# Exploring Curve Sketching

Questions to Play With:

First, show only the first derivative. Watch how the blue curve behaves as the tangent line moves along the green curve.

When the tangent line to the green curve has a positive slope, where is the blue curve in relation to the x-axis?

When the tangent line to the green curve has a negative slope, where is the blue curve in relation to the x-axis?

At what points on the green curve is the blue curve crossing the x-axis?

Think about why it makes sense that the blue curve (first derivative) and the green curve (original function) are related in this way.

Now, show only the second derivative. Watch how the gold curve behaves as the tangent line moves along the green curve.

When the gold curve is above the x-axis, what shape is the green curve?

When the gold curve is below the x-axis, what shape is the green curve?

As the gold curve crosses the x-axis, where is the tangent line located on the green curve?

Think about why it makes sense that the gold curve (second derivative) and the green curve (original function) are related in this way.