Taylor Series

Which of the following is the fourth-degree Taylor Polynomial for  $f\left(x\right)=x\sin x$  about  $x=0$  .

•  $P_4\left(x\right)=x-\frac{1}{6}x^3+\frac{1}{5!}x^5-\frac{1}{7!}x^7$  

•  $P_4\left(x\right)=-x^2+\frac{1}{6}x^4$  

•  $P_4\left(x\right)=x^2-\frac{1}{6}x^4$  

•  $P_4\left(x\right)=2x^2-4x^4$  

The graph of the function of  $f\left(x\right)$  is shown.  Which of the following could be a portion of the graph of the Taylor Polynomial  $P\$ of degree 13 for  $f\left(x\right)$  about  $x=0$   .

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Let  $T_4\left(x\right)$  be the fourth-degree Taylor Polynomial for  $f\left(x\right)=\left(x+1\right)^5$  about  $x=0$  .  Which of the following statements is true?

•  $T_4\left(x\right)=1+5x+10x^2+10x^3+5x^4$  provides a good approximation for f(x) only for values of x that are close to x = 0.

•  $T_4\left(x\right)=1+5x+10x^2+10x^3+5x^4$  provides a good approximation for f(x) for all real values of x.

•  $T_4\left(x\right)=1+5x+20x^2+60x^3+120x^4$  provides a good approximation for f(x) only for values of x that are close to x = 0.

•  $T_4\left(x\right)=1+5x+20x^2+60x^3+120x^4$   provides a good approximation for f(x) for all real values of x.

Let  $f$  be the function defined by  $f\left(x\right)=\ln\left(x+4\right)$  .  The third-degree Taylor Polynomial for  $f$  centered about  $x=-3$  is

•  $x-3-\frac{\left(x-3\right)^2}{2}-\frac{\left(x-3\right)^3}{3}$  

•  $x-3-\frac{\left(x-3\right)^2}{2}+\frac{\left(x-3\right)^3}{3}$  

•  $x+3+\frac{\left(x+3\right)^2}{2}+\frac{\left(x+3\right)^3}{3}$  

•  $x+3-\frac{\left(x+3\right)^2}{2}+\frac{\left(x+3\right)^3}{3}$  

The coefficient of the   $x^2-$  term in the Taylor Polynomial for  $y=x^{\frac{2}{3}}$  around  $x=8$  is

•  $-\frac{1}{144}$  

•  $-\frac{1}{72}$  

•  $-\frac{1}{9}$  

•  $\frac{1}{144}$  

What is the coefficient of the   $\left(x-3\right)^9$ in the Taylor Series expansion of  $e^x$  at  $x=3$  ?

•  $9e^3$  

•  $\frac{e^3}{9!}$  

•  $\frac{e^9}{3!}$  

•  $\frac{e^3}{9}$  

Let  $f$  be the function that has derivatives of all orders for all real numbers.  Assume  $f\left(0\right)=6,\ f'\left(0\right)=-5,\ f''\left(0\right)=14,\ and\ f'''\left(0\right)=36$  Write a third-order Taylor Polynomial for  $f$  at  $x=0$   and use it to approximate  $f\left(0.5\right)$  .

•  $6-\frac{x}{5}+\frac{x^2}{7}+\frac{x^3}{6};\ \ f\left(0.5\right)\approx5.957$  

•  $6-5x+7x^2+6x^3;\ \ f\left(0.5\right)\approx6.000$  

•  $6x-5x^2+7x^3+6x^4;\ \ f\left(0.5\right)\approx3.000$  

•  $6+7x-5x^2+6x^3;\ \ \ f\left(0.5\right)\approx9.000$  

Tags

CCSS.HSF.LE.A.2