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x=f(t), y=f(t), a<t<b
has the "initial point" (f(a), g(a)) and the terminal point (f(b), g(b)).
0<t<2pi
x^2+y^2=cos^2(t)+sin^2(t)=1
dy/dt =(dy/dx)*(dx/dt)
If dx/dt does not equal zero, we can solve for dy/dx.
The parametric curve will have a horizontal tangent when dy/dt = 0
The parametric curve will have a vertical tangent when dx/dt =0 (provided that dy/dt do not equal zero at this point). No 0/0's.
d^2/dx^2=(d/dx)(dy/dx)=[d/dt(dy/dx)]/(dx/dt)
To find d^2y/dx^2 go back to dy/dx = (dy/dt)/(dx/dt) and replace y with dy/dx.
Determine concavity of a parametric curve
To do this, calculate the second derivative.
or simply:
=the integral of g(t)f'(t)dt from alpha to beta
L=fnInt(SQRT(1+(dy/dx)^2)))dx
In parametric terms we modify this:
(r, theta+pi)
(-r, theta) =
r^2=x^2+y^2
x=r*cos(theta), y=r*sin(theta)
r=2 produces a circle centered at O.
or more generally: F(r, theta)=0, consists of all points P that have at least one polar representation (r, theta), whose coordinates satisfy the equation.
x=r*cos(theta)=f(theta)cos(theta) , y =r*sin(theta) = f(theta)sin(theta)
To find a tangent line to a polar curve r=f(theta), we regard theta as a parameter and write its parametric equation as:
We locate the vertical tangents setting dx/dtheta equal to zero.
We locate the horizontal tangents by finding the points where dy/dtheta = 0, provided that dx/dtheta do not equal zero.
L=fnInt(SQRT(r^2+(dr/d-theta)))d-theta
Arc Length Polar
if we interchange x and y we obtain
An equation of the parabola with focus (0,p) and directrix y=-p is x^2=4py
The point located halfway between the focus and the directrix lies on the parabola; it is called the vertex.
A parabola is the set of points in a plane that are equidistant from a fixed point F (called the focus) and a fixed line (called the directrix).
x^2+(y-p)^2=|y+p|^2=(y+p)^2
We get an equivalent equation by squaring and simplifying.
x^2=4py
An equation of the parabola with the focus (0,p) and the directrix y=-p is...
It opens upward if p>0 and downward if p<0.
a=(1)/(4p), then the standard equation of a parabola becomes y=ax^2
What is a sequence?
A sequence can be thought of as a list of numbers written in a definite order:
Notice that for every positive integer N there is a corresponding number aN, and so...
a sequence can be defined as a function whose domain is the set of positive integers. But we typically write aN instead of f(n) (or f(x)) for the value of the function of the number n.
If limN->infinity aN exists, we say the sequence converges (or is convergent).
if we can make the terms aN as close to L as we like by taking N sufficiently large.
if n>N then the absolute value (aN-L)< epsilon.
if for every epsilon >0 there is a corresponding integer N such that:
then lim n-> infinity aN=L.
If lim x-> infin f(x)=L and f(n) = aN when N is an integer: