The PDF has "Calculus is the study of the properties of functions." Better would be: Calculus is some mathematics for analyzing systems that change continuo

The PDF has "Calculus is the study of the properties of functions." Better would be: Calculus is some mathematics for analyzing systems that change continuously, usually over time. The first motivation, by Newton, was to analyze the motion of the planets. Since then, calculus is central in nearly all of of physics, most of engineering, a lot of chemistry, much of the rest of mathematics, and also probability and statistics. Calculus works, first, with real valued functions of a real variable. An example would be the speed of a baseball as a function of time. Later calculus works with real valued functions of several real variables. The first important concept in calculus is 'rate of change' or the 'first derivative' of a function. The second important concept is the integral which in effect 'adds up' changes. Integration can be used both to define and to find areas and volumes, e.g., the volume or the surface area of a sphere. Or if apply a force of 500 pounds to a car weighing 3000 pounds, how much time will the car need to accelerate from 0 to 60 MPH? The fundamental theorem of calculus is that, with mild assumptions, if take the first derivative of a function and integrate that, then get back the original function. Sometimes we have some information about the first derivative of a function and want to find the function; in this way we have a start on the subject differential equations. Some of the elementary parts of differential equations are also in a first course in calculus. Here is a simple differential equation: Suppose t denotes time in days and our revenue per day is y(t). Suppose we know the revenue now, that is, at t = 0. So, we know y(0). Suppose we believe that we will have saturated our market when our revenue reaches b per day. Suppose we believe that the rate of growth of revenue is directly proportional to both the number of our current customers and also the number of our target customers we do not yet have. Calculus denotes the rate of growth as y'(t). Then for some constant k we have y'(t) = k y(t) (b - y(t)) Then with integration in calculus, we can find y(t). What we get, depending on k, is a lazy S curve that rises slowly, then more rapidly, then slowly, and approaches b from below. So, this is an elementary case of differential equations. At one point, this differential equation and its solution kept FedEx from going out of business. Differential equations are important in calculating trajectories of space craft, how to get a fighter jet to 60,000 feet in least time, the analysis of analog circuits in electronic engineering, Maxwell's equations in electricity and magnetism, the differential geometry of general relativity, Newton's second law, the heat equation, the stiffness of beams, etc. Calculus is one of the main pillars of mathematics, science, engineering, technology, and civilization.