Calculus AP Physics C

Origins of Calculus Basic algebra had been the tool used to analyze motion for thousands of years, but there were many situations in which algebra was not technical enough. Isaac Newton and Gottfried Leibniz developed a sophisticated language of numbers and symbols called Calculus to explain more complicated motion. Newton began his work first but it was Leibniz who first published his findings. Naturally, both led the other towards accusations of plagiarism.

What does Calculus Do? Calculus is simply very advanced algebra and geometry that has been tweaked to solve more sophisticated problems. What separates AP Physics 1 from AP Physics C is that we can use calculus to analyze motion that algebra just cant handle. We are mostly concerned with calculating rates of change and areas under graphs. If a line is straight this task is simple, but what about a curve? Calculus lets us “zoom in” on a curve and apply algebra to a very small section of the curve.

Slopes of Tangent Lines Consider the function to the right. Is the slope of this function constant? NO! How would you find the slope at one point on the function? Calculus! The slope of a line tangent to a point on the function is the slope of the function at that point. To find this slope (rate of change), we must take what is called a derivative.

Derivatives The derivative of a function gives a new function that tells you the slope anywhere along the original function. This means that taking a derivative does not necessarily tell you how steep a function is at a particular point. To find the slope at that point, plug in the x coordinate to your derivative function and BOOM AWESOME MATH.

Limit Definition of a Derivative In Calculus, you will first learn to take derivatives by using what is called the limit definition. This involves evaluating the slope of a secant line as it becomes closer and closer to a tangent line. This is very important from a math standpoint, but we will skip straight to the shortcut.

Derivative Rules The set of rules to the right give us a starting point for taking derivatives of functions. As the year goes on, we will add various things to this list (trig functions, exponential and log functions). Remember, taking a derivative gets you a new function. To find the slope at a point, evaluate the derivative at that point.

Examples 𝑥 𝑡 =3𝑡, 𝑡=3 𝑣 𝑡 = 4𝑡 3 + 2𝑡 2 −𝑡, 𝑡=5 𝑦 𝑥 = 2 𝑥 3 , 𝑥=1 Take the derivatives of the following functions, and evaluate them at the points indicated. 𝑥 𝑡 =3𝑡, 𝑡=3 𝑣 𝑡 = 4𝑡 3 + 2𝑡 2 −𝑡, 𝑡=5 𝑦 𝑥 = 2 𝑥 3 , 𝑥=1

Integrals The opposite of a derivative is called an integral. An integral sums up a function that has been evaluated over some range of values. Graphically, an integral represent the area bound by a function and the horizontal axis. Integrals are essentially the “reverse” of a derivative. We are taking the anti-derivative. In Calculus, you will go through various methods to calculate integrals, but in physics we will again move ahead towards the shortcut.

Integral Rules There are many techniques for integration, but most of the integrals we evaluate in physics will be using the power rule. The +c is very important; it tells us that there are possibly initial conditions that were lost when a derivative was originally taken. Sometimes integrals will need to be evaluated over a range of values, we call these our integration limits. There are only TWO things you will be asked to do. DERIVE – Simply find a function, which do not require limits. EVALUATE – Find the function and solve using a given set of limits.

Examples Integrate the following functions, and evaluate the functions over the limits. 0 5 3 𝑡 2 +2𝑑𝑡 −2 2 10 𝑡 2 −2𝑡𝑑𝑡

Summing it Up. Get it. Summing. An Integral is a sum Summing it Up. Get it? Summing? An Integral is a sum. Whatever I thought it was funny.