/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Determine the third Taylor polyn... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 1

Determine the third Taylor polynomial of the given function at \(x=0\). $$f(x)=\sin x$$

Short Answer

Expert verified
The third Taylor polynomial of \(f(x) = \sin x\) at \(x = 0\) is \(P_3(x) = x - \frac{x^3}{6}\)

Step by step solution

01

- Understand the Taylor Series Formula

Recall the Taylor series formula for a function centered at 0: \[ P_n(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \dots + \frac{f^{(n)}(0)x^n}{n!} \]We need the third Taylor polynomial, meaning we will use terms up to the third derivative.
02

- Evaluate the Function and Its Derivatives at 0

Compute the first few derivatives of \(f(x) = \sin x\) and evaluate them at \(x=0\):\(f(x) = \, \sin x\)\(f(0) = \, \sin 0 = 0\)\(f'(x) = \, \cos x\)\(f'(0) = \, \cos 0 = 1\)\(f''(x) = -\sin x\)\(f''(0) = -\sin 0 = 0\)\(f'''(x) = -\cos x\)\(f'''(0) = -\cos 0 = -1\)
03

- Substitute the Values into the Taylor Series Formula

Using the values found in Step 2, substitute them into the Taylor series formula:\[ P_3(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} \]Substitute the known values:\[ P_3(x) = 0 + 1 \cdot x + \frac{0 \cdot x^2}{2!} + \frac{(-1) \cdot x^3}{3!} \]Simplify the polynomial:\[ P_3(x) = x - \frac{x^3}{6} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor series formula
The Taylor series is a way to represent a function as an infinite sum of terms. Each term is calculated from the values of the function's derivatives at a single point. Here, we'll focus on the Taylor series centered at 0, also known as the Maclaurin series.
The formula looks like this:
\begin {equation}P_n(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \text{...} + \frac{f^{(n)}(0)x^n}{n!}\text{.}\text{\text {(\text{\text{(
In this formula:
  • \(P_n(x)\) is the nth Taylor polynomial of \(f(x)\)
  • \(f(0)\) is the value of the function at 0
  • \(f'(0)\) is the value of the first derivative at 0
  • \(f''(0)\) is the value of the second derivative at 0
  • The list goes on depending on how many terms you want to include
  • \((n!)\) signifies the factorial of \(n\)
This formula helps us approximate functions at specific points with polynomials, making calculations easier in many cases.
Function derivatives
To build a Taylor polynomial, you need to find the derivatives of the function and evaluate them at the point where the polynomial should be centered. Let's look at these steps for the sine function:
  • First derivative: \(f(x) = \, \text {sin } x\), so \(f'(x) = \, \text {cos } x\). Evaluating at \(x=0\): \(f'(0) = \, \text{cos } 0 = 1\).
  • Second derivative: \(f''(x) = -\text {sin } x\). Evaluating at \(x=0\): \(f''(0) = -\text {sin } 0 = 0\).
  • Third derivative: \(f'''(x) = -\text {cos } x\). Evaluating at \(x=0\): \(f'''(0) = -\text {cos } 0 = -1\).
Remember, each derivative level corresponds to a different term in the Taylor polynomial. We stop at the third derivative for this case because we're looking for the third Taylor polynomial.
Sin function
The sine function \( \text {sin } x \) is one of the fundamental trigonometric functions. Here are some important properties related to the Taylor series:
  • The sine function oscillates between -1 and 1 over its input range.
  • It is periodic with a period of \( 2\text{蟺} \), meaning \( \text {sin}(x + 2\text{蟺}) = \text {sin }(x) \).
  • The derivatives of the sine function cycle every four terms, as shown in the previous section.
    To summarize, the Taylor series for \( \text {sin } x \) around 0 (the third Taylor polynomial) is as follows:\begin {equation}P_3(x) = x - \frac{x^3}{6}\text{.}This gives a polynomial that approximates the sine function near x=0. Polynomials are often easier to work with than trigonometric functions in calculations, which make these expansions very useful in practice.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Elimination of a Drug A patient receives 2 milligrams of a certain drug each day. Each day the body eliminates \(20 \%\) of the amount of drug present in the system. After extended treatment, estimate the total amount of the drug present immediately before a dose is given.

The sum of the first \(n\) odd numbers is \(n^{2}\); that is, $$ \sum_{x=1}^{n}(2 x-1)=n^{2} . $$ Verify this formula for \(n=5,10\), and 25 .

Rely on the fact that $$ \lim _{n \rightarrow \infty} \frac{|x|^{n+1}}{(n+1) !}=0 $$ The proof of this fact is omitted. Let \(R_{n}(x)\) be the \(n\) th remainder of \(f(x)=\cos x\) at \(x=0\). (See Section 11.1.) Show that, for any fixed value of \(x,\left|R_{n}(x)\right| \leq|x|^{n+1} /(n+1) !\), and hence, conclude that \(\left|R_{n}(x)\right| \rightarrow 0\) as \(n \rightarrow \infty .\) This shows that the Taylor series for \(\cos x\) converges to \(\cos x\) for every value of \(x\).

Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) $$\sum_{k=1}^{\infty} \frac{1}{(3 k)^{2}}$$

Find the Taylor series at \(x=0\) of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at \(x=0\) of \(\frac{1}{1-x}, e^{x}\), or \(\cos x .\) These series are derived in Examples 1 and 2 and Check Your Understanding Problem \(2 .\) $$\frac{x}{1+x^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.