Parametric Equations, Polar Coordinates, and Vector-Valued Functions

There are many different kinds of functions in math because not everything in the world exists on a plane with two variables. So far, everything we have been doing has been on the Cartesian plane: ℝ^2. This is also known as the xy-plane. However, some functions that model the world around us are better graphed using other types of planes, which we will explore in this unit. This unit makes up 11-12% of the AP Calculus BC Exam.

As you are reading through this guide, pay special attention to the formulas mentioned. This unit is very formula-heavy, and ideally you should have all these formulas memorized, but some of them you can derive on the exam.

#9.1 Defining and Differentiating Parametric Equations

Parametric functions are a way to express a relationship between variables in the form of an equation that involves time. We will often use parametric functions to express the position of an object moving in space, or to describe the shape of a curve.

A parametric equation is typically written in the form:

x = f(t) y = g(t)

where x and y are the coordinates of a point on the curve, and t represents time. By changing the value of t, we can trace out the entire curve defined by the parametric equations. In a parametric function, both the x and y variables are dependent variables, and time is the independent variable.

To find the derivative of a parametric function, we need to find the derivative of x(t) and y(t) and set y'(t) over x'(t). When we do this, the dt's cancel out and we are left with the derivative dy/dx.

#9.2 Second Derivatives of Parametric Equations

As with equations in the Cartesian plane, we can take the second derivative of a parametric function. The process for finding the second derivative is a bit different than the process you are used to. We use the chain rule after finding the first derivative to arrive at this equation for the second derivative of a parametric function:

Notice how inside the parentheses, the formula states we need to find dy/dx, not dy/dt. This means that to find the second derivative, you must first find the first derivative with respect to x, then take the derivative of the first derivative (usually using the quotient rule), then set all of that over the first derivative of x(t). As you can see, there are quite a few steps involved, but with some practice, you will master second derivatives in no time.

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