CALCULUS OF PARAMETRIC EQUATIONS

In the parametric representation of a curve, when you talk about the first derivative, you must specify which one you mean. There are dy / dt, dx / dt, and dy / dx. The meaning of each of these is summarized as follows:

• dy / dt : This is the instantaneous rate of change of y with respect to t. If you want to know the relative maximum or minimum values of y as a function of t, you would use this derivative. In many applications, this derivative also has physical meaning. For instance, if (x(t),y(t)) is the position of a object moving in space as a function of time t, then dy / dt would represent the vertical velocity. Likewise, d ² y / dt ² would represent the vertical acceleration.
• dx / dt : This is the instantaneous rate of change of x with respect to t. If you want to know the relative maximum or minimum values of x as a function of t, you would use this derivative. Again, if (x(t),y(t)) is the position of a object moving in space as a function of time t, then dx / dt would represent the horizontal velocity. Likewise, d ² x / dt ² would represent the horizontal acceleration.
• dy / dx : This is the instantaneous rate of change of y with respect to x. If you want to know the relative maximum or minimum values of y as a function of x, you would use this derivative. This would give you the relative maximum or minimum values you are used to working with. However, it is possible that a relative maximum or minimum value of y with respect to t not be a relative maximum or minimum value of y with respect to x. dy / dx will be zero or undefined when either dy / dt or dx / dt is zero or undefined. dy / dx will always give you the value of the slope of the tangent line to the curve for each value of t.

THEOREM: Let C be a smooth curve defined by (x(t),y(t)), then

All the above derivatives and differential arc length will be a function of t. To evaluate them or to integrate them in the case of arc length, you must use t values.

1. Let a curve be defined by

1. Graph the function for t on [-1,1]
2. For what values of t does the graph cross the origin.
3. Find dy / dt
4. Find dx / dt
5. Find dy / dx
6. Find the equation of the tangent line for those values of t found in part b. Add the the graphs of the tangent line(s) to your graph in part a.
7. Find all relative extrema of the graph.
8. Find d ² y / dx ²
9. Find the concavity of the graph for those values of t found in part b
10. Find ds.
11. Calculate the arc length of the curve on [ -1 / 2 , 1 / 3].

2. Let a curve be defined by x(t) = (1 + 2cost) cost and y(t) = (1 + 2cost) sint.

1. Graph the function for t on [- ,]
2. For what values of t does the graph cross the origin.
3. Find dy / dt
4. Find dx / dt
5. Find dy / dx
6. Find the equation of the tangent line for those values of t found in part b. Add the the graphs of the tangent line(s) to your graph in part a.
7. Find all relative extrema of the graph.
8. Find d ² y / dx ²
9. Find the concavity of the graph for those values of t found in part b
10. Find ds.
11. Calculate the arc length of the curve on [- ,].

3. Find the area of the region inside the curve defined by x(t) = 4cost and y(t)=3sint.

1. Graph the function on [- ,]
2. What symmetry can you use to simplify the problem?
3. Set up a representative vertical rectangle and add it to the graph.
4. In terms of t, what is the height of the rectangle?
5. In terms of t, what is the base of the rectangle?
6. What limits on t will give you the portion of the curve you want?
7. Set up the integral and evaluate the area.

In a similar fashion, you can evaluate surface area and volume of revolutions, centroids, work, etc. You will do more of this in Calculus III.

HOMEWORK ASSIGNMENT:

REQUIRED PROBLEMS	Section 9.3: 2, 10, 16, 24, 36, 40, 42
SUGGESTED PROBLEMS	Section 9.3: 7, 11, 21, 31, 37, 49, 51, 57, 67

An informative web site:

• http://archives.math.utk.edu/visual.calculus/5/areaparam.3/ Example of an area for parametric equations by Larry Husch at University of Tennessee.

READING ASSIGNMENT BEFORE NEXT WORKSHEET: Section 9.4