BC 10.5 (Alternating Power Series Error)

A function f has derivatives of all orders at x=0. It is known that

 $f\left(0\right)=3,\ f'\left(0\right)=-1,\ f''\left(0\right)=\frac{2}{5},\ and\ f'''\left(0\right)=-\frac{4}{3}$ .  If the second degree Macluarin series is used to approximate f(1).  Assuming that the Maclaurin series is alternating, what is the maximum error in this approximation? 

•  $\frac{2}{9}$  

•  $\frac{4}{9}$  

•  $\frac{1}{6}$  

•  $\frac{4}{11}$  

The function g has derivatives of all orders, and the Maclaurin series for f is $\sum_{n=0}^{\infty}\left(-1\right)^{n+1}\left(\frac{x^{3n+2}}{3n+1}\right)$  .  If the first two nonzero terms are used to approximate  $f'\left(1\right),$  what is the maximum error in that approximation?

•  $\frac{1}{7}$  

•  $\frac{8}{7}$  

•  $\frac{7}{9}$  

•  $\frac{8}{9}$  

Let $f\left(x\right)=xe^{-x}$ .  If the first two nonzero terms for the Taylor series centered about x=0 are used to approximate  $f\left(2\right)$  , what is the maximum error in this approximation?

Let $f\left(x\right)=x\cos2x.$   If  $g\left(x\right)=\int_0^xf\left(t\right)dt$  and the first two nonzero terms of the Maclaurin series for g are used to approximate  $\int_0^1f\left(t\right)dt,$  what is the maximum error in this approximation? (write the answer as an improper fraction and simplify fully)

Let $\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n}}{\left(2n\right)!}$   be the Maclaurin series for f(x).  Which of the following gives the smallest value of n for which  $\left|P_n\left(\pi\right)-f\left(\pi\right)\right|<0.01$  ?  

• 7

• 6

• 5

• 4

• 3

The maximum error in the 3rd degree Taylor Polynomial  approximation centered at x=0 for  $g\left(1\right)$   is $\frac{1}{5}$ .  Assuming that the Taylor Polynomial of  $g$   is an alternating series, what is the value of  $g^4\left(0\right)?$   (write your answer as an improper fraction)