a. find the linear approximating polynomial for the following function centered at the given point a.\nb…

a. find the linear approximating polynomial for the following function centered at the given point a.\nb. find the quadratic approximating polynomial for the following function centered at the given point a.\nc. use the polynomials obtained in parts a. and b. to approximate the given quantity.\nf(x)=e^{-x}, a = 0; approximate e^{-0.6}\na. p_1(x)=□

a. find the linear approximating polynomial for the following function centered at the given point a.\nb. find the quadratic approximating polynomial for the following function centered at the given point a.\nc. use the polynomials obtained in parts a. and b. to approximate the given quantity.\nf(x)=e^{-x}, a = 0; approximate e^{-0.6}\na. p_1(x)=□

Answer

Explanation:

Step1: Recall linear - approximation formula

The linear approximating polynomial (p_1(x)) of a function (y = f(x)) centered at (a) is given by (p_1(x)=f(a)+f^{\prime}(a)(x - a)). First, find (f(x)) and its derivative. Given (f(x)=e^{-x}), then (f^{\prime}(x)=-e^{-x}).

Step2: Evaluate (f(a)) and (f^{\prime}(a))

When (a = 0), (f(0)=e^{-0}=1) and (f^{\prime}(0)=-e^{-0}=- 1).

Step3: Find the linear - approximating polynomial

Substitute (f(0) = 1), (f^{\prime}(0)=-1) and (a = 0) into the formula (p_1(x)=f(a)+f^{\prime}(a)(x - a)). We get (p_1(x)=1+( - 1)(x - 0)=1 - x).

Answer:

(1 - x)