zuai-logo

What is a Taylor series?

An infinite sum of terms expressed in terms of the function's derivatives at a single point.

Flip to see [answer/question]
Flip to see [answer/question]

All Flashcards

What is a Taylor series?

An infinite sum of terms expressed in terms of the function's derivatives at a single point.

What is a Maclaurin series?

A Taylor series centered at x=0x=0.

Define f(n)(a)f^{(n)}(a) in the context of Taylor series.

The nthn^{\text{th}} derivative of the function f(x)f(x) evaluated at x=ax=a.

What is the nthn^{\text{th}}-order Taylor polynomial?

The nthn^{\text{th}} partial sum of the Taylor series.

What is a Taylor Approximation?

Using a Taylor polynomial to estimate the value of a function at a specific point.

What is the general formula for a Taylor series approximation of f(x)f(x) at x=ax=a?

∑n=0inftyf(n)(a)n!(x−a)n\sum_{n=0}^infty \frac{f^{(n)}(a)}{n!}(x-a)^n

Expand the Taylor series formula.

f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+f′′′(a)3!(x−a)3+...+f(n)(a)n!(x−a)nf(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+...+\frac{f^{(n)}(a)}{n!}(x-a)^n

What is the formula for the nthn^{th} term of a Taylor series?

f(n)(a)n!(x−a)n\frac{f^{(n)}(a)}{n!}(x-a)^n

What is the Maclaurin series formula?

∑n=0inftyf(n)(0)n!xn\sum_{n=0}^infty \frac{f^{(n)}(0)}{n!}x^n

What is the formula for the third-degree Maclaurin polynomial for e5xe^{5x}?

1+5x+252x2+1256x31+5x+\frac{25}{2}x^2+\frac{125}{6}x^3

How do you find a Taylor polynomial approximation?

  1. Find derivatives of f(x)f(x). 2. Evaluate derivatives at x=ax=a. 3. Plug values into Taylor series formula. 4. Simplify.

Steps to find a Maclaurin polynomial.

  1. Find derivatives of f(x)f(x). 2. Evaluate derivatives at x=0x=0. 3. Plug values into Maclaurin series formula. 4. Simplify.

How to find the third-degree Taylor polynomial for f(x)=ln(x)f(x) = ln(x) about x=1x = 1?

  1. Find f′(x)f'(x), f′′(x)f''(x), f′′′(x)f'''(x). 2. Evaluate at x=1x=1. 3. Use the Taylor polynomial formula up to the third degree. 4. Simplify: (x−1)−12(x−1)2+13(x−1)3(x-1)-\frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3.

How to find the fifth-degree Maclaurin polynomial for f(x)=cos(x)f(x) = cos(x)?

  1. Find derivatives up to order 5. 2. Evaluate at x=0x=0. 3. Use Maclaurin polynomial formula. 4. Simplify: 1−x22+x4241 - \frac{x^2}{2} + \frac{x^4}{24}.