Vector-Valued Functions: Ace AP Calculus BC Like a Pro

Defining and Differentiating Vector-Valued Functions

Abigail Young

6 min read

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Study Guide Overview

This study guide covers vector-valued functions. It begins with a review of vectors, including their magnitude, direction, and representation as $<x, y>$ . It then introduces vector-valued functions $r(t)=<f(t), g(t)>$ and how to find their derivatives $r'(t)$ . Finally, it provides practice problems on finding the derivative of vector-valued functions and applying them to particle motion (velocity vectors).

#9.4 Defining and Differentiating Vector-Valued Functions

So far in this unit, we’ve been delving deep into the functions of parametric equations. Now, we’re taking this knowledge and talking about vector-valued functions!

#↖️ Vector Review

A vector is a quantity that has both direction and magnitude. Magnitude refers to the length of a vector. All vectors have horizontal and vertical components. They are written and defined based on these two components. For example, a vector with a horizontal component of 5 and a vertical component of 4 would be written as $<5, 4>$ . While there are a couple of other methods of writing vectors, this kind of notation is used most frequently on the AP Calculus BC Exam!

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Example of Vector Notation

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Every vector is drawn with a head and a tail. The tail of the vector is the point at which the vector originates. If there is no additional information given, we can assume that the vector originates from (0,0) and that the tail of the vector is at (0,0). The head of the vector is the final point of the vector. In diagrams, the head of the vector is represented with an arrowhead.

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Illustration of a Vector

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The magnitude of the vector can be calculated based on the horizontal a...

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