Step by step
The question
Question: find the horizontal tangents, when x = 2t and y = t³ − 3t².
1 dx/dt and dy/dt
The power rule, with t: dx/dt = 2.
And dy/dt = 3t² − 6t.
2 dy/dx
The slope dy over dx is dy over dt, divided by dx over dt: the dt's cancel.
So dy/dx is (3t² − 6t)/2.
3 Where dy/dt = 0
A horizontal tangent has slope 0: the top, dy/dt, is 0, and the bottom is not.
Every term has an t in it: take it out.
A product is 0 when one of its parts is 0.
So t = 0, or t = 2.
Check dx/dt is not 0 there, and find each point.
So the horizontal tangents touch at (0, 0) and (4, −4).
Careful!
Horizontal means the top, dy/dt, is 0.
dx/dt = 0 gives vertical tangents.
Draw the curve: for each t, the point (x, y).
The curve is flat at these points.
4 Check
Let's check with a calculator: move t a tiny bit around 0, and divide the change in y by the change in x.
About 0: flat.
Right!
Let's look back.
First: dx/dt and dy/dt.
Then: dy/dx.
Then: where dy/dt = 0.
Last: check.