Unit 15
Calculus.
How fast something changes, the derivative, and how much it adds up to, the integral, joined by the fundamental theorem.
Before this unit
The unit, lesson by lesson
Lessons
4 lessons · 17 practice questions · about 2 hours
Average rate of change
A line has one slope, and a curve does not. Measure the average rate between two points with a secant, and see what happens as the two points close in.
The derivative
The tangent at a point and its slope, the derivative. Why the derivative of x² is 2x, the rule for any quadratic function, and why the slope is zero at the vertex.
The area under a curve
The area under a line is a triangle, and under a curve you estimate it with narrow rectangles. The antiderivative gives it exactly, and area below the axis counts as negative.
Slope and area undo each other
One curve with a hill and a dip, a tangent moving along it and area piling up beneath it. The fundamental theorem joins the two questions, because the rate at which the area grows is the height of the curve.