Lecture 7 Taylor Series Validity Power Series Vectors

Lecture 7: Taylor Series Validity, Power Series, Vectors, Dot Products, and Cross Product

Part I: Validity of Taylor Series

Taylor Series Failure •

Taylor series remainder term •

Part II: Power Series

Objectives • Know what power series are and how to determine the radius of convergence of power series • Know how power series can be manipulated within their radius of convergence Corresponding sections of Simmons: 14. 2, 14. 3

Power Series •

Radius of Convergence •

Manipulating Power Series •

Uniqueness of Power/Taylor Series •

Part III: Vectors

Objectives • Know what vectors are • Know how find the magnitude, direction, dot product, and cross product of vectors. Corresponding sections in Simmons: 18. 1, 18. 2, 18. 3

Vectors •

Picture for vectors • 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Scalar Multiplication of Vectors and Vector Direction •

Vector Addition and Subtraction •

Displacement and Position Vectors •

Dot Product •

Properties of the dot product •

Geometric Picture of the dot product •

Connection between algebraic and geometric pictures • R A c P b C B a Q

Condition for Perpendicular Vectors •

Cross Product •

Properties of the cross product •

Geometric Picture of the Cross Product •