Unit 10B Review

What are the first four terms for the series represented by

 $x\sin x$  ?

•  $x^2-\frac{x^4}{3!}+\frac{x^6}{5!}-\frac{x^8}{7!}$  

•  $x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}$  

•  $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}$  

•  $x^2+\frac{x^4}{3!}+\frac{x^6}{5!}+\frac{x^8}{7!}$  

The series expansion $x^3+\frac{x^6}{2}+\frac{x^9}{6}+\frac{x^{12}}{24}+...$ is equivalent to 

•  $e^{x^3}-1$  

•  $e^{x^3}$  

•  $\cos\left(x^3\right)-1$  

•  $\sin x-x$  

Find the sum of the series $1-\frac{e^2}{2!}+\frac{e^4}{4!}-\frac{e^6}{6!}+\frac{e^8}{8!}+...$  

•  $\cos\left(e\right)$  

•  $\sin\left(e\right)$  

•  $e^{\cos e}$  

•  $e^e$  

What is the interval of convergence for the power series $\sum_{n=0}^{\infty}\frac{\left(-1\right)^n\left(n+1\right)}{n\cdot3^n}\left(x-2\right)^n$ ? 

•  $-1<x<5$  

•  $0<x<4$  

•  $-2<x<2$  

•  $-5<x<1$  

What is the summation notation for the series represented by $e^{2x}$  

•  $\sum_{n=0}^{\infty}\frac{\left(2x\right)^n}{n!}$  

•  $\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$  

•  $\sum_{n=0}^{\infty}\left(2x\right)^n$  

•  $\sum_{n=0}^{\infty}\frac{\left(2x\right)^n}{\left(2n\right)!}$  

What is the coefficient of $x^{12}$ in the Maclaurin series for $\frac{\cos\left(3x^3\right)}{4}$ ?

•  $\frac{3^4}{4!}$  

•  $\frac{3^4}{4\cdot4!}$  

•  $\frac{3^4}{4^4\cdot4!}$  

•  $\frac{3^3}{4^4}$  

The third-degree Taylor polynomial for a function f about x=2 is $\frac{9}{32}\left(x-2\right)^3+\frac{1}{2}\left(x-2\right)^2+\frac{3}{4}\left(x-2\right)+1$ .  What is the value of  $f'''\left(2\right)$  ? 

•  $24$  

•  $\frac{27}{16}$  

•  $\frac{3}{64}$  

•  $\frac{64}{5}$  

What is the series expansion for $xe^{-2x}$ ? 

•  $x-2x^2+2x^3-\frac{4}{3}x^4+...$  

•  $1-2x+2x^2-\frac{4}{3}x^3+...$  

•  $x+2x^2+2x^3+\frac{4}{3}x^4+...$  

•  $1+2x+2x^2+\frac{4}{3}x^3+...$  

Find the radius of convergence for $\sum_{n=1}^{\infty}\left(\frac{2x}{5}\right)^n$  

•  $\frac{5}{2}$  

•  $\frac{2}{5}$  

• 10

• 1

The third-degree Taylor polynomial centered at x=4 for a function f is given by $P_3\left(x\right)=5-\left(x-4\right)+\left(x-4\right)^3$ . Find  $f'''\left(4\right)$  

• 0

• 6

• 1

• 5