 ## MATH1901 Quizzes

Quiz 8: Taylor polynomials
Question 1 Questions
Find the Taylor polynomial of degree 3 about $x=0$ for the function $f\left(x\right)={e}^{-x}$. Exactly one option must be correct)
 a) $1-x+\frac{1}{2}{x}^{2}-\frac{1}{6}{x}^{3}$ b) $1+x+\frac{1}{2}{x}^{2}-\frac{1}{6}{x}^{3}$ c) $1-x+{x}^{2}-{x}^{3}$ d) $1+x+{x}^{2}+{x}^{3}$ e) $1-x+\frac{1}{2}{x}^{2}-\frac{1}{3}{x}^{3}$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the Taylor polynomial of degree 3 about $x=0$ for the function $f\left(x\right)=ln\left(1+x\right)$. Exactly one option must be correct)
 a) $1-x+{x}^{2}-{x}^{3}$ b) $x-{x}^{2}+{x}^{3}$ c) $1-x+\frac{1}{2}{x}^{2}-\frac{1}{6}{x}^{3}$ d) $x-\frac{1}{2}{x}^{2}+\frac{1}{3}{x}^{3}$ e) $x-\frac{1}{24}{x}^{2}+\frac{1}{120}{x}^{3}$

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

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the Taylor polynomial of degree 3 about $x=0$ for $f\left(x\right)=sin2x$. Exactly one option must be correct)
 a) $2x-8{x}^{3}$ b) $x-\frac{{x}^{3}}{6}$ c) $2x+\frac{4{x}^{3}}{3}$ d) $2x-\frac{4{x}^{3}}{3}$ e) $x+\frac{{x}^{3}}{6}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Find the Taylor polynomial of degree 3 about $x=0$ for $f\left(x\right)=xcosx$. Exactly one option must be correct)
 a) $x-{x}^{2}+\frac{{x}^{3}}{3}$ b) $x-{x}^{2}-\frac{{x}^{3}}{3}$ c) $x-\frac{{x}^{3}}{2}$ d) $x-{x}^{3}$ e) $x-\frac{{x}^{3}}{6}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choice (e) is incorrect
Find the Taylor polynomial of degree 4 about $x=1$ for $f\left(x\right)=\frac{1}{x}$. Exactly one option must be correct)
 a) $1-x+{x}^{2}-{x}^{3}+{x}^{4}$ b) $1+x+{x}^{2}+{x}^{3}+{x}^{4}$ c) $1-\left(x-1\right)+{\left(x-1\right)}^{2}-{\left(x-1\right)}^{3}+{\left(x-1\right)}^{4}$ d) $1+\left(x-1\right)+{\left(x-1\right)}^{2}+{\left(x-1\right)}^{3}+{\left(x-1\right)}^{4}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Find the Taylor polynomial of degree 3 about $x=\pi ∕4$ for $f\left(x\right)=tanx$. Exactly one option must be correct)
 a) $0.01+\left(x-\pi ∕4\right)+0.02{\left(x-\pi ∕4\right)}^{2}+0.33{\left(x-\pi ∕4\right)}^{3}$ b) $1+2\left(x-\pi ∕4\right)+2{\left(x-\pi ∕4\right)}^{2}+\frac{8}{3}{\left(x-\pi ∕4\right)}^{3}$ c) $1+2\left(x-\pi ∕4\right)+2{\left(x-\pi ∕4\right)}^{2}+2{\left(x-\pi ∕4\right)}^{3}$ d) $1+2\left(x-\pi ∕4\right)+{\left(x-\pi ∕4\right)}^{2}+\frac{1}{6}{\left(x-\pi ∕4\right)}^{3}$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
If $f\left(x\right)=1+3x-5{x}^{2}+4{x}^{3}+2{x}^{4}$, which of the following are true statements? (Zero or more options can be correct)
 a) The Taylor polynomial of degree 6 about $x=3$ for $f\left(x\right)$ is equal to $f\left(x\right)$. b) The Taylor polynomial of degree 6 about $x=3$ for $f\left(x\right)$ does not exist. c) The remainder term ${R}_{5}\left(x\right)$ for $f\left(x\right)$ is equal to zero. d) The remainder term ${R}_{3}\left(x\right)$ for $f\left(x\right)$ is equal to zero. e) The remainder term ${R}_{3}\left(2\right)$ for $f\left(x\right)$ is equal to 32.

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
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 True.
Correct!
1. True
2. False
3. True
4. False
5. True
Find the difference between the Taylor polynomial of degree 4 about the point $0$ for $sinx$ evaluated at $x=1$, and $sin1$. Exactly one option must be correct)
 a) $\frac{cos1}{5!}$ b) $\frac{cosc}{5!}$ for some $c$ between $0$ and $1$. c) $\frac{-cosc}{5!}$ for some $c$ between $0$ and $1$. d) $\frac{cos1}{5!}{x}^{n+1}$ for some $x$ between $0$ and $1$. e) $\frac{-cos1}{5!}{x}^{n+1}$ for some $x$ between $0$ and $1$.

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the coefficient of ${x}^{n}$ in the Taylor polynomial of degree $n$ $\left(n\ge 2\right)$ for $f\left(x\right)=\sqrt{1+x}$ about $x=0$. Exactly one option must be correct)
 a) $\frac{{\left(-1\right)}^{n}3.5.7.\dots \left(2n-1\right)}{{2}^{n}n!}$ b) $\frac{{\left(-1\right)}^{n}3.5.7.\dots \left(2n-1\right)}{{2}^{n}}$ c) $\frac{{\left(-1\right)}^{n+1}3.5.7.\dots \left(2n-3\right)}{{2}^{n}}$ d) $\frac{{\left(-1\right)}^{n+1}3.5.7.\dots \left(2n-3\right)}{{2}^{n}n!}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!