## MATH1901 Quizzes

Quiz 9: Taylor series and Eulers formula
Question 1 Questions
Find the coefficient of ${x}^{2}$ in the Taylor series about $x=0$ for $f\left(x\right)={e}^{-{x}^{2}}$. Exactly one option must be correct)
 a) 1/4 b) -1 c) 1/2 d) -2 e) 1

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the coefficient of ${x}^{3}$ in the Taylor series about $x=0$ for $f\left(x\right)=sin2x$. Exactly one option must be correct)
 a) -2/3 b) -4/3 c) 4/3 d) -8/3 e) 2/3

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which one of the following is the Taylor series for $sinx$? Exactly one option must be correct)
 a) $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n-1}}{\left(2n-1\right)!}$ b) $\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n+1}{x}^{2n+1}}{\left(2n+1\right)!}$ c) $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n+1}}{\left(2n+1\right)!}$ d) $\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n+1}}{\left(2n+1\right)!}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Which of the following are true statements? (Zero or more options can be correct)
 a) Any function $f\left(x\right)$ is equal to its Taylor series for all real $x$. b) There exist functions $f\left(x\right)$ which are equal to their Taylor series for all real $x$. c) There exist functions $f\left(x\right)$ which are equal to their Taylor series for some, but not all, real numbers $x$. d) A function $f\left(x\right)$ can never equal its Taylor series. The Taylor series is only ever an approximation to the function.

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be True.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be False.
Correct!
1. False
2. True
3. True
4. False
${\sum }_{n=0}^{\infty }{\left(-x\right)}^{n}$ is the Taylor series for which function? Exactly one option must be correct)
 a) $sinx$ b) $cosx$ c) $\frac{1}{1-x}$ d) $\frac{1}{1+x}$ e) ${e}^{x}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Find the Taylor series about $x=0$ for $f\left(x\right)=\frac{1}{{\left(1-x\right)}^{2}}$. Exactly one option must be correct)
 a) $\sum _{n=0}^{\infty }\left(n+1\right){x}^{n}$ b) $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\left(n+1\right){x}^{n}$ c) $\sum _{n=0}^{\infty }n{x}^{n}$ d) $\sum _{n=0}^{\infty }{x}^{2n}$ e) $\sum _{n=0}^{\infty }-{x}^{2n}$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
The simplest possible cartesian form of $8{e}^{i\pi }$ is Exactly one option must be correct)
 a) 8 b) 8$\pi$ c) $8-i$ d) $8i$ e) $-8$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
Note ${e}^{i\pi }=cos\pi +isin\pi =-1+0i$ so $8{e}^{i\pi }=-8$.
If $z=-1+i$ then $z$ expressed in exponential notation equals Exactly one option must be correct)
 a) $\sqrt{2}cis\frac{\pi }{4}$ b) $\sqrt{2}{e}^{i3\pi ∕4}$ c) $\frac{1}{\sqrt{2}}{e}^{-i\pi ∕4}$ d) $exp\frac{\pi }{4}$ e) None of the above

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{llll}\hfill \sqrt{2}{e}^{i3\pi ∕4}& =\sqrt{2}\left(cos\frac{3\pi }{4}+isin\frac{3\pi }{4}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\sqrt{2}\left(-\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-1+i.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find $uv$ if $u=2exp\left(\right-\frac{\pi }{2}i\left)\right$ and $v=\frac{1}{2}exp\left(\right\frac{5\pi }{3}i\left)\right$. (Zero or more options can be correct)
 a) $exp\left(\right\frac{5{\pi }^{2}}{6}\left)\right$ b) $exp\left(\right\frac{7\pi }{6}i\left)\right$ c) $exp\left(\right-\frac{5\pi }{6}i\left)\right$ d) $exp\left(\right\frac{13\pi }{6}i\left)\right$

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be True.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be False.
Correct!
1. False
2. True
3. True
4. False
Which of the following are equal to ${\left(1+i\right)}^{10}$? (Zero or more options can be correct)
 a) $0$ b) $32exp\left(\right\frac{5\pi i}{2}\left)\right$ c) $32i$ d) $\sqrt{2}exp\left(\right\frac{5\pi i}{2}\left)\right$ e) $exp\left(\right\frac{5\pi i}{2}\left)\right$

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be True.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be False.
There is at least one mistake.
For example, choice (e) should be False.
Correct!
1. False
2. True
3. True
4. False
5. False