Term
| What is an nth order Taylor Polynomial generated by f(x) at x=a? |
|
Definition
| An nth order Taylor Polynomial generated by f(x) at x=a is a polynomial that shares the same y-coordinate and the same first n derivatives as f(x) at x=a. |
|
|
Term
| What is a Taylor Polynomial used for? |
|
Definition
| A Taylor Polynomial is used to approximate a function. It's like a tangent line on steroids. |
|
|
Term
| What is the difference between a Taylor Polynomial and a Taylor Series |
|
Definition
| A Taylor Polynomial has a finite number of terms. A Taylor Series has an infinite number of terms. |
|
|
Term
|
Definition
| A Taylor Series is a power series that converges to a function on some interval of x. |
|
|
Term
| What information do you get from a Taylor Series? |
|
Definition
| You get the y-coordinate and an infinite number of derivatives of f(x) at x=a. |
|
|
Term
What is the nth derivative (at the center) of
[image]? |
|
Definition
| The nth derivative of this series is 1. |
|
|
Term
| What is a Maclaurin Series? |
|
Definition
| A Maclaurin Series is a Taylor Series centered at 0. |
|
|
Term
| How do you use a Taylor Polynomial to approximate a function? |
|
Definition
| Find the equation of the desired Taylor Polynomial, then substitute the value of x. |
|
|
Term
| What is the relationship between a tangent line at x=a and a Taylor Series at x=a? |
|
Definition
| The tangent line is equivalent to the first order Taylor Polynomial. |
|
|
Term
| What are the six methods for creating a power series that converges to a function? |
|
Definition
1) Use the sum of a geometric series.
2) Differentiate a convergent series.
3) Integrate a convergent series.
4) Substitute an expression of x for x.
5) Multiply (or divide) by an expression of x.
6) Use the Taylor Series general term to create a Taylor series from scratch. |
|
|
