Midterm 2 + Previous Topics Flashcards
(24 cards)
What is the Taylor Series?
A function f(x) is equal to the sum of its derivatives at a point a: f(x) = sum from n=0 to infinity of f^(n)(a) * (x-a)^n / n!
What is a Maclaurin Series?
A Taylor Series centered at a = 0
Define Power Series.
A sum of the form sum from n=0 to infinity of c_n*(x-a)^n; converges within a radius R
What is the Interval of Convergence?
The set of x-values for which a power series converges
What is an Alternating Series?
A series whose terms alternate in sign: a_1 - a_2 + a_3 - …
What does the Alternating Series Test state?
If a_n is decreasing and lim a_n = 0, then the series sum (-1)^n * a_n converges
What is a Taylor Polynomial?
A polynomial approximation of f(x) near x = a using n derivatives
What is the Remainder/Error Term?
The error between a function and its Taylor Polynomial: R_n(x) = f(x) - P_n(x)
Define Polar Coordinates.
A coordinate system using (r, theta), where r is the radius and theta is the angle
Convert Polar to Cartesian coordinates.
x = rcos(theta), y = rsin(theta)
Convert Cartesian to Polar coordinates.
r = sqrt(x^2 + y^2), theta = arctan(y/x)
What is the Area in Polar Coordinates?
(1/2) * integral from a to b of [r(theta)]^2 dtheta
What is a Parametric Curve?
A curve defined by x = f(t), y = g(t) where t is a parameter
What is the Arc Length of a Parametric Curve?
L = integral from a to b of sqrt((dx/dt)^2 + (dy/dt)^2) dt
Define Vectors.
An object with both magnitude and direction, written <x, y> or <x, y, z>
How is Vector Addition performed?
Add corresponding components: <x1, y1> + <x2, y2> = <x1+x2, y1+y2>
What is Scalar Multiplication?
Multiply each component by the scalar: k * <x, y> = <kx, ky>
Define the Dot Product.
<x1, y1> · <x2, y2> = x1x2 + y1y2; result is a scalar
What is the relationship between Dot Product and Angle?
a · b = |a||b|cos(theta); used to find angle between vectors
What are Orthogonal Vectors?
Two vectors a and b are orthogonal if a · b = 0
How is Vector Projection calculated?
proj_b(a) = (a · b / |b|^2) * b
What is the Cross Product?
Only in 3D: a × b = a vector perpendicular to both a and b
How is a Line in 3D defined?
Defined by a point and a direction vector: r(t) = r0 + tv
