Points, Vectors, and Functions Exercises

Answer

• We want to trace the unit circle starting from the usual point and going in the same direction as usual. However, we want to trace it very slowly, taking 8π radians to go around instead of 2π. Take the usual parameterization and make the period of each equation 4 times longer:

• Since we want to take 8π radians to go around the full circle, we need 0 ≤ t ≤ 8π.

Answer

• We want to start at (-1, 0) and go around the circle clockwise. The value of y will be doing the same thing it always does: going from 0 up to 1, down to -1, then back up to 0.

We can leave the equation for y unchanged.

The value of x usually goes from 1 to -1 and back to 1.

Instead, we want to go from -1 to 1 and back to -1.

That is, we want x to be the negative of what it usually is. We need to stick a negative sign on the front of the equation for x.

The new parameterization is

x(t) = -cos t

y(t) = sin t.

Since we're going around the circle at the normal speed, we have 0≤ t ≤ 2π.

Answer

• We want to start at (0, 1) and go around the circle counterclockwise from t = 0 to t = 4π.

From this example we know that switching the "normal" equations traces out the circle clockwise startingfrom (0, 1).

0≤ t ≤ 2π

Start with these equations. We want y to stay the same, moving from 1 down to -1 and back up to 1:

However, we want x to be the opposite. We want x to first move to -1, then to + 1, then back to 0:

If we stick a negative sign in front of the x equation and leave the y equation unchanged, we'll draw the circle in the correct direction. The equations

x(t) = -sin t

y(t) = cos t

for 0 ≤ t ≤ 2π produces the circle starting at the correct point and drawn in the correct direction:

There's one thing left to address: the speed. Right now we're going from 0 to 2π to draw the circle. That's not long enough. We want to take from 0 to 4π. To fix this, double the period of each equation. The final parameterization is

for 0 ≤ t ≤ 4π.