In Lecture: More 10.1 (parametric equations), Office hours today: 2:00-3:00 in PDL C-326

Today:

Read Textbook 10.1, 2.1-2.2WORK ON HOMEWORK (in Webassign). Hwk 1 closes Friday night (parametric eqs. + some tangents)

To Do:

Quiz section tomorrow Office hoursMath Study Center (in CMU B-014, 9:30am-9:30pm)CLUE (MGH Commons, see link on website for schedule)

Have Questions?

I Parametric Curves

Wednesday, April 1

What curve is parameterized by the following parametric equations?

A) a lineB) a parabola

C) an ellipse

D) a circle

E) a sinusoidal curve

F) no idea

Question 2:

What curve is parameterized by the following parametric equations?

(hint: find the Cartesian equation of this curve)

.

Question 1:

Example:

Suppose an object moves counterclockwise along the unit circle, at constant velocity, traveling a full rotation per minute, starting at the bottom of the circle at time .

Let's find the parametric equations for its motion.

radius of the circle

where:

.

The standard parameterization ofUniform Circular Motion is:

What kind of curve and motion is given by each of the following parameterizations:

a)

b)

f)

d)

(Try it: http://graphsketch.com/parametric.php)

.

More examples:

g)

Which of the following parameterizations are correct for an object moving on a circle of radius 3, centered at (0,0), starting at the top of the circle, and moving in clockwise direction?

A)

C)

D)

B)

Question:

Suppose that the position of one particle at time t is given by

and the position of a second particle is given by

a) Sketch the paths of the two particles, and find their intersection points.

b) Do the particles collide?

One more Example:

Parametric equations are any set of equations and that give the and coordinates of points on a curve separately, in terms of a parameter .

1.

(a) Select various values of and evaluate the functions to find the corresponding points .(b) Plot these points and indicate the corresponding time and direction of movement.

2. To graph a curve given by parametric equations:

(a) Solving for in one equation and substituting in the other to get an eq. in x and y.(b) Use some identity to combine the equations (ex: ).

3. To find the xy-equation for the curve traced out by the motion, try to eliminate the parameter by:

Extra data: a parameterized curve encodes not only the coordinates of the points on the curve, but also a certain motion along the curve.

Can study more complicated curves, which do not have nice Cartesian equations

4. Why study parameterized curves?

5. Types of parametrizations we studied:

a) Uniform Linear Motion (motion along a line at constant speed):

where are the coordinates at and are the horizontal and vertical velocities (rates of change of each coordinate with respect to time)

b) Uniform Circular Motion (An object moving on a circular path at constant velocity

radius of the circle

where:

Summary