Taylor Polynomials

Authored by Anthony Clark

Mathematics

9th Grade

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11 questions

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MULTIPLE CHOICE QUESTION

1 min • 1 pt

Fill in the General Form of a Taylor Polynomial with the correct pieces:

$1=f^n\left(c\right);\ 2=n!;\ 3=\left(x-c\right)$

$1=f^n\left(c\right);\ 2=n!;\ 3=x$

$1=f^n\left(0\right);\ 2=n!;\ 3=\left(x-c\right)$

$1=n!;\ 2=f^n\left(c\right);\ 3=\left(x-c\right)$

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Given f(1)=3; f'(1)=-2; f''(1)=4; f'''(1)=-12, the first four terms of the Taylor Polynomial for f(x) are

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

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

$3-2\left(x-1\right)+4\left(x-1\right)^2-12\left(x-1\right)^3$

$3-2x+4x^2-12x^3$

MULTIPLE CHOICE QUESTION

1 min • 1 pt

This is a third-degree Taylor polynomial for a function f about x = 1. What is the value of $f''\left(1\right)$ ?

$\frac{40}{3}$

240

480

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Suppose a function f is approximated with a fourth-degree Taylor polynomial about x = 1. If $\left|f^{\left(5\right)}\left(x\right)\right|<0.01$ between x = 1 and x = 3, then the maximum error incurred using $T_4\left(x\right)$ to compute f(3) is

2.667

0.266

0.026

0.002

MULTIPLE CHOICE QUESTION

1 min • 1 pt

The graph of the function f(x) is shown. Which of the following could be a portion of the graph of the Maclaurin polynomial of degree 13 for f(x)?

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Which of the following is the fourth-degree Taylor Polynomial for $f\left(x\right)=x\sin x$ about $x=0$ .

$P_4\left(x\right)=x-\frac{1}{6}x^3+\frac{1}{5!}x^5-\frac{1}{7!}x^7$

$P_4\left(x\right)=-x^2+\frac{1}{6}x^4$

$P_4\left(x\right)=x^2-\frac{1}{6}x^4$

$P_4\left(x\right)=2x^2-4x^4$

MULTIPLE CHOICE QUESTION

1 min • 1 pt

The coefficient of $\left(x-\frac{\pi}{4}\right)^3$ in the Taylor series about $\frac{\pi}{4}$ of $f\left(x\right)=\cos x$ is

$\frac{\sqrt[]{3}}{12}$

$\frac{1}{12}$

$\frac{1}{6\sqrt[]{2}}$

$\frac{1}{3\sqrt[]{2}}$

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Let $f$ be the function defined by $f\left(x\right)=\ln\left(x+4\right)$ . The third-degree Taylor Polynomial for $f$ centered about $x=-3$ is

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

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

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

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

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Let $f$ be the function that has derivatives of all orders for all real numbers. Assume Write a third-order Taylor Polynomial for $f$ at $x=0$ and use it to approximate $f\left(0.5\right)$ .

$6-\frac{x}{5}+\frac{x^2}{7}+\frac{x^3}{6};\ \ f\left(0.5\right)\approx5.957$

$6-5x+7x^2+6x^3;\ \ f\left(0.5\right)\approx6.000$

$6x-5x^2+7x^3+6x^4;\ \ f\left(0.5\right)\approx3.000$

$6+7x-5x^2+6x^3;\ \ \ f\left(0.5\right)\approx9.000$

10.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

The coefficient of the $x^2-$ term in the Taylor Polynomial for $y=x^{\frac{2}{3}}$ around $x=8$ is

$-\frac{1}{144}$

$-\frac{1}{72}$

$-\frac{1}{9}$

$-\frac{1}{2}$

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Taylor Polynomials

Fill in the General Form of a Taylor Polynomial with the correct pieces:

Given f(1)=3; f'(1)=-2; f''(1)=4; f'''(1)=-12, the first four terms of the Taylor Polynomial for f(x) are

This is a third-degree Taylor polynomial for a function f about x = 1. What is the value of f′′(1)f''\left(1\right)f′′(1) ?

Suppose a function f is approximated with a fourth-degree Taylor polynomial about x = 1. If ∣f(5)(x)∣<0.01\left|f^{\left(5\right)}\left(x\right)\right|<0.01​f(5)(x)​<0.01 between x = 1 and x = 3, then the maximum error incurred using T4(x)T_4\left(x\right)T4​(x) to compute f(3) is

The graph of the function f(x) is shown. Which of the following could be a portion of the graph of the Maclaurin polynomial of degree 13 for f(x)?

Which of the following is the fourth-degree Taylor Polynomial for f(x)=xsin⁡xf\left(x\right)=x\sin xf(x)=xsinx about x=0x=0x=0 .

The coefficient of (x−π4)3\left(x-\frac{\pi}{4}\right)^3(x−4π​)3 in the Taylor series about π4\frac{\pi}{4}4π​ of f(x)=cos⁡xf\left(x\right)=\cos xf(x)=cosx is

Let fff be the function defined by f(x)=ln⁡(x+4)f\left(x\right)=\ln\left(x+4\right)f(x)=ln(x+4) . The third-degree Taylor Polynomial for fff centered about x=−3x=-3x=−3 is

Let fff be the function that has derivatives of all orders for all real numbers. Assume Write a third-order Taylor Polynomial for fff at x=0x=0x=0 and use it to approximate f(0.5)f\left(0.5\right)f(0.5) .

The coefficient of the x2−x^2-x2− term in the Taylor Polynomial for y=x23y=x^{\frac{2}{3}}y=x32​ around x=8x=8x=8 is

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Discover more resources for Mathematics

This is a third-degree Taylor polynomial for a function f about x = 1. What is the value of $f''\left(1\right)$ ?

Suppose a function f is approximated with a fourth-degree Taylor polynomial about x = 1. If $\left|f^{\left(5\right)}\left(x\right)\right|<0.01$ between x = 1 and x = 3, then the maximum error incurred using $T_4\left(x\right)$ to compute f(3) is

Which of the following is the fourth-degree Taylor Polynomial for $f\left(x\right)=x\sin x$ about $x=0$ .

The coefficient of $\left(x-\frac{\pi}{4}\right)^3$ in the Taylor series about $\frac{\pi}{4}$ of $f\left(x\right)=\cos x$ is

Let $f$ be the function defined by $f\left(x\right)=\ln\left(x+4\right)$ . The third-degree Taylor Polynomial for $f$ centered about $x=-3$ is

Let $f$ be the function that has derivatives of all orders for all real numbers. Assume Write a third-order Taylor Polynomial for $f$ at $x=0$ and use it to approximate $f\left(0.5\right)$ .

The coefficient of the $x^2-$ term in the Taylor Polynomial for $y=x^{\frac{2}{3}}$ around $x=8$ is