1.6. By manipulating Taylor series, determine the constant C for an error expansion of (1.3) of the form wj−u' (xj) ~ Ch¼u (5) (x;), where u (5) denotes the fifth derivative. Based on this value of C and on the formula for u(5) (x) with u(x) = esin(x), determine the leading term in the expansion for w; - u'(x;) for u(x) = esin(x). (You will have to find maxε[-T,T] |u(5) (x)| numerically.) Modify Program 1 so that it plots the dashed line corresponding to this leading term rather than just N-4. This adjusted dashed line should fit the data almost perfectly. Plot the difference between the two on a log-log scale and verify that it shrinks at the rate O(h6).

Calculate the first few terms of the Fourier series of -sin(2-x), 0≤x < sin(2-x), -

Let f(x) = cos (4x³) - 1 x5 Evaluate the 7th derivative of f at x = 0. f(7) (0) = 0 Hint: Build a Maclaurin series for f(x) from the series for cos(x).

, Let f(x) = cos (2x²) - 1 x² -. Evaluate the 10th derivative of fat x = 0. f(¹0) (0) = Hint: Build a Maclaurin series for f(x) from the series for cos(x).

(2) Let f (x) bea function which is represented by the power series f(r) = (-1)" - 1)". n3 n=1 What is the fifth derivative of f at x 1?

Let g(x) = 4x sin(3x). Find the Taylor series for g(x).

3. (a) Use derivatives and the definition of the Taylor series to write out the degree three Taylor polynomial T3(x) expanded about X-value a=1 for the functionff(x) = ln (2 - x). (Caution: use a = 1; do not use a = 0.) (b)Use the degree three Taylor polynomial found in part (a) to estimate In(1.5).

b. Write the third-order Taylor Series for f(x) = 2 sin(x) + x³ about x, = 0 showing the terms up to and including the third derivative of f (x). Include your simplified third-order Taylor series approximation: f(x)Taylor =? · Hint 1: f(x)raylor = f (xo) + f'(xo(x – xo) + f(m) (xo) (x – xo)"“ %3D f"(xo) (x – xo)² + f(3) (xo) (x – xo) 3 + 2! 3! + п! Hint 2: 3!=6 Hint 3: f'(x) = 2 cos(x) + 3x² f"(x) = –2sin(x) + 6x f()(x) = -2 cos(x) + 6 c. Use the result above to evaluate f(2)Taylor (show only four decimals) d. Calculate the true error for the Taylor Series approximation to the value at f(2) Hint 1: Using a calculator set to Radian, the true value of f (2) can be calculated as: 9.8186 Hint 2: The true error is defined as |f(x) – f(x)Taylor|

Find the power series representation for the derivative of the function

Suppose the function f(x) has a Taylor series at x = 4 that converges to f(x) for all r € R, and the n-th derivative of f(x) at = 4 is f(")(4) = (-1)"n! 3" (n + 1) Define T,(x) as the n-th degree Taylor polynomial of f at x = 4. Show T;(5) approximates f(5) with error less than or equal to 5103 5103