Calculus Is Easier than You Think
Calculus is the mathematics of motion. In this volume we restrict ourselves to the simplest type of motion, constant velocity motion, and we see that calculus for constant velocity motion is simple indeed.
Calculus in a nutshell is the formula distance equals velocity times time, or distance = velocity * time. Surprisingly, you won't discover this in your high school or college calculus courses. Let's look in more detail:
Differential calculus is the calculation of derivatives, or, in plain English, velocities. Well, velocity = distance / time does it (for constant velocity motion).
Integral calculus is the calculation of the area under a velocity curve for a given interval. A velocity curve (for constant velocity motion) is a horizontal line, and the area under the curve is given by the formula for the area of a rectangle, that is, height * length. The height of the velocity curve is the velocity, the length of the interval is the time, and hence, the value of the integral is velocity * time.
The Fundamental Theorem of Calculus states that the integral of a velocity curve for a time interval is the distance traveled in the interval. Well, we've seen that the value of the integral is velocity * time, so the Fundamental Theorem of Calculus is velocity * time = distance (for constant velocity motion).
Differential equations are, in plain English, velocity equations. If we have an equation for, that is, know, the velocity for constant velocity motion, and we know the starting position of the motion, it is easy to use the formula distance = velocity * time to compute the position of the motion, that is, to solve the differential equation.
Is all this a parlor trick? No, the calculus of constant velocity motion really is the foundation for all calculus, and Calculus Without Tears is the only calculus method that presents the fundamentals in this easy to understand way, in CWT Volume 1, Constant Velocity Motion. We go beyond constant velocity motion in Volumes 2 and 3.