What are multivariables, and what are they doing on the AP Calculus test? In this short review article you’ll discover what you need to know about multivariables for the AP Calculus BC exam!

## The AP Calculus Exams and Multivariables

By definition, a **multivariable** function is simply a function that has more than one (*multiple*) input and/or output variable. Happily, the only kinds of multivariable functions you might encounter on the test are those with one input — *t* — and two outputs — *x* and *y* — often called **parametric** or **vector** functions.

Moreover, multivariables only make an appearance on the AP Calculus BC exam, not the AB. (For a complete list of topics found on the BC exam, check out What Topics are on the AP Calculus BC Exam?)

### Parametric and Vector Functions

How do we track the motion of a rocket as it blasts off from the surface of the Earth? We must keep track of many different things. For each instant of time, *t*, the rocket will be at some horizontal distance, *x* = *f*(*t*) away from the station. But the height also changes with time according to a separate function, *y* = *g*(*t*).

A **parametric function** is a set of two (or more) functions all defined in terms of the same **parameter** (usually *t* for time). We usually express a parametric function as a list, *x* = *f*(*t*), *y* = *g*(*t*).

A **vector function** is the same concept except in the form of a **vector**: (*f*(*t*), *g*(*t*)).

### Example

Here is a typical parametric function:

And the same expression written as a vector function:

Both of the above functions have the same graph, which is an ellipse. Most graphing calculators are capable of graphing parametric functions.

## Derivatives of Parametric Functions

If *x* = *f*(*t*) and *y* = *g*(*t*) define a parametric function, then we find the derivative by the formula,

### Example

Find the slope of the ellipse (3 cos *t*, 2 sin *t*) at *t* = π/3.

Plugging in the given *t* = π/3,

## Velocity and Acceleration Vectors

If (*f*(*t*), *g*(*t*)) define the position of an object in terms of time *t*, then the following expressions define the **velocity** and **acceleration** vectors of the function at time *t*.

Note that the velocity vector involves the first derivatives while the acceleration vector involves second derivatives.

### Example

With our running example, the ellipse equations, we can derive expressions for the velocity and acceleration of an object traveling along the ellipse.

## Length of Parametric Curves

The formula for the **length** of a parametric curve *x* = *f*(*t*) and *y* = *g*(*t*) from *t* = *a* to *t* = *b* is:

### Example

Set up the formula for the length of the semi-ellipse defined by (3 cos *t*, 2 sin *t*) where 0 ≤ t ≤ π.

Using the formula for the length of a curve:

## Summary

Multivariables only show up on the AP Calculus BC exam, and only in the form of parametric or vector functions. The topics you should be aware of include:

- Graphing and interpreting graphs of parametric functions
- Finding the slope of a parametric function at a given point
- Finding velocity and acceleration vectors
- Computing the length of a parametric curve

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##### About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!

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