Calculus

Three ways to ask about a curve

A graph can answer different questions. What value does it approach near a missing point? How steep is it at one instant? How much area lies beneath it? The courses separate these questions before showing how they fit together. The notation matters, but so does explaining what a calculation measures. A slope is not an area, and the value of a function at a point need not be its limit there. Tables, formulas and movable graphs give different kinds of evidence for an answer.

Nearby values before instantaneous rates

Begin with Limits. Its units introduce the two questions behind calculus, limits of functions, one-sided and infinite limits, limit laws and continuity. A function with a hole makes the distinction between approaching a point and reaching its value visible. Check values from both sides rather than trusting a single row of a table.

Continue with Derivatives. The difference quotient becomes a tangent slope as the separation between points approaches zero. The course then treats the derivative as a function, develops the power, product, quotient and chain rules, and uses derivatives for rates of change and maxima and minima. Read the sign of the derivative alongside the original curve instead of treating differentiation as a rule-copying exercise.

Finish with Integrals. Rectangles first approximate an area; Riemann sums and the midpoint rule make the approximation systematic. The definite integral and the Fundamental Theorem of Calculus connect accumulation with differentiation. The final unit considers net change and average value, so an integral has a meaning beyond the space under a drawing.

Move the points, then explain the numbers

In the secant lab for x² at a = 1, move the second point and compare the slope 2 + h with the limiting slope 2. In the derivative lab, drag along x³ − 3x and watch the point on the derivative curve follow the current slope. In the integration lab, change the number of rectangles and inspect how the area estimate changes. Try predicting the result before touching a control, then use the displayed quantities to explain any disagreement.

Conditions that a shortcut can hide

A limit can exist even when the function value is missing or different. Continuity does not guarantee differentiability: corners and vertical tangents need attention. A zero derivative is not automatically a maximum or minimum, as a flattened cubic shows. The derivative of a product is not the product of the derivatives, and a chain rule calculation needs the inside derivative. Keep these distinctions beside the worked questions. They explain why an apparently familiar formula can produce the wrong answer when its conditions are ignored.