{"id":10064,"date":"2017-06-07T11:47:46","date_gmt":"2017-06-07T18:47:46","guid":{"rendered":"https:\/\/magoosh.com\/hs\/?p=10064"},"modified":"2017-06-06T11:56:12","modified_gmt":"2017-06-06T18:56:12","slug":"ap-calculus-bc-review-vector-valued-functions","status":"publish","type":"post","link":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-bc-review-vector-valued-functions\/","title":{"rendered":"AP Calculus BC Review: Vector-Valued Functions"},"content":{"rendered":"

One thing that sets the AP Calculus BC exam apart from the AB exam is the topic of vector-valued functions. The BC test has them, while the AB does not. In this article we will review how to graph, find derivatives and integrals, and interpret the meaning of vector-valued functions.<\/p>\n

What are Vector-Valued Functions?<\/h2>\n

A vector-valued function<\/strong> is like a typical function y<\/em> = f<\/em>(x<\/em>), except that there is more than one output<\/strong> value. In fact, a vector<\/strong> may be thought of as a list of multiple values, such as (1, 4, -2).<\/p>\n

On the AP Calculus BC exam, you will only encounter vector-valued functions having two outputs. You will see these as a pair of functions, x<\/em> = f<\/em>(t<\/em>) and y<\/em> = g<\/em>(t<\/em>), or equivalently, (f<\/em>(t<\/em>), g<\/em>(t<\/em>)).<\/p>\n

For example, (3 cos t<\/em>, 5 sin t<\/em>) is a vector-valued function. It specifies the x<\/em>-value (3 cos t<\/em>) and y<\/em>-value (5 sin t<\/em>) for any given input t<\/em>. <\/p>\n

\"Graph
The graph of a vector-valued function, x=3cos(t), y = 5sin(t).<\/figcaption><\/figure>\n

When there is a single input variable (like t<\/em> in the above example), then a vector-valued function is essentially the same thing as a set of parametric equations<\/strong>. <\/p>\n

Both of these concepts are basic types of multivariable functions<\/strong>. Check out AP Calculus Review: Multivariables<\/a> for more.<\/p>\n

Graphing<\/h2>\n

Let’s take a closer look at the example given above, (3 cos t<\/em>, 5 sin t<\/em>). How do you graph it?<\/p>\n

Well what I did to create the graph shown above was to use graphing software.<\/p>\n

In fact, most graphing calculators are capable of graphing vector functions. However, this feature will most likely be under term parametric<\/em>. Here is a good parametric graphing tutorial<\/a>.<\/p>\n

However what do you do on the parts of the exam that do not permit a calculator?<\/p>\n

Sketching the Graph Without a Calculator<\/h3>\n

Most often you will not need a detailed graph to answer any particular question. However, it’s good to have a few techniques up your sleeve to help you visualize a vector function when you need to.<\/p>\n

Some situations come up so often that you should probably memorize them.<\/p>\n