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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 ycoordinate and the same first n derivatives as f(x) at x=a. 


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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. 


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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. 


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Definition
A Taylor Series is a power series that converges to a function on some interval of x. 


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What information do you get from a Taylor Series? 

Definition
You get the ycoordinate and an infinite number of derivatives of f(x) at x=a. 


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What is the nth derivative (at the center) of [image]? 

Definition
The nth derivative of this series is 1. 


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What is a Maclaurin Series? 

Definition
A Maclaurin Series is a Taylor Series centered at 0. 


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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. 


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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. 


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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. 

