(a) Find the third Maclaurin polynomial for
(b) Find the third Taylor polynomial about
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- For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. f(x)=2x3x, between x=1 and x=1.arrow_forwardLet f(x) = , find L, (x) of the second Lagrange interpolating polynomial, if x, =2, x, =2.5, X2 =4 Select one: a. (x - 6.5)x + 10, b. (0.05x - 0.425)x + 1.15 (-4x + 24)x – 32 3 (x – 4.5)x + 5 O d. 3arrow_forwardOne 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 = ? > > > ||arrow_forward
- Determine the third Taylor polynomial of the function at x = a. f(x) = x 2 + x + 1, a = 4 O 21 + 9(x - 4) + 9(x - 4)² + 21(x - 4)³ O 5+ 9(x - 4) + 13(x - 4)2 O 21 + 9(x - 4) + (x - 4)2 O 1+ 3(x - 4)+ 3(x - 4)2 + (x - 4)3 A Moving to the next question prevents changes to this answer. e Type here to search PrtScn DII F3 F7 F5 F6 F4arrow_forwardUsing 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 3.36812arrow_forward4. Consider the polynomial f(x) = x² + 1. (a) Show that f (x) is irreducible in (Z/3Z) [x]. (b) Show that f(x) is reducible in (Z/2Z) [x], and factor it as a product of irreducibles.arrow_forward
- Find the Maclaurin polynomial of degree two for the function f(x) = sec (9x). Select one: O a. O b. O c. ○ d. O O e. 81 P₂(x) = 1 + $12² x ²2 P₂(x)=x- P₂(x) = 1+ 81 81 81 P₂(x) = 1-5/1²x²2 81 P₂(x) = x + ² x ² 2arrow_forwardLet T(x) denote the Chebyshev polynomial of degree k. Express the composite polynomial Tn(Tm(x)) of degree n · m in terms of Chebyshev polynomials.arrow_forwardVerify that the third Maclaurin polynomial for f (x) = ex sin x is equal to the product of the third Maclaurin polynomials of f (x) = ex and f (x) = sin x (after discarding terms of degree greater than 3 in the product).arrow_forward
- 1. Consider the polynomial f(x) = (x – 1)(x+2)(x- 3)(x+5). A. What are the roots of f? B. Suppose you were to multiply the polynomial out; what would the leading coefficient of f be? C. Suppose you were to multiply the polynomial out; what would the constant term of f be?arrow_forwardDefine f(n)= II (1 = II ( ¹ - ² ) = (¹ - ²¹ ) · ( ¹ − 3 ) · (¹ – ↓ ) --- (¹ - -/-), - 32 1=2 where n € Z+ and n > 2. Here, Z+ is the set of all positive integers. Find a rational polynomial that is equal to f(n). n ƒ(n) = II (¹-²1) i=2 = Justify this equality by induction. (1)arrow_forwardFind the numerical solution of the following fourth-degree polynomial has a real zero (14) f(x) = (219 + d)x4 + (9 + d)x3 + (4 + d)x2 − 211x − d, in [0, 1]. Attempt to approximate the zero to within 10−4 using the Secant methodarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage