Free Printable Unit 10B Review

1.

What are the first four terms for the series represented by

x\sin x

?

a)

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

b)

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

c)

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

d)

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

2.

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

a)

$e^{x^3}-1$

b)

$e^{x^3}$

c)

$\cos\left(x^3\right)-1$

d)

$\sin x-x$

3.

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

a)

$\cos\left(e\right)$

b)

$\sin\left(e\right)$

c)

$e^{\cos e}$

d)

$e^e$

4.

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

a)

$-1<x<5$

b)

$0<x<4$

c)

$-2<x<2$

d)

$-5<x<1$

5.

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

a)

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

b)

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

c)

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

d)

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

6.

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

a)

$\frac{3^4}{4!}$

b)

$\frac{3^4}{4\cdot4!}$

c)

$\frac{3^4}{4^4\cdot4!}$

d)

$\frac{3^3}{4^4}$

7.

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 $f'''\left(2\right)$ ?

a)

$24$

b)

$\frac{27}{16}$

c)

$\frac{3}{64}$

d)

$\frac{64}{5}$

8.

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

a)

$x-2x^2+2x^3-\frac{4}{3}x^4+...$

b)

$1-2x+2x^2-\frac{4}{3}x^3+...$

c)

$x+2x^2+2x^3+\frac{4}{3}x^4+...$

d)

$1+2x+2x^2+\frac{4}{3}x^3+...$

9.

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

a)

$\frac{5}{2}$

b)

$\frac{2}{5}$

c)

10

d)

1

10.

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 $f'''\left(4\right)$

a)

0

b)

6

c)

1

d)

5

11.

Find the interval of convergence for $\sum_{n=0}^{\infty}\frac{\left(2x\right)^n}{3n+1}$

a)

$-\frac{1}{2}\le x<\frac{1}{2}$

b)

$-\frac{1}{2}<x<\frac{1}{2}$

c)

$-\frac{2}{3}<x\le\frac{2}{3}$

d)

$-\frac{2}{3}<x<\frac{2}{3}$

12.

$\sum_{n=0}^{\infty}\frac{\left(x-2\right)^n}{n\cdot2^n}$

What is the interval of convergence for the series above?

a)

$0\le x<4$

b)

$0<x<4$

c)

$0<x\le4$

d)

$0\le x\le4$

13.

Find the sixth-degree term in the Maclaurin series for $f\left(x\right)=\frac{\sin x}{x},\ x\ne0$

a)

$\frac{x^6}{7!}$

b)

$-\frac{x^6}{7!}$

c)

$\frac{x^6}{6!}$

d)

$-\frac{x^6}{6!}$

14.

Find the second-degree Taylor polynomial centered at $x=e$ for $f\left(x\right)=e^x$

a)

$e^e+e^e\left(x-e\right)+\frac{e^e\left(x-e\right)^2}{2!}$

b)

$1+e^e\left(x-e\right)+\frac{e^e\left(x-e\right)^2}{2!}$

c)

$1+\left(x-e\right)+\frac{\left(x-e\right)^2}{2!}$

d)

$e^e+\left(x-e\right)+\frac{\left(x-e\right)^2}{2!}$

15.

The series expansion $x^2+\frac{x^4}{2}+\frac{x^6}{6}+\frac{x^8}{24}+...$ is equivalent to...

a)

$e^{x^2}-1$

b)

$\cos\left(x^2\right)$

c)

$x^2e^x$

d)

$x^2\cos x$

16.

Let f be a function having derivatives of all orders for x>0 such that $f\left(3\right)=2,\ f'\left(3\right)=-1,\ f''\left(3\right)=6$ , and $f'''\left(3\right)=12$ . Which of the following is a third-degree Taylor polynomial for f about x=3?

a)

$2-x+6x^2+12x^3$

b)

$2-x+3x^2+2x^3$

c)

$2-\left(x-3\right)+3\left(x-3\right)^2+4\left(x-3\right)^3$

d)

$2-\left(x-3\right)+3\left(x-3\right)^2+2\left(x-3\right)^3$

17.

What is the radius of convergence for the series $\sum_{n=0}^{\infty}\frac{\left(x-4\right)^{2n}}{3^n}$

a)

$2\sqrt{3}$

b)

3

c)

$\sqrt{3}$

d)

$\frac{\sqrt{3}}{2}$

e)

0

18.

The power series $\sum_{n=0}^{\infty}a_n\left(x-3\right)^n$ converges at x=5. Which of the following must be true?

a)

The series diverges at x=0

b)

The series diverges at x=1

c)

The series converges at x=1

d)

The series converges at x=2

e)

The series converges at x=6

19.

What is the coefficient of $x^6$ in the Taylor series for $e^{3x^2}$ about x=0?

a)

$\frac{1}{1440}$

b)

$\frac{81}{160}$

c)

$\frac{9}{4}$

d)

$\frac{9}{2}$

e)

$\frac{27}{2}$

20.

The third-degree Taylor polynomial for a function f about x=4 is $\frac{\left(x-4\right)^3}{512}-\frac{\left(x-4\right)^2}{64}+\frac{\left(x-4\right)}{4}+2$ . What is the value of $f'''\left(4\right)$ ?

a)

$-\frac{1}{64}$

b)

$-\frac{1}{32}$

c)

$\frac{1}{512}$

d)

$\frac{3}{256}$

e)

$\frac{81}{256}$

21.

What are all values of x for which the series $\sum_{n=1}^{\infty}\frac{\left(-1\right)^n}{n}\left(x+\frac{3}{2}\right)^n$ converges?

a)

$-\frac{5}{2}<x\le-\frac{1}{2}$

b)

$-\frac{5}{2}\le x<-\frac{1}{2}$

c)

$-\frac{1}{2}<x<\frac{1}{2}$

d)

$x\le-\frac{1}{2}$

22.

Let $P=3-3x^2+6x^4$ be the fourth-degree Taylor polynomial for the function f about x=0. What is the value of $f^{\left(4\right)}\left(0\right)$ ?

a)

0

b)

$\frac{1}{4}$

c)

6

d)

24

e)

144

23.

Which of the following is the Maclaurin series for $e^{3x}$ ?

a)

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

b)

$\sum_{n=0}^{\infty}\frac{3^{1+n}x^n}{n!}$

c)

$\sum_{n=0}^{\infty}\left(-1\right)^n\frac{\left(3x\right)^n}{n!}$

d)

$\sum_{n=0}^{\infty}\frac{3x^n}{n!}$

e)

$\sum_{n=0}^{\infty}\frac{\left(3x\right)^n}{n!}$

24.

What is the coefficient of $x^2$ in the Taylor series for $\sin^2x$ about x=0?

a)

-2

b)

-1

c)

0

d)

1

e)

2

25.

The sum of the series $1+\frac{2^1}{1!}+\frac{2^2}{2!}+\frac{2^3}{3!}+...$ is

a)

$\ln2$

b)

$e^2$

c)

$\cos2$

d)

$\sin2$

e)

nonexistent

26.

What is the radius of convergence for the power series $\sum_{n=0}^{\infty}\frac{\left(x-4\right)^n}{2\cdot3^{n+1}}$ ?

a)

$\frac{1}{3}$

b)

$\frac{3}{2}$

c)

3

d)

4

e)

6

27.

Let f be a function that has derivatives of all orders for all real numbers, and let P(x) be the third-degree Taylor polynomial for f about x=0. The Taylor series for f about x=0 converges at x=1 and $\left|f^{\left(n\right)}\left(x\right)\right|\le\frac{n}{n+1}$ for $1\le n\le4$ and all values of x. What is the Lagrange Error Bound for P(1)?

a)

$\frac{4}{5}$

b)

$\frac{4}{5}\cdot\frac{1}{4!}$

c)

$\frac{4}{5}\cdot\frac{1}{3!}$

d)

$\frac{3}{4}\cdot\frac{1}{4!}$

e)

$\frac{3}{4}\cdot\frac{1}{3!}$

28.

The function f has derivatives of all orders for all real numbers with $f\left(0\right)=3,\ f'\left(0\right)=-4,\ f''\left(0\right)=2$ , and $f'''\left(0\right)=1$ . Let g be the function given by $g\left(x\right)=\int_0^xf\left(t\right)dt$ . What is the third degree Taylor polynomial for g about x=0?

a)

$-4x+2x^2+\frac{1}{3}x^3$

b)

$-4x+x^2+\frac{1}{6}x^3$

c)

$3x-2x^2+\frac{1}{3}x^3$

d)

$3x-2x^2+\frac{2}{3}x^3$

e)

$3-4x+x^2+\frac{1}{6}x^3$

29.

Let f be a function with second derivative $f''\left(x\right)=\sqrt{1+3x}$ . The coefficient of $x^3$ in the Taylor series for f about x=0 is

a)

$\frac{1}{12}$

b)

$\frac{1}{6}$

c)

$\frac{1}{4}$

d)

$\frac{1}{2}$

e)

$\frac{3}{2}$

30.

Find the 3rd term of the Maclaurin series for $f\left(x\right)=e^{-\frac{x}{2}}$ .

a)

$\frac{x^2}{8}$

b)

$-\frac{x^2}{4}$

c)

$-\frac{x^2}{8}$

d)

$\frac{x^2}{4}$

31.

Find the 4th term of the Maclaurin series for $f\left(x\right)=\cos\left(2x\right)$ .

a)

$-\frac{4x^6}{45}$

b)

$\frac{4x^6}{45}$

c)

$-\frac{x^6}{360}$

d)

$\frac{x^6}{360}$

32.

$1+x+x^2+x^3+...$

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

33.

$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

34.

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

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

35.

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

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

36.

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

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

37.

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n+1}}{\left(2n+1\right)!}$

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

38.

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n}}{\left(2n\right)!}$

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

39.

$\sum_{n=0}^{\infty}x^n$

a)

$e^x$

b)

$\sin x$

c)

$\cos x$

d)

$\frac{1}{1-x}$

40.

Interval of convergence for the series that represents $f\left(x\right)=e^x$

a)

$-1<x<1$

b)

$-\infty<x<\infty$

41.

Interval of convergence for the series that represents $f\left(x\right)=\sin x$

a)

$-1<x<1$

b)

$-\infty<x<\infty$

42.

Interval of convergence for the series that represents $f\left(x\right)=\cos x$

a)

$-1<x<1$

b)

$-\infty<x<\infty$

43.

Interval of convergence for the series that represents $f\left(x\right)=\frac{1}{1-x}$

a)

$-1<x<1$

b)

$-\infty<x<\infty$

44.

$1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+...$

a)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^n}{n!}$

b)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n+1}x^n}{n!}$

c)

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

d)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{n+1}}{n!}$

45.

$f\left(x\right)=\left(e^x\right)^2-1$

a)

$2x-2x^2+\frac{8x^3}{3!}-\frac{16x^4}{4!}+...$

b)

$2x+2x^2+\frac{8x^3}{3!}+\frac{16x^4}{4!}+...$

c)

$x^2+\frac{x^4}{2!}+\frac{x^6}{3!}+\frac{x^8}{4!}+...$

d)

$x^2-\frac{x^4}{2!}+\frac{x^6}{3!}-\frac{x^8}{4!}+...$

46.

$2x+2x^2+\frac{8x^3}{3!}+\frac{16x^4}{4!}+...$

a)

$\sum_{n=2}^{\infty}\frac{2^{n-1}x^{n-1}}{\left(n-1\right)!}$

b)

$\sum_{n=1}^{\infty}\frac{2^{n-1}x^{n-1}}{\left(n-1\right)!}$

c)

$\sum_{n=0}^{\infty}\frac{2^nx^n}{\left(n+1\right)!}$

d)

$\sum_{n=1}^{\infty}\frac{2^nx^n}{\left(n-1\right)!}$

47.

What is the fourth term of the power series for $x^4\sin\left(x^5\right)$ .

a)

$-\frac{x^{39}}{7!}$

b)

$-\frac{x^{19}}{7!}$

c)

$-\frac{x^{29}}{7!}$

d)

$-\frac{x^{49}}{7!}$

48.

What is the fourth term of the power series for $2\sin\left(3x^2\right)-5$ .

a)

$\frac{486x^{10}}{5!}$

b)

$-\frac{4374x^{12}}{7!}$

c)

$-\frac{1458x^{12}}{5!}$

d)

$-\frac{162x^{10}}{7!}$

49.

For x>0, the power series $1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+...+\frac{\left(-1\right)^nx^{2n}}{\left(2n+1\right)!}+...$ converges to which of the following?

a)

$\cos x$

b)

$\sin x$

c)

$\frac{\sin x}{x}$

d)

$e^x-e^{x^2}$

e)

$1+e^x-e^{x^2}$

50.

$\int_0^x\sin\left(t^6\right)dt=$

a)

$\frac{x^2}{2}-\frac{x^4}{4}+\frac{x^6}{6}-...$

b)

$\frac{x^2}{2}-\frac{x^4}{4\cdot3!}+\frac{x^6}{6\cdot5!}-...$

c)

$\frac{x^7}{7}-\frac{x^{19}}{19}+\frac{x^{31}}{31}-...$

d)

$\frac{x^7}{7}-\frac{x^{19}}{19\cdot3!}+\frac{x^{31}}{31\cdot5!}-...$

51.

Find the Maclaurin series for $\frac{8}{5-x}$ (note this comes from a geo series!)

a)

$\sum_{n=0}^{\infty}8\left(5x\right)^n$

b)

$\sum_{n=0}^{\infty}8\left(-5x\right)^n$

c)

$\sum_{n=0}^{\infty}\frac{8}{5}\left(\frac{x}{5}\right)^n$

d)

$\sum_{n=0}^{\infty}\frac{8}{5}\left(-\frac{x}{5}\right)^n$

52.

Find the Maclaurin series for $\frac{6}{4+3x}$ (note this is from a geo series!)

a)

$\sum_{n=0}^{\infty}\frac{3}{2}\left(-\frac{3x}{4}\right)^n$

b)

$\sum_{n=0}^{\infty}\frac{3}{2}\left(\frac{3x}{4}\right)^n$

c)

$\sum_{n=0}^{\infty}\frac{2}{3}\left(\frac{x}{2}\right)^n$

d)

$\sum_{n=0}^{\infty}\frac{2}{3}\left(-\frac{x}{2}\right)^n$

53.

We didn't do this one this year. Click the smiley face!

Find the Taylor series for $\frac{4}{1+x}$ centered at $x=5$

a)

$\sum_{n=0}^{\infty}\left[-1\left(-\frac{x-5}{4}\right)^n\right]$

b)

$\sum_{n=0}^{\infty}\left[-1\left(\frac{x-5}{4}\right)^n\right]$

c)

$\sum_{n=0}^{\infty}\frac{2}{3}\left(\frac{x-5}{6}\right)^n$

d)

$\sum_{n=0}^{\infty}\frac{2}{3}\left(-\frac{x-5}{6}\right)^n$

e)

:)

54.

We didn't do this one. Click the smiley face!

Find the Taylor series for $\frac{2}{2+x}$ centered at $x=-3$

a)

$\sum_{n=0}^{\infty}\left[-2\left(x+3\right)^n\right]$

b)

$\sum_{n=0}^{\infty}\left[-2\left(-\left(x+3\right)\right)^n\right]$

c)

$\sum_{n=0}^{\infty}2\left(\frac{x+3}{2}\right)^n$

d)

$\sum_{n=0}^{\infty}2\left(-\frac{x+3}{2}\right)^n$

e)

:)

55.

$\frac{d}{dx}\left(\frac{1}{1-x}\right)=$

a)

$1+2x+3x^2+4x^3+...$

b)

$1-2x+3x^2-4x^3+...$

c)

$1+x+x^2+x^3+...$

d)

$1-x+x^2-x^3+...$

56.

$e-\frac{e^3}{3!}+\frac{e^5}{5!}-\frac{e^7}{7!}+...=$

a)

$\sin e$

b)

$\cos e$

c)

$e^e$

d)

$\frac{1}{1-e}$

57.

What is the coefficient of $x^4$ in the Maclaurin series for $f\left(x\right)=e^{2x}-\cos\left(x^2\right)$

a)

$\frac{7}{6}$

b)

$\frac{3}{4}$

c)

$\frac{1}{2}$

d)

$\frac{1}{4}$

58.

The power series $\sum_{n=1}^{\infty}\frac{\left(x-5\right)^n}{2^nn^2}$ has a radius of convergence 2. At which of the following values of x can the alternating series test be used with this series to verify convergence at x?

a)

6

b)

4

c)

2

d)

0

e)

-1

59.

For $-1<x<1$ if $f\left(x\right)=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}x^{2n-1}}{2n-1}$ , then $f'\left(x\right)=$

a)

$\sum_{n=1}^{\infty}\left(-1\right)^{n+1}x^{2n-2}$

b)

$\sum_{n=1}^{\infty}\left(-1\right)^nx^{2n-2}$

c)

$\sum_{n=1}^{\infty}\left(-1\right)^{2n}x^{2n}$

d)

$\sum_{n=1}^{\infty}\left(-1\right)^nx^{2n}$

60.

Let $f\left(x\right)=\cos\left(3x+\frac{\pi}{6}\right)$ . Let $P_4\left(x\right)$ be the fourth degree Maclaurin polynomial for f(x). Use the Lagrange Error Bound to find the maximum error for this polynomial at $x=\frac{1}{6}$

a)

$\frac{3^5\sin\left(\frac{1}{2}+\frac{\pi}{6}\right)\left(\frac{1}{6}\right)^5}{5!}$

b)

$\frac{3^5\left(\frac{1}{6}\right)^5}{5!}$

c)

$\frac{1\left(\frac{1}{6}\right)^5}{5!}$

d)

$\frac{\sin\left(\frac{1}{2}+\frac{\pi}{6}\right)\left(\frac{1}{6}\right)^5}{5!}$

Unit 10B Review

Total questions: 60

Worksheet time: 2hrs 0mins

Similar Resources on Wayground