Mathematics, 29.05.2021 05:50 TH3L0N3W0LF

The third-degree Taylor polynomial about x = 0 of In(1 - x) is

Answers: 3

Show answers

Answers

Answer from: tshort2000oyreb1

$\displaystyle P_3(x) = -x - \frac{x^2}{2} - \frac{x^3}{3}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Algebra I

FunctionsFunction Notation

Calculus

Derivatives

Derivative Notation

Derivative Rule [Quotient Rule]: $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Derivative Rule [Chain Rule]: $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

MacLaurin/Taylor Polynomials

Approximating Transcendental and Elementary functionsMacLaurin Polynomial: $\displaystyle P_n(x) = \frac{f(0)}{0!} + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ... + \frac{f^{(n)}(0)}{n!}x^n$ Taylor Polynomial: $\displaystyle P_n(x) = \frac{f(c)}{0!} + \frac{f'(c)}{1!}(x - c) + \frac{f''(c)}{2!}(x - c)^2 + \frac{f'''(c)}{3!}(x - c)^3 + ... + \frac{f^{(n)}(c)}{n!}(x - c)^n$

Step-by-step explanation:

*Note: I will not be showing the work for derivatives as it is relatively straightforward. If you request for me to show that portion, please leave a comment so I can add it. I will also not show work for elementary calculations.

Step 1: Define

Identify

f(x) = ln(1 - x)

Center: x = 0

n = 3

Step 2: Differentiate

[Function] 1st Derivative: $\displaystyle f'(x) = \frac{1}{x - 1}$ [Function] 2nd Derivative: $\displaystyle f''(x) = \frac{-1}{(x - 1)^2}$ [Function] 3rd Derivative: $\displaystyle f'''(x) = \frac{2}{(x - 1)^3}$

Step 3: Evaluate Functions

Substitute in center x [Function]: $\displaystyle f(0) = ln(1 - 0)$ Simplify: $\displaystyle f(0) = 0$ Substitute in center x [1st Derivative]: $\displaystyle f'(0) = \frac{1}{0 - 1}$ Simplify: $\displaystyle f'(0) = -1$ Substitute in center x [2nd Derivative]: $\displaystyle f''(0) = \frac{-1}{(0 - 1)^2}$ Simplify: $\displaystyle f''(0) = -1$ Substitute in center x [3rd Derivative]: $\displaystyle f'''(0) = \frac{2}{(0 - 1)^3}$ Simplify: $\displaystyle f'''(0) = -2$

Step 4: Write Taylor Polynomial

Substitute in derivative function values [MacLaurin Polynomial]: $\displaystyle P_3(x) = \frac{0}{0!} + \frac{-1}{1!}x + \frac{-1}{2!}x^2 + \frac{-2}{3!}x^3$ Simplify: $\displaystyle P_3(x) = -x - \frac{x^2}{2} - \frac{x^3}{3}$

Topic: AP Calculus BC (Calculus I/II)

Unit: Taylor Polynomials and Approximations

Book: College Calculus 10e

Answer from: Quest

17 1/3

step-by-step explanation:

Answer from: Quest

y=3n + 1

step-by-step explanation:

this is wrong because it says the second number is one less and i added one instead

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 14:30, gwynolegario

The first five terms of a linear sequence are given below. 7 , 12 , 17, 22 , 27 , what is the next term of the sequence?

Answers: 1

continue

Mathematics, 21.06.2019 19:00, princessjsl22

The pyramid shown has a square base that is 24 centimeters on each side. the slant height is 16 centimeters. what is the lateral surface area?

Answers: 2

continue

Mathematics, 21.06.2019 19:00, Mariaisagon9050

Will give brainliest plz hurry lexie surveyed 4 teenagers to find out approximately how many hours, per week, they read for fun, and how many siblings they have. brooke said she reads about 10 h a week for fun, and that she has 5 siblings. celia said she has 1 brother and 1 sister and reads for fun about 3 h each week. henry said he has 2 brothers and reads for fun about 30 min each week. jude said he doesn't read for fun and has 3 brothers and 3 sisters. use this information to complete the data table. express your answers as whole numbers or decimals. name hours of fun number of siblings brooke celia __ henry jude

Answers: 2

continue

Mathematics, 22.06.2019 00:00, nataliajaquez02

Jessie and bob are financing $425,500 to purchase a house. they obtained a 30/8 balloon mortgage at 6.55%. what will their balloon payment be?

Answers: 3

continue

You know the right answer?

The third-degree Taylor polynomial about x = 0 of In(1 - x) is...