Calculus for AP Physics


AP Physics C is a calculus-based physics course, but the fact is that you can do pretty well, grade-wise, in AP Physics C - particularly the mechanics part - without any knowledge of calculus. Of course it won't hurt you to know what is going on with all those strange mathematical symbols and concepts, and there will definitely be problems on the AP Test that require a knowledge of calculus. Fortunately, you can get by really well - even on the AP Test - with very rudimentary calculus skills that can be learned in a few minutes.

You need to know something about:


A derivative is a rate of change, which, geometrically, is the slope of a graph. In physics, velocity is the rate of change of position, so mathematically velocity is the derivative of position. Acceleration is the rate of change of velocity, so acceleration is the derivative of velocity. Net force is the rate of change of momentum, so the derivative of an object's momentum tells you the net force on the object. These are only a few of the applications of the derivative in physics.

Derivative Rules:

Finding the derivative of a function ("differentiating" in calculus language) is a rule-based operation. In other words, you need to recognize what derivative rule applies, and then apply it. In order to recognize what derivative rule applies, you need to know some derivative rules. The tables below list the derivative rules that you will use in this course, and shows some examples of their use. These rules are stated using "t" as a variable (the derivative is "with respect to" t, in calculus language), since most of the functions that we will use are functions of time. If you are taking a derivative whose variable is "s," simply substitute "x" for "t" in the derivative rule.

Definite Integrals:

A definite integral represents an area, and evaluating a definite integral ("integrating" in calculus language) is the inverse of finding a derivative - like subtraction is the inverse of addition. In physics, the area under a velocity vs. time graph represents displacement, so the definite integral of velocity gives displacement. The area under an acceleration vs. time graph equals change in velocity, so the definite integral of acceleration tells you the change in velocity. The area under a force vs. position graph equals work done by the force, so the definite integral of force (with respect to position) tells you the work done by the force. There are many, many more applications of the definite integral in physics.

area as an integral

The diagram above shows the relationship between the definite integral notation and the area it represents. "a" and "b" - called "limits of integration" go at the bottom and top of the big "S". "f(t)" is the function being integrated (the "integrand"), and "dt" says that "t" is the variable being used.

The notation is read "the definite integral from a to b of f of t, dt".

Rules for Definite Integrals:

Just like you can subtract 5 from 12 by thinking, "What do I have to add to 5 to get 12?", you can evaluate definite integrals by thinking "What function would I have to differentiate to get the function in this integral?" That function is called the "integral" or "antiderivative." The mathematical symbol for the antiderivative looks like the definite integral without the limits of integration.

last update November 26, 2007 by JL Stanbrough