Taylor’s formula with n=1 and a=0 gives the linearization of a funct

Taylor’s formula with ${\displaystyle n=1}$ and ${\displaystyle a=0}$ gives the linearization of a function at ${\displaystyle x=0}$ With ${\displaystyle n=2}$ and ${\displaystyle n=3}$ we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions:
a) For what values of x can the function be replaced by each approximation with an error less than ${\displaystyle {10}^{-2}}$
b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval? Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and interval.
Step 1: Plot the function over the specified interval.
Step 2: Find the Taylor polynomials ${\displaystyle P_1\left(x\right),P_2\left(x\right)\text{and}{P}_3\left(x\right)}$ at ${\displaystyle x=0}$
Step 3: Calculate the ${\displaystyle (n+1)}$ st derivative ${\displaystyle f^{(n+1)}\left(c\right)}$ associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of c over the specified interval and estimate its maximum absolute value, M.
Step 4: Calculate the remainder ${\displaystyle R_n\left(x\right)}$ for each polynomial. Using the estimate M from Step 3 in place of ${\displaystyle f_{(n+1)}\left(c\right)}$ plot ${\displaystyle R_n\left(x\right)}$ over the specified interval. Then estimate the values of x that answer question (a).
Step 5: Compare your estimated error with the actual error ${\displaystyle E_n\left(x\right)=|f\left(x\right)-P_n\left(x\right)|}$ by plotting ${\displaystyle E_n\left(x\right)}$ over the specified interval. This will help answer question (b).
Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5.
${\displaystyle f\left(x\right)={(1+x)}^{\frac{3}{2}},-\frac{1}{2}\le x\le 2}$