' 2y(o)13-yol. The solutions are y0 and 3 Find the equilibrium solutions and determine which are stable and which are unstable. 0 0 (unstable); y-3 (stable) y y-3 (unstable); y- 0 (stable) y3 (stable); y- 0 (unstable) y-0 (stable);...
CALCULUS Consider the function f : R2 → R, defined by ï. Exam 2018 (a) Find the first-order Taylor approximation at the point Xo-(1, -2) and use it to find an approximate value for f(1.1, -2.1 (b) Ca...
CALCULUS Consider the function f : R2 → R, defined by ï. Exam 2018 (a) Find the first-order Taylor approximation at the point Xo-(1, -2) and use it to find an approximate value for f(1.1, -2.1 (b) Calculate the Hessian ã (x-xo)' (H/(%)) (x-xo) at xo (1,-2) (c) Find the second-order Taylor approximation at Xo (1,-2) and use it to find an approximate value for f(1.1, -2.1) Use the calculator to compute the exact value of the function f(1.1,-2.1) 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...
1. Use the method of undetermined coefficients to compute the coefficients of a finite difference approximation for u'(E) using the values u(0),u (1) and u(2). Choose the coefficients such that the formula is exact for polynomials with degree less or equal to 2. Can you use these ecoefficients to get an approximation for a first derivative based on function values v(r),v(x+h)and v(x +2h)? At which point z and for which functions v(a) is this approximation equal to '()? Determine the...
need help with 28,29,30
Write the formula for Newton's method and use the given initial approximation to compute the approximations X1 and x2. Round to six decimal places. 28) f(x) = e-x-ixo = In 4 Use a calculator to compute the first 10 iterations of Newton's method when applied to the function with the given initial approximation. Make a table for the values. Round to six decimal places. 29) f(x) = 3x - cos x; x0 = 1 Use Newton's...
4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0) = y'(1) = 0 Let h = 1/n, X; = jh, where j = 0,1,..., n and u; y(x;). Consider two "exterior" mesh points 2-1 = -h and 2n+1 = 1+h. Write out an 0(ha) approximate linear tridiagonal system for {u}. Hint: Let u-1 = y(x-1) = y(-h) and Un+1 = y(2n+1) = y(1 + h). Then using f(a+h) – f(a – h). f'(a)...
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...
Use a step size of 0.1 and round your answers to five decimal places if needed. Use Euler's method to approximate the solution x1o for the IVP y' Ty, y(0)-1. The Euler approximation for xio is Find all equilibrium solutions of y' 2y(o)13-yol. The solutions are y0 and 3 Find the equilibrium solutions and determine which are stable and which are unstable. 0 0 (unstable); y-3 (stable) y y-3 (unstable); y- 0 (stable) y3 (stable); y- 0 (unstable) y-0 (stable);...
CALCULUS Consider the function f : R2 → R, defined by ï. Exam 2018 (a) Find the first-order Taylor approximation at the point Xo-(1, -2) and use it to find an approximate value for f(1.1, -2.1 (b) Calculate the Hessian ã (x-xo)' (H/(%)) (x-xo) at xo (1,-2) (c) Find the second-order Taylor approximation at Xo (1,-2) and use it to find an approximate value for f(1.1, -2.1) Use the calculator to compute the exact value of the function f(1.1,-2.1) 2....
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