Let f(x) = e* cosx. a) Find the 2nd order Taylor polynomial P₂(x) for f about xo=0.

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Let f(x)= xe* . a) Find the third Taylor polynomial P,(x) for the function f(x) about x; 0. Use this polynomial to approximate f(0.8). b) Use Taylor's inequality to find the error bound for the truncation error and compare it to the true error. c) Given value ĩ = 0.8, with corresponding error Ař = 0.01, compute the resulting error Af (X) in the function f(x).

3. Let T,(x)= f(a)+ f '(a)(x-a)+ f"(a) (x-a)² + 2 f'(a) (x-a) be the Taylor polynomial of degree three for the 6 function f(x), centered at the point x=a. a) Find T, '(x), T, "(x), and T,"'(x). b) Evaluate T '(x), T, "(x), and T,"'(x) at x=a to verify that: T;(a) = f(a), T, "(a) = f'(a), T",(a) =f"(a), and T,"'(a) = f"(a). %3D