Applications of Calculus

The integral symbol, ∫, in the heading for this section indicates that it is meant to be read by students in calculus-based physics. Students in an algebra-based physics course should skip these sections. The calculus-related sections in this book are meant to be usable by students who are taking calculus concurrently, so at this early point in the physics course I do not assume you know any calculus yet. This section is therefore not much more than a quick preview of calculus, to help you relate what you're learning in the two courses.

Newton was the first person to figure out the tangent-line definition of velocity for cases where the x - t graph is nonlinear. Before Newton, nobody had conceptualized the description of motion in terms of x - t and v - t graphs. In addition to the graphical techniques discussed in this chapter, Newton also invented a set of symbolic techniques called calculus. If you have an equation for x in terms of t, calculus allows you, for instance, to find an equation for v in terms of t. In calculus terms, we say that the function v(t) is the derivative of the function x(t). In other words, the derivative of a function is a new function that tells how rapidly the original function was changing. We now use neither Newton's name for his technique (he called it "the method of fluxions") nor his notation. The more commonly used notation is due to Newton's German contemporary Leibnitz, whom the English accused of plagiarizing the calculus from Newton. In the Leibnitz notation, we write

to indicate that the function v(t) equals the slope of the tangent line of the graph of x(t) at every time t. The Leibnitz notation is meant to evoke the delta notation, but with a very small time interval. Because the dx and dt are thought of as very small Δx's and Δt's, i.e., very small differences, the part of calculus that has to do with derivatives is called differential calculus.

Differential calculus consists of three things:

• The concept and definition of the derivative, which is covered in this book, but which will be discussed more formally in your math course.

• The Leibnitz notation described above, which you'll need to get more comfortable with in your math course.

• A set of rules for that allows you to find an equation for the derivative of a given function. For instance, if you happened to have a situation where the position of an object was given by the equation x = 2t⁷, you would be able to use those rules to find dx/dt = 14t⁶. This bag of tricks is covered in your math course.