Question

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work.

x = cos θ,         y = cos 3θ

Answer 1

Step 1:

The curve is , .

Find the points on the curve where the tangent line is horizontal or vertical.

Slope of the horizontal tangent line is 0.

Slope of the vertical tangent line is .

Find the slope of the curve.

Consider .

Differentiate on each side with respect to .

Consider .

Differentiate on each side with respect to .

Step 2:

Slope of the tangent line is first derivative of the curve.

Slope of the curve is

Substitute  and in the above equation.

Slope of the tangent line is .

Answer 2

Contd......

Step 3:

Slope of the horizontal tangent line is 0.

Consider .

Consider .

Now substitute in and  .

Now substitute in and .

The points on the curve where the tangent line is horizontal are and .

Step 4:

Slope of the vertical tangent line is.

Since the slope is not defined, there is no vertical tangent lines to the given curve.

Step 5:

To check the result, graph the curve.

The graph of the curve , is :

Observe the graph of the curve notice that, the curve has horizontal tangent lines at the points and .

Solution:

The points on the curve where the tangent line is horizontal are and .

There is no vertical tangent lines to the given curve.