(a) Let h() be defined by 1. h(x) h2, х > а where h h2 and a are constants. Find the first and second derivatives of...
Estimate the second derivative of the following function using stencils for the FORWARD and CENTRAL derivatives for an order of accuracy of O(h2) for each. Use a step size of h -1. fo)x-2x2 +6 Second derivative, Forward Difference Approximation, o(h2)- Second derivative, Central Difference Approximation, O(h2) Which of the two methods is closer to the true value? (Forward/Central 12.5 points Differential Equation Estimate the second derivative of the following function using stencils for the FORWARD an derivatives for an order...
1. Consider the distribution Tylo]-)dx where f(x)H(x), and H() is the Heavi- side function. Find the distributional derivative of f (r 1. Consider the distribution Tylo]-)dx where f(x)H(x), and H() is the Heavi- side function. Find the distributional derivative of f (r
Let f be a function having derivatives of all orders for all real numbers. Selected values of f and its first four derivatives are shown in the table above. (a) Write the second-degree Taylor polynomial for f about x = 0 and use it to approximate f(0.2). (b) Let g be a function such that g(x) =f(x3). Write the fifth-degree Taylor polynomial for g', the derivative of g, about x = 0. (c) Write the third-degree Taylor polynomial for f about x =...
Let X and Y have the joint pmf defined by (х, у) (1,2) (0,0) (0,1) (0,2) (1,1) (2,2) 2/12 1/12 3/12 1/12 1/12 4/12 Pxy (x, y) Find py (x) and p, (y) а. b. Are X and Y independent? Support your answer. Find x,y,, and o, С. d. Find Px.Y
Consider the standard second order finite difference approximation to the second derivative h2 Let;") =e.mjh for , = 1, M-1.h=r/M. (b) Prove that the farthest eigenvalue from the origin approaches -/h2 as h0 and use this to explain why the ODE systenm du(t) is stiff Consider the standard second order finite difference approximation to the second derivative h2 Let;") =e.mjh for , = 1, M-1.h=r/M. (b) Prove that the farthest eigenvalue from the origin approaches -/h2 as h0 and use...
15 4 23 Let fbe a function having derivatives of all orders for all real numbers. Selected values of fand its first four derivatives are shown in the table above. 6. a) Write the second-degree Taylor polynomial for faboutx0 and use it to approximate f(0.2). (b) Let g be a function such that g(x) -fx Write the fifth-degree Taylor polynomial for g', the derivative of g, about x = 0 We were unable to transcribe this image 15 4 23...
4. For f(x) = e-* and h = 0.10 where, C = 1.** a) Use centered approximations to estimate the first and second derivatives of f(x) at x = 2. Use the east accurate formulas available. (10 pts) b) Using the most acurate forward and backward difference formulas, estimate the first derivative of f(x) at x 2. (10 pts) Forward Difference First Derivative 7.) - SD Error OM or) = -1.) + 40..) - 3 ) 2h Second Derivative 'w...
Given a second order linear homogeneous differential equation а2(х)у" + а (х)У + аo(х)у — 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, V2. But there are times when only one function, call it y, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the a2(x) F 0 we rewrite...
a. 4. Let h(x) = x4 – 6x3 + 12x2. Find h'(x) and h"(x). b. Find the open intervals on which h is concave upward and concave downward. Give the points of inflection for h as ordered pairs. c. a. 5. Let g(x) = x4 – 2x3 + 3. x3 This function is defined, differentiable, for all real numbers except x = where g has a vertical asymptote. b. Find g'(x), given any other value of x. c. Suppose we...
5. Find the derivative matrices of the following composition of functions. (а) fog where f (x, у) — 2х — 3у, g(u, v) - (usin u, U sin u) (Ъ) f.g where f(х, у, 2) %3D (x? + у? +2?,х— у+2:), g() %3 (2, 13, 2/4) (с) fog wherе f (x, у, z) 3D (хуz, ху + xz — yz) where g(u, v, w) %3D (uu, uw, vw) 5. Find the derivative matrices of the following composition of functions. (а)...