Calc Week 8 Flashcards
parametric equations and curves
Let I be an interval and f: I -> R be a continuous function whose domain is I. Also let g: I-> R be continuous.
The set of points of the form
(x,y) = (f(t), g(t)) for some t in I is the graph of the parametric equations
x=f(t)
y=g(t)
Here, t is called a parameter
A curve is a graph along with its parametric equations
graph of parametric equations
The set of points of the form
(x,y) = (f(t), g(t)) for some t in I is the graph of the parametric equations
x=f(t)
y=g(t)
parameter
Here, t is called a parameter
notations
set builder
{(cos(t),sin(t)) | 0 <= t <= 2pi}
{(x,y) | x^2 + y^2 = 1}
{ x= t^2
y = t+ 1 for t in I =….
(x,y) = (…,…)
plotting parametric equations
1.make a table of t’s, x=…, and y = …
2. fill out table to find points for each t
3.plot, draw line
what do parametric equations allow us to do
plot curves which are difficult to graph using y = f(x) or x = f(y) because do not pass vertical line test
singularity
the curve crosses over itself
convert from cartesian to parametric from
(x,y) = (input (usually t), function) for interval
convert from parametric to cartesian form
eliminate the variable t
solve for t
plug in t to find x or y
for interval…
ellipse equation
1/a^2 (x-h)^2 + 1/b^2 (y-k)^2 = 1
center (h,k)
horizontal radius a
vertical radius b
smooth
a curve C given by the parametric equations
x = f(t)
y = g(t)
for t ranging over an interval I is said to be smooth on I if
f’ and g’ are continuous on I and
never simultaneously zero on the interior of the interval (not including endpoints)
derivative equation for parametric equations
dy/dx(t) = dy/dx / dx=dt = g’(t) / f’(t)
criteria for derivative to exist
functions f and g are both differentiable and f’(t) != 0
tangent line equation
tangent line of C at t=s is the line
y - g(s) = g’(s)/f’(s) (x - f(s))
normal line equation
normal line to C at t=s is the line
y - g(s) = -f’(s)/g’(s) (x - f(s))
