Quiz about Taylor Polynomial

Question 1

What is the 1st order Taylor Polynomial for the function $$f\left(x\right)=\cos\ x$$ , with a center at $$\frac{\pi}{6}$$ ?

Accepted Answer

$$\frac{\sqrt[]{3}}{2}-\frac{1}{2}\left(x-\frac{\pi}{6}\right)$$

Answer

$$-\frac{1}{2}\left(x-\frac{\pi}{6}\right)$$

Answer

$$\frac{\sqrt[]{3}}{2}+\frac{1}{2}\left(x-\frac{\pi}{6}\right)$$

Answer

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

Question 2

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 3

Let $$f$$ be the function that has derivatives of all orders for all real numbers. Assume $$f\left(0\right)=6,$$ Write a third-order Taylor Polynomial for $$f$$ at $$x=0$$ and use it to approximate $$f\left(0.5\right)$$ .

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

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

Accepted Answer

0.002

Answer

2.667

Answer

0.266

Answer

0.026

Question 5

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)?

Accepted Answer

Option C is correct

Answer

A.

Answer

B.

Answer

D.

Question 6

This is a third-degree Taylor polynomial for a function f about x = 1. What is the value of $$f$$ ?

Accepted Answer

480

Answer

$$\frac{40}{3}$$

Answer

80

Answer

240

Question 7

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 8

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

Accepted Answer

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

Answer

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

Answer

$$\frac{1}{12}$$

Answer

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

Question 9

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

Accepted Answer

$$-\frac{1}{144}$$

Answer

$$-\frac{1}{72}$$

Answer

$$-\frac{1}{9}$$

Answer

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

Question 10

Find the approximate value of $$e^2$$ using the fifth-degree Maclaurin polynomial for $$f\left(x\right)=e^x.$$

Accepted Answer

7.267

Answer

7.135

Answer

7.356

Answer

7.389

Question 11

The maximum error incurred by approximating the sum of the series \frac{1}{3!}-\frac{2}{4!}+\frac{3}{5!}-\frac{4}{6!}+... by the sum of the first four terms is

Accepted Answer

0.000992

Answer

0.000148

Answer

0.00119

Answer

0.00555

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