How can I build a polynomial that looks like $f$ ?

Taylor Polynomials About $x=0$

12-16.

A POLYNOMIAL APPROXIMATION OF $f(x) = \cos(x)$

In Lesson 12.1.1, you approximated a function given information about its derivatives. However, what if you are not given the derivative information outright?

Generate an eighth-degree Taylor polynomial, $p_{8}$ , to approximate $f(x) = \cos(x)$ about $x = 0$ . An efficient way to determine the coefficient of each term is by organizing the derivatives in a table such as the one below.

$f(x)=$ $\cos(x)$	$f(0)=1$
$f^\prime (x)=$ $-\sin(x)$	$f^\prime(0)=0$
$f^{\prime\prime}(x) =$	$f^{\prime\prime}(0)=0$
$\ \ \ \ \vdots$
$f^8(x)=$

Use your graphing calculator to graph $y = p_8(x)$ and $y = f(x)$ on the same set of axes. On what interval is $p_{8}$ a good approximation of $f(x) = \cos(x)$ ?
Use sigma notation to write an equation for $p_{8}\left(x\right)$ .
Taylor polynomials are often used to approximate values of another function, $f$ , at specific values of $x = b$ where $b$ is located near the “center.” As with any approximation, a certain amount of error is expected. That error is determined by $| f ( b ) - p _ { n } ( b ) |$ .

Calculate the error if $p_{8}$ is used to approximate:
1. $\cos(0.5)$
1. $\cos(2)$
1. $\cos(3)$

12-17.

TWO WAYS TO FIND A POLYNOMIAL APPROXIMATION FOR $f(x) = \sin(x)$

Review each method outlined below. Then choose your favorite method to create a seventh-degree Taylor polynomial about $x = 0$ for $f(x) = \sin(x)$ . Explain your choice.

Method 1: Repeat the process you used in problem 12-16. Determine the initial value of the function and its first seven derivatives at $x = 0$ . Then write a seventh-degree Taylor polynomial, $p_7(x)$ , that approximates $f$ .

Method 2: You already know the equation of an eighth-degree Taylor polynomial for $f(x) = \cos(x)$ , centered at $x = 0$ . Use the fact that $\frac { d } { d x }\cos(x) = -\sin(x)$ and the Power Rule to write a seventh-degree polynomial that approximates $f(x) = \sin(x)$ .

12-18.

Let $p_7(x)$ represent a seventh-degree Taylor polynomial for $f(x) = e^x$ centered at $x = 0$ .

Represent $p_{7}\left(x\right)$ with sigma notation and in expanded form.
Sketch $y = p_7(x)$ and $y = f(x)$ over $–5 ≤ x ≤ 5$ and $–10 ≤ y ≤ 10$ . On what interval about $x = 0$ will $p_{7}$ approximate $f$ within $0.05$ units? In other words, on what interval is the error, $| f ( x ) - p _ { 7 } ( x ) | < 0.05$ ?

12-19.

Given $f ( x ) = \frac { 1 } { 1 - x }$ : Homework Help ✎

Write a fourth-degree Taylor polynomial, $p_4(x)$ , about $x = 0$ for $f(x)$ near $x = 0$ .
Write $p_4(x)$ using sigma notation.

12-20.

Examine the graphs of $p _ { 3 } ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 6 }$ and $q _ { 3 } ( x ) = 3 + ( x - 5 ) + \frac { ( x - 5 ) ^ { 2 } } { 2 } + \frac { ( x - 5 ) ^ { 3 } } { 6 }$ . 12-20 HW eTool (Desmos). Homework Help ✎

What is the relationship between the graphs of $y = p_3(x)$ and $y = q_3(x)$ ?
You are a contestant on “The Strongest Link” and you must name the coordinates of some point on $y = q_3(x)$ . What point do you name?

12-21.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int \frac { 1 } { x \operatorname { ln } ( 3 x ) } d x$

$\int e ^ { x } \operatorname { sin } ( 2 x ) d x$

$\int \frac { d x } { x ^ { 2 } - x }$

$\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } d x$

12-22.

Consider the infinite series below. For each series, decide if it converges conditionally, converges absolutely, or diverges and justify your conclusion. State the tests you used. Homework Help ✎

$\displaystyle \sum _ { k = 0 } ^ { \infty } \frac { k ^ { 2 } } { \sqrt { k ^ { 5 } + 7 } }$

$\displaystyle \sum _ { j = 2 } ^ { \infty } \frac { 1 } { j \sqrt { \operatorname { ln } ( j ) } }$

$\displaystyle \sum _ { j = 1 } ^ { \infty } j ^ { 10 } ( \frac { 9 } { 10 } ) ^ { j }$

$\displaystyle \sum _ { n = 4 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } - 1000 } }$

12-23.

A moving particle has position $(x(t), y(t))$ at time $t$ . At $t = 1$ , its position is $\left(6, 0\right)$ and the velocity vector at any time $t > 0$ is $\langle 1 - \frac { 2 } { t ^ { 2 } } , 1 + \frac { 1 } { t ^ { 2 } } \rangle$ . Homework Help ✎

What is the acceleration vector at $t = 2$ ?
For what time $t > 0$ does the velocity vector have a slope of $5$ ?

12-24.

Multiple Choice: The graph at right shows the growth of a population $P$ of sea lions introduced near a remote island after $t$ years. A reasonable differential equation to model this growth is: Homework Help ✎

$\frac { d P } { d t } = \frac { 1 } { 800 } P ( 400 - P )$

$\frac { d P } { d t } = \frac { 1 } { 400 } P ( 800 - P )$

$\frac { d P } { d t } = \frac { 1 } { 800 } P$

$\frac { d P } { d t } = \frac { 1 } { 400 } ( 800 - P )$

$\frac { d P } { d t } = \frac { 1 } { 800 } ( 400 - P )$

First quadrant, x axis labeled, t, scaled from 0 to 5, y axis labeled, P, scaled from 0 to 800, continuous curve with the following approximate critical points, starting @ (0, comma 10), passing through (1, comma 25), changing from concave up to concave down @ (2, comma 400), passing through (3, comma 750), leveling off @ (3.5, comma 790).

12-25.

Given the logistic differential equation from problem 12-24, determine the size of the population when it is changing most rapidly. What is the significance of this value? Homework Help ✎

12-26.

Multiple Choice: The arc length of $y = \tan(x)$ over $0 \leq x \leq \frac { \pi } { 4 }$ , is given by: Homework Help ✎

$\int _ { 0 } ^ { \pi / 4 } \sqrt { 1 + \operatorname { sec } ^ { 4 } ( x ) } d x$

$\int _ { 0 } ^ { \pi / 4 } \sqrt { 1 + \operatorname { tan } ^ { 2 } ( x ) } d x$

$\int _ { 0 } ^ { \pi / 4 } \operatorname { sec } ( x ) d x$

$\int _ { 0 } ^ { \pi / 4 } \operatorname { sec } ^ { 2 } ( x ) d x$

$\int _ { 0 } ^ { \pi / 4 } ( 1 + \operatorname { sec } ^ { 4 } ( x ) ) d x$

12-27.

Multiple Choice: Which of the following equations can be used to determine where $r(θ) = 5 - 4\cos(θ)$ has a horizontal tangent? Homework Help ✎

$\sin(θ) = 0$

$5\sin(θ) = 4\sin(2θ)$

$5\sin(θ) = 2\sin(2θ)$

$5\cos(θ) = 2\cos(2θ)$

$5\cos(θ) = 4\cos(2θ)$

Calculus Third Edition

How can I build a polynomial that looks like $f$ ?

Taylor Polynomials Centered at $\boldsymbol{x = 0}$

Calculus Third Edition

Taylor Polynomials Centered at x=0\boldsymbol{x = 0}

Taylor Polynomials Centered at $\boldsymbol{x = 0}$