Mathematics, 11.07.2019 00:00 2023greenlanden

Find the second degree taylor polynomial for f(x)= sqrt(x^2+8) at the number x=1

Answers: 1

Show answers

Answers

Answer from: kahmir15

$\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Algebra I

Functions

Function Notation

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

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

Taylor Polynomials

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

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) = √(x² + 8)

Center: x = 1

n = 2

Step 2: Differentiate

[Function] 1st Derivative: $\displaystyle f'(x) = \frac{x}{\sqrt{x^2 + 8}}$ [Function] 2nd Derivative: $\displaystyle f''(x) = \frac{8}{(x^2 + 8)^\bigg{\frac{3}{2}}}$

Step 3: Evaluate

Substitute in center x [Function]: $\displaystyle f(1) = \sqrt{1^2 + 8}$ Simplify: $\displaystyle f(1) = 3$ Substitute in center x [1st Derivative]: $\displaystyle f'(1) = \frac{1}{\sqrt{1^2 + 8}}$ Simplify: $\displaystyle f'(1) = \frac{1}{3}$ Substitute in center x [2nd Derivative]: $\displaystyle f''(1) = \frac{8}{(1^2 + 8)^\bigg{\frac{3}{2}}}$ Simplify: $\displaystyle f''(1) = \frac{8}{27}$

Step 4: Write Taylor Polynomial

Substitute in derivative function values [Taylor Polynomial]: $\displaystyle P_2(x) = \frac{3}{0!} + \frac{\frac{1}{3}}{1!}(x - c) + \frac{\frac{8}{27}}{2!}(x - c)^2$ Simplify: $\displaystyle P_2(x) = 3 + \frac{1}{3}(x - c) + \frac{4}{27}(x - c)^2$ Substitute in center c: $\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2$

Topic: AP Calculus BC (Calculus I + II)

Unit: Taylor Polynomials and Approximations

Book: College Calculus 10e

Answer from: Quest

the expected value of the random variable : 0, 1 and 2

step-by-step explanation:

Answer from: Quest

Suredude why not right

Another question on Mathematics

Mathematics, 21.06.2019 13:00

The surface area (sa) of a cube with a as the length of each of its sides is given by the formula . if the surface area is known, how can you rewrite the formula to find its side?

Answers: 2

Answer

Mathematics, 21.06.2019 19:30

The volume of a cube-shaped shipping container is 13,824 cubic inches. find the edge length of the shipping container.

Answers: 1

Answer

Mathematics, 21.06.2019 22:30

What is the graph of the absolute value equation ? y=|x|-5

Answers: 1

Answer

Mathematics, 21.06.2019 23:30

Aparking garage charges $1.12 for the first hour and 0.50 for each additional hour. what is the maximum length of time tom can park in the garage if we wishes to play no more than $5.62

Answers: 1

Answer

You know the right answer?

Find the second degree taylor polynomial for f(x)= sqrt(x^2+8) at the number x=1...

Questions

Mathematics, 03.02.2021 01:20

By substituting the given values for x calculate the y value for the following equation y=x+2

Mathematics, 03.02.2021 01:20

The length of the rectangle is 6 feet. The width is three times the length. What is the total area of the rectangle?...

Mathematics, 03.02.2021 01:20

A lamppost is 12 feet high and casts a 6-foot shadow. At the same time of day, a lightpole directly behind the lamppost casts a 25-foot s...