Chapter 10.2, Problem 39E

Determine whether the following polynomials are even, odd, or neither. (A) f (x) = 5' - 2a- 4x + 1 (B) f (x) = -9a1 - 2x' + 5x + 6x (C) f(x) = -8r°-7x-3r² - 6 7x-3x2

Also, our definition of the first 4 Legendre polynomials is: 1 Po(x) V2 P1 (x) = V5/2 (1 – 31) P2(1r) = 2 V7/2 P3 (x) (3x – 5x³) 2 Write x2 as a linear combination of the first 4 Legendre polynomials, i.e. write x? = El = 0°c¢Pe and tell me what the values of the cę coefficients. (Spoiler alert: Some of the coefficients are zero. Some aren't.) If you have to do some integrals here, feel free to use Wolfram Alpha or Mathematica to do them. Now do the same for x. What is the projection of x onto the first 4 Legendre polynomials?

Now suppose b +0. The characteristic polynomial of A is (a-X)(d-A)-b² = x² − (a+d)λ + ad-b². From this it follows that the discriminant is (a + d)² - 4(ad-b²) = (a − d)² + 4b². Since b + 0, this quantity is always positive and hence the characteristic polynomial has two distinct real roots. Thus A has two distinct real eigenvalues, X₁ and X₂. Now explain why we may choose eigenvectors V₁ for ₁ and v₂ for X2 that each have unit length and are orthogonal. Finally, define P and D using the data you've conjured so far such that A = PDP ¹ where P is an orthogonal matrix and D is diagonal. Delete this red text.

Write the polynomial p(x) = x³ – 3x² + x − 9 as a Taylor polynomial centered at 20 = 1.

Find the Maclaurin polynomial of order 3 generated by f. f (x) = e3z? O 2 (x) = 1+ 3x² + §æª + O 2 (x) = 1+ 3x + a² + §æ3 O P3 (x) = 1+ æ² + 3xª + 9 O P3 (x) = 1 + x + 5a²

Using Newton's Divided Difference, the polynomial that interpolates x + sinx at the points x, = 1,x2 = 2 ,x3 = 3,x4 = 4 , and xs = 5 can be written as P(x) = c, + c,(x - 1) + c,(x – 1)(x – 2) + cz (x – 1)(x – 2)(x – 3) +C4 (х — 1)(х — 2)(х — 3)(х — 4). What is c, ? A 1.84147 B 0.004970 -0.649329 3.36812

Determine the degree of each of the following polynomial functions. a. f(x) = 1+ x + x² + x³ Degree of f: b. g(x) = x8 – 272 Degree of g: c. h(x) = x³ + 2x' + 1 Degree of h: d. j(x) x2 – 16 Degree of j: Submit

3R

One of the two sets of functions, f1, f2, f3, or g1, 92, 93, is graphed in the figure to the left below; the other set is graphed in the right figure. Points A and B each have x = 0. Taylor polynomials of degree 2 approximating these functions near x = 0 are as follows: f1(x) - 5 +x + 2x? 91(x) - 2 - x + 2x2 f2(x) 2 5 + x – x2 92(x) z 2 + x + 2x? f3(x) 2 5 + x + x? 93(x) z 2 – x + x². II. B (a) Match the functions to the appropriate figure: f matches || g matches | v (b) What are the coordinates of the points A and B? A = В - (c) Match each function with the graphs (a)--(c) in the appropriate figure. fi = ? f2 ? f3 = ? 91 = ? 92 = ? 93 = ? > > > ||