Consider the three points (-1,0), (0,1), (2,0) 1. Construct a second degree polynomial P(a) that interpolates the g...
Please answer problem 4, thank you. 2. The polynomial p of degree n that interpolates a given function f at n+1 prescribed nodes is uniquely defined. Hence, there is a mapping f -> p. Denote this mapping by L and show that rl Show that L is linear; that is, 3. Prove that the algorithm for computing the coefficients ci in the Newton form of the interpolating polynomial involves n long operations (multiplications and divisions 4. Refer to Problem 2,...
Problem 3. (8 points) Given that the interpolation polynomial of the points (-3,2), (-2,1),(-1,-1), (0,1), (1,0), (2,0), (3, 1) is 191 13 5 781 , 53 Q(x) = -3602 + 30++ Find a polynomial curve passing through these seven points and additionally the point (4,0). Write your polynomial in standard form anx" +...+212 +00 +1. 360" + en
7. Find the polynomial of degree 4 through the points (1, 2), (-1,0), (2, 15), (0,1) and (-2, 11), and check that it works.
//NOTE: This question has a graph included with points at (-2,0) (1,0) (2,0). all the zeros. The polynomial P(x) polotted below is a cubic. From the polt below, it is easy to determine the three factors of P(x). With a little more work, you can also determine the leading coefficient of P(x). (a) Find a factorization of P(x) which includes the unkown leading coefficient a and the three factors you can read from the plot, like P(x) =a(factor 1)(factor 2)(factor...
Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following interpolating polynomials, and use MATLAB to graph both the interpolating polynomials and the data points: a) The piecewise linear Lagrange interpolating polynomialx) b) The piecewise quadratic Lagrange interpolating polynomial q(x) c) Newton's divided difference interpolation pa(x) of degree s 4 Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
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).
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Consider the points: P (-1,0, -1), Q (0,1,1), and R(-1,-1,0). 1.) Compute PQ and PR. 2.) Using the vectors computed above, find the equation of the plane containing the points P, Q, and R. Write it in standard form. 3.) Find the angle between the plane you just computed, and the plane given by: 2+y+z=122 Leave your answer in the form of an inverse trigonometric function.
Find a polynomial function P of the lowest possible degree, having real coefficients, a leading coefficient of 1, and with the given zeros. 2 + 3i. - 1 and 2 The polynomial of lowest degree is P(x) =