Parametric Curves

Question 1

What is a vector function (example)

Answer 1

Like the path of an object through time
Gamma (vector): R-> R2 or R3
γ(t) = (x(t), y(t))

Question 2

What are vector functions sometimes called

Answer 2

Parametric curves

Question 3

How does the information in parametric curves compare with normal functions

Answer 3

They contain way more info because they have time

Like a circle has direction

Question 4

How are parametric curves useful

Answer 4

They provide a vocabulary to describe curves that are next to impossible to describe in cartesian coordinates

Question 5

How do we extend ideas of limits, continuity, differentiation, etc to vector functions?

Answer 5

We apply single variable definition to each element
Ex. Derivative is tangent vector
Ex. Magnitude of tangent vector is speed (like sqrt(x^2+y^2))

Question 6

Arclength

Answer 6

Arclength = S(b,a)speed•time

=S(b,a)||dγ/dt||dt

Question 7

Chain rule for parametric curves

Answer 7

dy/dx = (dy/dt)/(dx/dt) if dx/dt isn’t 0
dy/dx is magnitude of slope of tangent line
(Regular one is tangent vector)