taylor - Taylor’s Formula in Single-Variate Calculus...

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Unformatted text preview: Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! Taylor’s Formula The Beast, Two-Variable Version! Shubhodeep Mukherji Multivariate Calculus Independent Study Science Academy of South Texas March 11, 2011 Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! Taylor Polynomials Math! Recall linearization from your Calculus I days Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! Taylor Polynomials Math! Recall linearization from your Calculus I days It is a polynomial approximation of a function around the desired point Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! Taylor Polynomials Math! Recall linearization from your Calculus I days It is a polynomial approximation of a function around the desired point Represents a function as a series of polynomials Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! Taylor Polynomials Math! Recall linearization from your Calculus I days It is a polynomial approximation of a function around the desired point Represents a function as a series of polynomials Taylor Polynomial of Order n Suppose f ( x ) has derivatives of all orders throughout some interval containing the point x = a . If so, then the n th order taylor polynomial generated by f ( x ) at x = a is the polynomial P n ( x ) = f ( a )+ f ( a )( x- a )+ f 00 ( a ) 2! ( x- a ) 2 + ··· + f ( k ) ( a ) k ! ( x- a ) k + ··· + f ( n ) ( a ) n ! ( x- a ) n . Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! The Power of a Taylor Polynomial !!! Math! Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! The Error Term Math! Since the taylor polynomial is an approximation, there is bound to be an error Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!! The Error Term Math! Since the taylor polynomial is an approximation, there is bound to be an error This error term is also called the remainder of order n Shubhodeep Mukherji Taylor’s Formula Taylor’s Formula in Single-Variate Calculus Basics of Multi-Cal Second Derivative Test Linear Approximation The Fun Stuff !!!...
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This note was uploaded on 08/03/2011 for the course ECON 101 taught by Professor Profeessor during the Spring '11 term at Aachen University of Applied Sciences.

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taylor - Taylor’s Formula in Single-Variate Calculus...

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