Slope and area undo each other.

Step 1 of 4

A curve with a hill

So far you have worked with parabolas. The curve here is more involved: $f(x) = 0.2x^3 - 1.2x^2 + 1.8x + 0.5$$f(x) =$$0.2x^3 - 1.2x^2 + 1.8x + 0.5$⁠, with a hill and a dip. The brass line is the tangent at the point, and the blue slices under it are the rectangles of a Riemann sum.

Move Point x and find where the tangent lies level on the hill.

Step 2 of 4

Downhill

The derivative of this curve is $f'(x) = 0.6(x - 1)(x - 3)$$f'(x) =$$0.6(x - 1)(x - 3)$⁠. It is zero at the top of the hill and the bottom of the dip, and changes sign there.

Step 3 of 4

Narrow slices

Point x is now locked at $4$⁠, and the slices are $1$⁠ wide. The slice sum is shown on the board, and the exact area under the curve from $0$⁠ to $4$⁠ is $3.6$⁠.

Step 4 of 4

The fundamental theorem

Drag Thinner slices to Exact area. The pink curve is $A(x)$⁠, the area collected from $0$⁠ up to the point. A tangent touches it too, with a slope triangle $1$⁠ wide under it. Compare the triangle’s rise with the mint line from the axis up to the first curve, and move Point x.