Although there is no upper limit on the subject matter of calculus, the bottom is defined by a number of factors, including notations, calculations, equations, and proofs. But it turns out that the real foundation consists of ideas, and that you can bypass all of the rest in order to concentrate on the questions that calculus is designed to answer, which can be represented in simple visual geometry. Later on, I can show you how to do some of the apparatus in Turtle Art, too, such as limits of infinite sequences, sums of infinite series, and graphical solutions to the simple differential equations that define various common functions.

A line that touches a curve in only one point either intersects the curve or is tangent to it. In the case of a tangent, the direction of the tangent line is defined to be the direction of the curve at the point of tangency, as in this Turtle Art example. Later on we will want to measure that direction, but we don’t have to do that in order to understand the definition.

It then turns out that we can graph a measure of the direction of a curve, giving a new curve and a new function.

It is visually obvious, but still needs to be proved, that a continuous function that is negative at one point and positive at another must be 0 somewhere in between. This is the First Mean Value Theorem.

It is also reasonably obvious that a continuous smooth curve that is 0 at two points must have one or more points in between where its tangent is level. This is the Second Mean Value Theorem, which follows from the First.

Similarly we can talk about the highest and lowest points on a curve, and points where the curve levels out and then resumes its earlier direction.

These are the essential topics in differential calculus, from which all of the rest follows.

The other major problem at the foundation of calculus is area inside a curve, such as the area of a circle. It is convenient to consider curves representing single-valued functions, and thus to calculate the area of half a circle at a time. In this diagram, we have a curve over an interval on the x axis, taken with the x axis and two vertical lines at the ends to delimit an area. In the classroom, we can draw such curves on paper, cut them out, and weigh them on sensitive scales. A little bit of arithmetic, dividing by the weight of a known area of the paper, allows us to find their area. Or we can make an outline in clay, fill it to a measured depth with water, and then pour the water into a measuring cup.

Some areas are easy to do with Euclidean geometry, such as the area of a triangle, which is one half times the base times the altitude.

Once we know how to integrate a function between two endpoints, we can allow one endpoint to vary, giving us a new function. Let us try this with the equation y = x.

It turns out that the two problems of direction of a curve and area inside a curve are inverses, in a very specific sense. Look at this diagram.

Having laid all of this out, we now have to start over and think about how to explain these ideas to preschool children. We will have to ask them how much of it makes sense to them, and see which explanations work for them.

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