Download Parametric Equations

Review
Parametric Equations
x  f (t)
 parametric equations of a curve with parameter
y  g(t) 
ex. Graph the following parametric equations and find the
Cartesian equation of the curve that the equations trace.
x  t 3  2, y  2t 2 , 0  t  3
t.

ex.
x  2cos2 sin t, y  sin sin t , 0  t  2
Polar Coordinates
x 2  y2  r 2
x  r cos
y  r sin 
y
tan  
x
Convert between polar and Cartesian
 7 
r,


5,

ex.   
 6 



ex. x, y   3 3, 3

ex. The Cartesian equation for
ex.
is:
:
ex. Plot the curve

r 2 16cos(2 )
Complex Numbers
z  x  iy
z  re
i
Cartesian form
Polar form
ex.
2i
z
4  2i
ex.
z  1  i
12
ex.
z  12 1 i
Basic Vectors
1. Find a vector between points A(3, 4) and B(-5, 1).
2. Find point C which is equally distant from point A as point B
in the direction of vector <-4, -3>.
3. Is vector AC orthogonal to AB? parallel? What is the angle
between AC and AB?
Understand dot and cross products.
Lines and Planes
The parametric equations of a line are:
x  a  v1t
y  b  v2 t
z  c  v 3t
The equation of a plane is
is the normal vector and
ex.
n1 x  n2 y  n3 z  n  p where n
p
is the vector formed by the point.
Vector-valued Functions
Be able to differentiate and integrate vector-valued functions.
y  0
Projectiles: maximum height
y 0
maximum range
If non constant gravity, be able to integrate from acceleration
at   0,g to find velocity and position.


Formulas
v
T
Unit tangent vector;
v
s  t v dt
t2
Arclength:
1

Unit normal vector:

Curvature:

T 
N
T 

T 
v
v t  at 
v t
or
at   aT T  aN N
Components of acceleration:
aT 
dv
dt

va
v

3
2
aN  v  
va
v