Slopes of tangent lines

a. Find all points where the following curves have vertical and horizontal tangent lines.

b. Find the slope of the lines tangent to the curve at the origin (when relevant).

c. Sketch the curve and all the tangent lines identified in parts (a) and (b).

r2 = 2 cos 2θ

Solution 17RE

Step 1:

Given = 2 cos 2θ

(a) First determine x and y in terms of r and θ in order to find dy/dx

Recall x = r cos θ and y = r sin θ where = 2 cos 2θ

So x =(2 cos 2 cos θ = 2 cos 2 cos θ

And y =( 2 cos 2θ) sin θ= ( Here the parameter is θ.)

Calculate the derivative provided that ≠ 0

dy/dθ = -2 cos 3

dx/dθ =

So

Step 2:

(b)Result for horizontal tangents:

So for dy/dθ = 0 = -2 cos 3

yields

Substitute values in r =we get,

Hence the curve has horizontal tangents at the points are

Result for vertical tangents:

So for dx/dθ = 0

yields

Now substituting the initial values of , for each approximate form of we get

Substitute values in we get,

Hence the curve has horizontal tangents at the points are

(,