Ch 10 Approximating Functions Using Series

10.3 Finding and Using the Taylor Series are questions 1-2

10.4 The Error in Taylor Polynomial Approximations questions 3-6

Please show work and answer for each question.

Find the first four nonzero terms of the Taylor series about 0 for the function below. (7 + x) List the polynomial terms in increasing order. + i X + +Find the first three terms of the Taylor series for f (x) = ed around 0. Use this information to approximate the integral dx. o (a) Find the first three terms of the Taylor series for f(x) = es. NOTE: Enter the exact answer. f (x) = (b) Use the previous result to approximate dx. 0 NOTE: Round your answer to three decimal places. f(x) dx ~Current Attempt in Progress Use the Lagrange Error Bound for Pr (x) to find a bound for the error in approximationg the quantity with a third-degree Taylor polynomial for the given function f(x) about x = 0. 2 0.35, f (x) = ex Round your answer to five decimal places. Error Bound = iUse the Lagrange Error Bound for Pr (x) to find a bound for the error in approximationg the quantity with a third-degree Taylor polynomial for the given function f(x) about x = 0. cos ( - 0.4), f (x) = cosx Round your answer to six decimal places. Error bound = iUse the Lagrange Error Bound for Pr (x) to find a bound for the error in approximationg the quantity with a third-degree Taylor polynomial for the given function f(x) about x = 0. In (1.5), f(x) = In(1 + x) Round your answer to five decimal places. Error bound = iThe Taylor polynomial Pn(x) about 0 approximates f (x) with error En(a) and the Taylor series converges to f (x). Find the smallest constant K given by the alternating series error bound such that [ EA(1) |