Download - Lecture 7: Taylor Series Validity, Power Series, Vectors, Dot Products, and Cross Product
Lecture 7: Taylor Series Validity, Power Series, Vectors, Dot
Products, and Cross Product
Part I: Validity of Taylor Series
Taylor Series Failure
• Consider the function if , if .• for all n, so the Taylor series is just 0
Taylor series remainder term• Recall that • When the equation fails, it is because of a
“remainder term” at infinity• Similarly, • so
• This process can be continued.
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
• Def: A power series is a series of the form • Examples:
• Taylor Series around
Radius of Convergence• The radius of convergence of a power series is
the number such that converges if and diverges if (the behavior at is undertermined)
• Example: For , • Some power series, such as , converge for all .
In this case, .• Some power series, such as , diverge for all . In
this case, .
Manipulating Power Series• As long as we are within the radius of convergence
of a power series, we can differentiate, integrate, and make substitutions in power series.
• Cool example:• for .• Plugging in for , for • Integrating gives for
Uniqueness of Power/Taylor Series• Fact: The Power/Taylor series for a given
function is unique.• Reason: Completely determined by the fact
that it has to match the nth derivative of the function
• Corollary: If we find a power series for a function by any means, it must be valid!
• Plugging in ,
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
• A vector is described by n coordinates .• We write• The magnitude of , written as , is
Picture for vectors
• A vector can be thought of as an arrow.
• is the length of the arrow
0 1 2 3 4-1-2-3-4
01234
-1-2-3-4
5 6-5-6
-5-6
56
𝑣=¿3,5>¿
Scalar Multiplication of Vectors and Vector Direction
• If then • The direction of a vector , denoted as , is • is the vector with length 1 which points in the
same direction as .
Vector Addition and Subtraction
• If and then • If and then
Displacement and Position Vectors
• Given points and , • The position vector is the displacement vector
from the origin.• If is the origin and then • Proposition: Given points P,Q,R,
Dot Product
• If and then • Example:
Properties of the dot product
• Linearity in and :• Commutativity:
Geometric Picture of the dot product
• , where is the angle between and .• Note that is the length of the projection of
onto
𝜃𝑣
�⃗�
projection of onto
Connection between algebraic and geometric pictures
• Law of cosines:
a
bc A
B C
• Proof: and algebra gives the result• Algebraically,
|�⃗�+𝑤|2=(�⃗�+�⃗� ) ∙ (�⃗�+�⃗� )=|⃗𝑣|2+|�⃗�|2+2 �⃗� ∙�⃗�• Take and
P Q
R
Condition for Perpendicular Vectors
• Corollary: if and only if
Cross Product• The cross product is defined for 3 dimensions.• Definition: If , , and then
• Example:
Properties of the cross product
• Linearity in and :• Anti-Commutativity:• Warning:
Geometric Picture of the Cross Product
• has magnitude equal to the area of the parallelogram with sides and
• is perpendicular to both and • Right-hand rule: To find the direction of , point
your right thumb in the direction of . Then point your other fingers in the direction of . will point in the direction which goes outward from your palm.