(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula b...
Given the data points (-3,5),(-2,5),(-1,3), (0, 1) (a) Find the interpolating polynomial passing through these points. (b) Using your polynomial from (a), evaluate P(1). (c) This polynomial interpolates the function f(x) = 24. Find an upper bound for the approximation in part (b).
12. Given the data set: We want to find the interpolating polynomial of degree 2 through these points. a) Write the interpolating polynomial in Lagrange form b) Write the interpolating polynomial in Newton form.
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r). 1. Consider the polynonial Pl (z) of degree 4...
(3 points) Find a formula for the polynomial of least degree through the points shown in the graph. f(x) = help (formulas) - - -2
Using Lagrange interpolation, find degree two interpolating polynomial if following points are known (0, 1, 5), (2, 0, −3), (1, 2, 8), (−2, −1, 10), (−1, 0, 5
Using Lagrange interpolation, find degree two interpolating polynomial if following points are known (0, 1, 5), (2, 0, −3), (1, 2, 8), (−2, −1, 10), (−1, 0, 5) (2,3,1)
6. (25 pts) Find the osculating polynomial, P, interpolating the following table of data, and evaluate P(1): -1 2 f(x) f'(x -4 2 1 5 -4 f"(x) -12 6. (25 pts) Find the osculating polynomial, P, interpolating the following table of data, and evaluate P(1): -1 2 f(x) f'(x -4 2 1 5 -4 f"(x) -12
4. For the following table, answer the questions. (1) Find the cubic Newton’s interpolating polynomial using the first four data points and estimate the function value at x=2.5 with the interpolating polynomial. (2) Find the quartic Newton’s interpolating polynomial using the five data points and estimate the function value at x=2.5 with the interpolating polynomial. (3) Find the bases functions of Lagrange interpolation, Li(x) (i=1,2,…,5), and estimate the function value at x=2.5 with the Lagrange interpolating polynomial. 3 5 1...
1,2,3, and 4 Here are some practice exercises for you. 1. Given f(x) e2, find the a. Maclaurin polynomial of degree 5 b. Taylor polynomial of degree 4 centered at 1 c. the Maclaurin series of f and the interval of convergence d. the Taylor series generated by f at x1 2. Find the Taylor series of g(x) at x1. 3. Given x -t2, y t 1, -2 t1, a. sketch the curve. Indicate where t 0 and the orientation...
Parts b and c please 4. (25 pts) Given a function f with three continious derivatives, and three equally spaced points zi z, = zi+h, エ3年エ1 + 21, we would like to approxinate f'(z), Let p(z) be the quadratic polynomial interpolating ()) (a) Write p in the Lagrange form. (b) Show the forward difference formula (c) Prove that this expression is a second order approximation off,(r), that is, show that where C depends on the third derivative of f. 4....