Free Printable Topics 10.11-10.13 Review

1.

What is the interval of convergence for

\sum_{n=0}^{\infty}\frac{n^2x^{n+1}}{3^n}

a)

$-3<x<3$

b)

$-3\le x<3$

c)

$-3<x\le3$

d)

$-3\le x\le3$

2.

What is the interval of convergence for $\sum_{n=0}^{\infty}\frac{\left(x-2\right)^n}{10^n}$ .

a)

(-8,12)

b)

(8,12)

c)

(-12,-8)

d)

(-12,8)

3.

What is the fourth degree term in the Maclaurin Polynomial for

f\left(x\right)=e^{3x}

a)

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

b)

$\frac{81\left(x-1\right)^4}{4!}$

c)

$\frac{27x^4}{4!}$

d)

$\frac{81x^4}{4!}$

4.

What is the fifth degree term in the Maclaurin Polynomial for

f\left(x\right)=e^{-x}

a)

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

b)

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

c)

$\frac{-ex^5}{5!}$

d)

$\frac{ex^5}{5!}$

5.

What is the fifth NONZERO term in the Maclaurin Polynomial for

f\left(x\right)=\sin x

a)

$-\frac{x^9}{9!}$

b)

$\frac{x^9}{9!}$

c)

$\frac{x^{11}}{11!}$

d)

$-\frac{x^{11}}{11!}$

6.

What is the third NONZERO term in the Maclaurin Polynomial for

f\left(x\right)=\cos\left(\pi x\right)

a)

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

b)

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

c)

$-\frac{\pi^4x^4}{4!}$

d)

$\frac{\pi^4x^4}{4!}$

7.

What is the third degree term in the Maclaurin Polynomial for

f\left(x\right)=\frac{1}{x+1}

a)

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

b)

$-\frac{6x^3}{3!}$

c)

$\frac{2x^3}{3!}$

d)

$\frac{6x^3}{3!}$

8.

What is the second degree term in the Maclaurin Polynomial for

f\left(x\right)=xe^x

a)

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

b)

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

c)

$\frac{3x^2}{2!}$

d)

There is no second degree term

9.

Given $f\left(0\right)=1$ , $f^{\left(n\right)}\left(0\right)=\frac{n^2}{3^n}$ for $n\ge1$ . What is the third term for this Maclaurin polynomial?

a)

$\frac{2x^2}{9}$

b)

$\frac{2x^2}{27}$

c)

$\frac{x^2}{18}$

d)

$\frac{x^2}{24}$

10.

What is the third degree term in the Taylor polynomial for

$f\left(x\right)=\frac{2}{x}$ and $c=1$

a)

$-2\left(x-1\right)^3$

b)

$\frac{2\left(x-1\right)^3}{3}$

c)

$-\frac{\left(x-1\right)^3}{6}$

d)

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

11.

What is the second degree term in the Taylor polynomial for

$g\left(x\right)=\sqrt{x}$ and $c=2$

a)

$-\frac{\left(x-2\right)^2}{16\sqrt{2}}$

b)

$-\frac{\left(x-2\right)^2}{4\sqrt{2}}$

c)

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

d)

$\frac{\left(x-2\right)^2}{12\sqrt{2}}$

12.

What is the fourth term in the Taylor polynomial for

$h\left(x\right)=\sin x$ and $c=\frac{\pi}{4}$

a)

$-\frac{\sqrt{2}\left(x-\frac{\pi}{4}\right)^3}{12}$

b)

$-\frac{\sqrt{2}\left(x-\frac{\pi}{4}\right)^4}{48}$

c)

$\frac{\sqrt{2}\left(x-\frac{\pi}{4}\right)^4}{48}$

d)

$\frac{\sqrt[]{2}\left(x-\frac{\pi}{4}\right)^{\text{3}}}{12}$

13.

$f\left(2\right)=0$ , $f^{\left(n\right)}\left(2\right)=\frac{2^n}{\left(n+1\right)!}$ for $n\ge1$ , what is the second degree term for the Taylor Polynomial centered at 2?

a)

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

b)

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

c)

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

d)

$\frac{4\left(x-2\right)^2}{3!}$

14.

Which of the following is the 2nd degree Maclaurin polynomial for the function $f\left(x\right)=e^{2x}$

a)

$P_2\left(x\right)=1+x+x^2$

b)

$P_2\left(x\right)=1+x+2x^2$

c)

$P_2\left(x\right)=1+2x+x^2$

d)

$P_2\left(x\right)=1+2x+2x^2$

15.

If the fourth term of a Taylor polynomial centered at c=2 is given by $12\left(x-2\right)^3$ , then $f^{\left(3\right)}\left(2\right)$ , the third derivative of f(x) at x=2, is which of the following?

a)

2

b)

6

c)

12

d)

72

16.

Which of the following power series, centered at 0, give the sum $S=\frac{12}{6+x}$ ?

a)

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

b)

$\sum_{n=0}^{\infty}12\left(-\frac{x}{6}\right)^n$

c)

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

d)

$\sum_{n=0}^{\infty}12\left(-x\right)^n$

17.

What is the radius of convergence for the power series $\sum_{n=1}^{\infty}\frac{\left(x-7\right)^n}{2^n\left(n-1\right)!}$

a)

0

b)

2

c)

7

d)

$\infty$

18.

Estimate the error in approximating $\sin\left(0.5\right)$ using the fifth degree Maclaurin polynomial by finding the Lagrange remainder.

a)

$1\cdot\frac{0.5^5}{5!}$

b)

$1\cdot\frac{0.5^6}{6!}$

c)

$1\cdot\frac{\cos^5\left(0.5\right)}{5!}$

d)

$1\cdot\frac{\sin^6\left(0.5\right)}{6!}$

19.

A student needs to calculate $\frac{1}{\sqrt{e}}$ on a four-function calculator. He uses a third degree Maclaurin polynomial for $f\left(x\right)=e^x$ and adds on the Lagrange error for this approximation using e as the maximum derivative. What is the result?

a)

0.61125

b)

0.98253

c)

1.22143

d)

1.65291

20.

Let f be a function given by $f\left(x\right)=\cos\left(3x-\frac{\pi}{10}\right)$ and let P(x) be the fourth degree Taylor polynomial with center 0. Is this statement true? $\left|f_4\left(\frac{1}{4}\right)-P_4\left(\frac{1}{4}\right)\right|<\frac{1}{500}$

a)

Yes, it's true.

b)

No, it's false.

21.

Let f be a function with derivatives shown in the table, and $\left|f^{\left(4\right)}\left(x\right)\right|\le50$ for all x in the interval [2,2.5]. Find $P_3\left(2.1\right)+R_3\left(2.1\right)$ to estimate the value of $f\left(2.1\right)$

a)

8.4832

b)

8.5213

c)

8.6192

d)

8.6978

22.

The power series $\sum_{n=1}^{\infty}a_n\left(x-3\right)^n$ converges when $x=1$ . Is the statement "The series diverges when $x=-2$ " Always true, Sometimes true, or Never true?

a)

Always true

b)

Sometimes true

c)

Never true

23.

The power series $\sum_{n=1}^{\infty}a_n\left(x-3\right)^n$ converges when $x=1$ . Is the statement "The series converges when $x=2$ " Always true, Sometimes true, or Never true?

a)

Always true

b)

Sometimes true

c)

Never true

24.

Find the interval of convergence for the power series $\sum_{n=1}^{\infty}\frac{\left(-1\right)^nnx^n}{n+1}$

a)

$-1<x<1$

b)

$-1\le x<1$

c)

$-1<x\le1$

d)

$-1\le x\le1$

25.

Find the radius of convergence for the series $\sum_{n=0}^{\infty}\left(\frac{x-1}{3}\right)^n$

a)

0

b)

1

c)

3

d)

$\infty$

26.

The power series $\sum_{n=0}^{\infty}a_n\left(x-2\right)^n$ converges at $x=6$ . Which of the following must be true?

a)

The series diverges at $x=-2$

b)

The series diverges at $x=7$

c)

The series converges at $x=-2$

d)

The series converges at $x=-1$

27.

What is the interval of convergence for $\sum_{n=2}^{\infty}\frac{n^2\left(x-3\right)^n}{4^n\left(n^2+2n-1\right)}$ ?

a)

$\left(-1,7\right)$

b)

$\left[-1,7\right)$

c)

$\left(-4,4\right)$

d)

$\left[-4,4\right)$

28.

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

29.

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 the third degree Taylor Polynomial for f about $x=3$ ?

a)

$2-\left(x-3\right)+6\left(x-3\right)^2+12\left(x-3\right)^3$

b)

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

c)

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

30.

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 $x=2$

d)

The series converges at $x=1$

31.

The geometric Maclaurin series for the function f is given by $f\left(x\right)=\sum_{n=0}^{\infty}\left(-\frac{x}{4}\right)^n$ . What is the value of $f\left(3\right)$ ?

a)

$-3$

b)

$-\frac{3}{7}$

c)

$\frac{4}{7}$

d)

$\frac{13}{16}$

e)

4

32.

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)

$\frac{81}{256}$

33.

Let $P\left(x\right)=3-3x^2+6x^4$ be a 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

34.

What is the interval of convergence of the power series $\sum_{n=1}^{\infty}\frac{\left(x-3\right)^n}{n\cdot2^n}$

a)

$1<x<5$

b)

$1\le x<5$

c)

$1\le x\le5$

d)

$2<x<4$

e)

$2\le x\le4$

35.

Let f be a function that has derivatives of all orders for all real numbers. Let $P_3\left(x\right)$ be a third degree Taylor polynomial for f about x=0. If $\left|f^{\left(n\right)}\left(0\right)\right|\le\frac{n}{n+1}$ , what is the Lagrange error bound for $P_3\left(1\right)$ ?

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)

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

36.

The power series $\sum_{n=0}^{\infty}a_n\left(x-1\right)^n$ converges at x=5. For which of the following values of x must the series converge?

a)

-5

b)

-3

c)

-1

d)

7

37.

What is the radius of convergence for the series $\sum_{n=1}^{\infty}\frac{\left(x-2\right)^n}{3^n\left(2n+1\right)^2}$ ?

a)

$\frac{1}{3}$

b)

1

c)

2

d)

3

38.

What is the interval of convergence for $\sum_{n=0}^{\infty}\frac{\left(-1\right)^n\left(x-1\right)^n}{3n}$ ?

a)

$0<x<2$

b)

$0<x\le2$

c)

$0\le x\le2$

d)

$0\le x<2$

39.

Find the Interval of Convergence and the Radius of Convergence

a)

-3<x<1

radius = 2

b)

-2<x<2

radius = 2

c)

-3<x<1

radius = 1

d)

-2<x<2

radius = 1

40.

f(x) = $\sum_{n=0}^{\infty}\frac{\left(-1\right)^n\left(x-3\right)^n}{2n}$ what is the interval of convergence for f(x)?

a)

$2<x<4$

b)

$2<x\le4$

c)

$2\le x\le4$

d)

$2\le x<4$

41.

f(x) = $\sum_{n=0}^{\infty}\frac{\left(-1\right)^n\left(\frac{x}{4}\right)^n}{n^2}$ what is the interval of convergence for f(x)?

a)

$-4<x<4$

b)

$-4<x\le4$

c)

$-4\le x\le4$

d)

$-4\le x<4$

42.

What are the endpoints of the interval of convergence for this series if its radius is 1?

a)

2 and 4

b)

-4 and -2

c)

1 and 3

d)

3 and 5

43.

Does this series converge at x = -4?

a)

yes

b)

no

44.

Does this series converge at x = -2?

a)

yes

b)

no

45.

What is the interval of convergence for this series?

a)

$-4≤x≤-2$

b)

$-4<x≤-2$

c)

$-4≤x<-2$

d)

$-4<x<-2$

46.

Does this series converge at x = -3?

a)

yes

b)

no

47.

What are the endpoints of the interval of convergence for this series if its radius is 1?

a)

2 and 4

b)

-4 and -2

c)

1 and 3

d)

-3 and -1

48.

Does this series converge at x = 2?

a)

yes

b)

no

49.

Does this series converge at x = 4?

a)

yes

b)

no

50.

What is the interval of convergence for this series?

a)

$2≤x≤4$

b)

$2≤x<4$

c)

$2<x≤4$

d)

$2<x<4$

51.

Let $P\left(x\right)=3x^2-5x^3+7x^4+3x^5$ be the fifth-degree Taylor polynomial for the function $f$ about $x=0$ . What is the value of $f^{'''}\left(0\right)$ ?

a)

$-30$

b)

$-15$

c)

$-5$

d)

$-\frac{5}{6}$

e)

$-\frac{1}{6}$

52.

The third-degree Taylor polynomial about $x=0$ of $\ln\left(1-x\right)$ is ?

a)

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

b)

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

c)

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

d)

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

e)

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

53.

The power series $\sum_{n=0}^{\infty\ }a_n\left(x-4\right)^n$ converges for $x=7$ and diverges for $x=-1$ . Which of the following must be true? Select all that apply.

a)

The series converges for $x=4$ .

b)

The series converges for $x=2$ .

c)

The series converges for $x=1$ .

d)

The series converges for $x=-2$ .

54.

The fourth-degree Taylor polynomial for f centered at $x=-2$ is given by $P_4\left(x\right)=1-2\left(x+2\right)+\frac{3}{5}\left(x+2\right)^2+\frac{7}{8}\left(x+2\right)^3-\frac{5}{12}\left(x+2\right)^4$ . What is $f^{\left(4\right)}\left(-2\right)$ ?

a)

$\frac{5}{12}$

b)

$-\frac{5}{12}$

c)

$-10$

d)

$10$

55.

The eighth-degree Taylor polynomial for f centered at $x=3$ , $P_8\left(x\right)$ , is used to approximate $f\left(3.5\right)$ . If $\left|f^{\left(9\right)}\left(x\right)\right|\le\frac{1}{12}$ for all x in the interval $\left[3,3.5\right]$ , give the Lagrange error bound.

a)

$\frac{12\left(0.5\right)^8}{8!}$

b)

$\frac{12\left(0.5\right)^9}{9!}$

c)

$\frac{\left(0.5\right)^8}{12\cdot8!}$

d)

$\frac{\left(0.5\right)^9}{12\cdot9!}$

56.

A portion of the graph of $y=\left|f^{\left(5\right)}\left(x\right)\right|$ is given. If the fourth-degree Maclaurin polynomial to f is used to approximate $f\left(1\right)$ , what is the least possible Lagrange error bound?

a)

$\frac{8}{5!}$

b)

$\frac{6.2}{5!}$

c)

1

d)

$\frac{1}{4!}$

Topics 10.11-10.13 Review

Total questions: 56

Worksheet time: 3hrs 48mins

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