# Solutions to Problem Set 31

Math 211-03

11-21-2017

[Parametric equations]

1. Two lines are given in parametric form:

Find the point where the lines intersect.

Set the two x-expressions equal:

Equate the expressions for y, then substitute :

Substitute this into and to get and .

The intersection point is .

2. Find the equation of the tangent line to

When , I have and . The point of tangency is .

Setting gives . The tangent line is

3. Find the equation of the tangent line to

When , I have and .

Setting gives . The tangent line is

4. Consider the parametric curve

(a) Find the value(s) of t for which the curve has a horizontal tangent line.

(b) Find the value(s) of t for which the curve has a vertical tangent line.

I have

(a) The tangent line is horizontal when , which occurs when .

(b) The tangent line is vertical when is undefined, which occurs when .

5. Find for the curve

When , I have .

6. Find for the curve

When , I have .

7. Find for the curve

When , I have

8. You can think of a parametric curve as being traced out by a particle moving in the x-y-plane.

The velocity vector of the curve is given by . It points in the direction of the particle's motion.

The speed is the length of the velocity vector, and is given by .

Find the velocity vector and the speed for

When , the velocity vector is .

At this point, the speed is .

One hears only those questions for which one is able to find answers. - Friedrich Nietzsche

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