AP Physics C is a calculusbased physics course, but the fact is that you can do pretty well, gradewise, 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.
Finding the derivative of a function ("differentiating" in calculus language) is a rulebased 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.
Rule in English 
Rule in Math. Notation 
Example 

The derivative of a constant is zero. 

If x(t) = 5, then v(t) = 0. If v(t) = 3, then a(t) = 0. 
The derivative of t is one. 

If x(t) = t, then v(t) = 1. 
The derivative of t to a power is the power times t to the "one less" power. 

If x(t) = t^{2}, then v(t) = 2t^{1} = 2t. (n = 2) If v(t) = t^{4}, then a(t) = 4t^{3}. (n = 4) If x(t) = t^{3}, then v(t) = 3t^{4}. (n = 3) 
The derivative of the sine of t is the cosine of t. 

If x(t) = sin t, then v(t) = cos t. 
The derivative of the cosine of t is the negative of the sine of t. 

If v(t) = cos t, then a(t) = sin t. 
Rule in English 
Rule in Math. Notation 
Example 

The derivative of a constant times a function equals the constant times the derivative of the function. 

If x(t) = 3t^{2}, then v(t) = 3(2t^{1}) = 6t. (c = 3 and u = t^{2}) If v(t) = 4sin t, then a(t) = 4cos t. (c = 4, u = sin t) 
The derivative of the sum (or difference)of two functions is the sum (or difference) of their derivatives. 

If x(t) = t + sin t, then v(t) = 1 + cos t. (u = t, w = sin t) If v(t) = t^{2}  4t, then a(t) = 2t^{1}  4(1) = 2t  4. (u = t^{2}, w = 4t) 
The derivative of a composite (one function inside another function) function equals the derivative of the "outside" function leaving the "inside" function alone, times the derivative of the "inside" function. (The Chain Rule) 

If x(t) = (t + 2)^{2}, then v(t) = 2(t + 2)^{1}(1 + 0) = 2(t + 2). (u = t^{2}, w = t + 2) If v(t) = sin(2t^{3}), then a(t) = cos(2t^{3})(2)(3t^{2}) = 6t^{2}cos(2t^{3}) (u = cos t, w = 2t^{3}) 
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.

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". 
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.
Rule in English 
Rule in Math. Notation 
Example 

The antiderivative of a constant is the constant times t. 


The antiderivative of t to a power is t to the powerplusone, divided by the power plus one. 


The antiderivative of the sine of t is the negative of cosine t. 


The antiderivative of the cosine of t is the sine of t. 


Rule in English 
Rule in Math. Notation 
Example 

The antiderivative of a constant times a function equals the constant times the antiderivative of the function. 


The antiderivative of the sum (or difference) of two functions equals the sum (or difference) of their antiderivatives. 


Rule In English 
Rule in Math. Notation 
Example 

If F(t) is an antiderivative of f(t), then the definite integral from a to b of f equals the function F evaluated when t equals b minus F evaluated when t equals a. 
