= (a) The fourth Taylor polynomial for f(x) = ln (sec(x)) at a co + c₁x + ₂x² + 3x³ + ₁x² where Co = :0 is = (b) The approximation of In(sec(-0.3)) by this fourth degree Taylor polynomial is
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- p"(x;) = 0. p'(xi) Let p(x) be a polynomial function of degree n with simple roots x1,..., xn. Show thatSuppose g is a function which has continuous derivatives, and that g(8) = -1, g'(8) = 2, g"(8) = 5, g"(8) = -5. (a) What is the Taylor polynomial of degree 2 for g near 8? P2(x) = (b) What is the Taylor polynomial of degree 3 for g near 8? P3(x) = (c) Use the two polynomials that you found in parts (a) and (b) to approximate g(7.9). With P2, 9(7.9) = With P3, g(7.9) -Suppose g is a function which has continuous derivatives, and that g(6) = -2, g' (6) = 5, g" (6) = 1, g" (6) = 5. (a) What is the Taylor polynomial of degree 2 for g near 6? P2(x) = (b) What is the Taylor polynomial of degree 3 for g near 6? P3(x) = (c) Use the two polynomials that you found in parts (a) and (b) to approximate g(6.1). With P2, g(6.1) With P3, g(6.1) 2
- Suppose g is a function which has continuous derivatives, and that g(8) = -2, g(8) = 3. g"(8) = -4, g"(8) = 4. %3D (a) What is the Taylor polynomial of degree 2 for g near 8? P:(z) = (b) What is the Taylor polynomial of degree 3 for g near 8? Pa(z) = (c) Use the two polynomials that you found in parts (a) and (b) to approximate g(8.1). With Pa, g(8.1) = With Ps, g(8.1) =Suppose g is a function which has continuous derivatives, and that g(8) = 3, g'(8) = 3, g"(8) = −2, g""(8) = 1. (a) What is the Taylor polynomial of degree 2 for near 8? P₂(x) = 3 + 3(x − 8) − (x − 8)². (b) What is the Taylor polynomial of degree 3 for Ps(x) = 3 + 3(x− 8) – (x−8)² = 2 + (x-8)³ 6 g near 8? (c) Use the two polynomials that you found in parts (a) and (b) to approximate g(8.1). With P2, g(8.1) ≈ With P3, g(8.1) ≈Let f(x) = e^xFind the polynomial H3(x) of degree at most 3 such thatH3(1) = f(1), H3(2) = f(2), H3'(1) = f'(1), H3'(2) = f'(2) by using a (generalized)divided-difference table. (Do not use decimal approximations for the number e.)
- Find the cubic polynomial f(x) such that f(2) = -29, f' = -27, f''(2) = -18, and f'''(2) = -6 f(x) = ?Find a second-degree polynomial f(x) = ax² + bx + c such that its graph has a tangent line with slope 10 at the point (2,7) and an x -intercept at (1,0).10. (a) Produce the linear Taylor polynomials to f(x) = in (x) on 1 < x < 2, expanding about x, = }. Graph the error. Produce the linear minimax approximation to f(x) = ln (x) on [1, 2]. (b) Graph the error, and compare it with the Taylor approximation.
- consider the polynomial f (x) = 7x4- (2 + 2x + 6x2 + 5x3) -8x5. In x = 0.7, use the approximation in divided differences centered for the second derivative of F (x). Use a rack of Δx = 0.2 step. F '' (x) ≈Consider the cubic polynomial T(X) = X - X. Find the equation of the tangent line of f(x) at the point (1, 0). Plot both f(x) and the tangent line on the same set of axes (label both). On a new set of axes, plot both T(X) and 7 (X) (label both graphs). What do you notice about 1(X) and f '(x)== 8. Suppose g is a function which has continuous derivatives, and that g(8) = 4, g'(8) = 1, g" (8) = 4, and g(3)(8) 5. = (a) What is the Taylor polynomial of degree 2 for g near 8? P2(x) = (b) What is the Taylor polynomial of degree 3 for g near 8? P3(x) = (c) Use the two polynomials that you found in parts (a) and (b) to approximate g(8.1). With P2, g(8.1)≈ With P3, 9(8.1)≈