What you have is the degree 2 term (almost). What's the sign?skyturnred said:Homework Statement
Here is the question:
I don't quite know what I did wrong. My method is below.
Homework Equations
The Attempt at a Solution
f(x)=√x
f'(x)=[itex]\frac{1}{2(x)^{1/2}}[/itex]
f''(x)=[itex]\frac{-1}{(2)(2)(x^{3/2}}[/itex]
a=4
f(a)=2
f'(a)=1/4
f''(a)=[itex]\frac{-1}{4(4^{3})^{1/2}}[/itex]
so wouldn't the second Taylor polynomial be what I put in the answer field above? Thanks.
SammyS said:
A Taylor series is a representation of a function as an infinite sum of terms, where each term is a polynomial expression of increasingly higher order. This series can be used to approximate the value of a function at a certain point or to express the function in a simpler form.
To express a function as a Taylor series, you need to find the derivatives of the function at a specific point (known as the center of the series) and plug them into the general formula for a Taylor series. The general formula is f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ..., where f'(a), f''(a), and f'''(a) represent the first, second, and third derivatives of the function at the point a, respectively.
Expressing a function as a Taylor series allows us to approximate the value of the function at any point in its domain. This can be useful for functions that are difficult to evaluate directly, as the Taylor series provides a simpler representation of the function. It also allows us to analyze the behavior of the function around a specific point.
The center of the Taylor series is important because it determines the point at which the function is being approximated. The closer the center is to the point of interest, the more accurate the approximation will be. If the center is too far from the point of interest, the Taylor series may not accurately represent the function.
To express the root x as a Taylor series about a=4, you first need to find the derivatives of the function at the point a=4. Then, plug these derivatives into the general formula for a Taylor series to get the series representation of the function. The resulting series will be a polynomial expression of increasingly higher order, with a=4 as the center.