Bennett Laxton Austin Mahlum Julia Callaghan Samantha Biehl
Bennett Laxton Austin Mahlum Julia Callaghan Samantha Biehl
Overview
Brook Taylor: The man behind the numbers 1685 - 1731 English mathematician best known for Taylor’s theorem and the Taylor series Also wrote On the Jewish Sacrifices and On the Lawfulness of Eating Blood
Taylor Polynomial Review The Taylor series can express a function in polynomial form, near some local value Taylor Series Expansion for a function of one variable:
Conditions Of Taylor Polynomial
Example of first-order Taylor polynomial
Example cont’d Blue line shows the approximation we just found. ^zooming out
1 st Degree Taylor Polynomial: 2 Variables
2 nd Degree Taylor Polynomial: 2 Variables
Verification of Partial Derivatives
Verification of Partial Derivatives
Verification of Partial Derivatives
Example 1
Example 1
Example 1 Graph
Example 2
Example 2
Example 2
Example 2 Graph
Multiple Variables
For the 2 nd Order Taylor polynomial
Hessian Matrix
Multiple Variables
Example cont’d
Applications Evaluating definite integrals Understanding asymptotic behavior Understanding the growth of functions Solving differential equations
Application: Evaluating a Definite Integral
Evaluating a Definite Integral Cont’d.
Summary
Works Cited http: //www. intmath. com/series-expansion/1 -taylorseries. php http: //www. ugrad. math. ubc. ca/coursedoc/math 101/no tes/series/apps. Taylor. html http: //mathinsight. org/taylors_theorem_multivariable_ introduction Stewart Calculus textbook
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- Slides: 34