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AP®MathsCollege BoardCalculus BCStudy GuidesUnit 9: Parametric Equations, Vector-Valued Functions & Polar CoordinatesVector-Valued FunctionsDefining Vector-Valued Functions

Defining Vector-Valued Functions (College Board AP® Calculus BC)

Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Defining vector-valued functions

What is a vector-valued function?

  • A vector-valued function is a pair of functions, x open parentheses t close parentheses and y open parentheses t close parentheses, that are grouped together using vector notation, open angle brackets space comma space close angle brackets, in the form

open angle brackets x open parentheses t close parentheses comma space y open parentheses t close parentheses close angle brackets

  • One input, t equals t subscript 0 , gives two outputs grouped together in a vector: open angle brackets x open parentheses t subscript 0 close parentheses comma space y open parentheses t subscript 0 close parentheses close angle brackets

    • x open parentheses t subscript 0 close parentheses is called the first component

    • y open parentheses t subscript 0 close parentheses is called the second component

Examiner Tips and Tricks

When working with vector-valued functions, answers must be given in vector notation.

Worked Example

Evaluate the vector-valued function open angle brackets t squared plus 2 comma fraction numerator space 20 over denominator square root of t end fraction close angle brackets when t equals 4.

Substitute t equals 4 into both components and simplify

open angle brackets 4 squared plus 2 comma fraction numerator space 20 over denominator square root of 4 end fraction close angle brackets equals open angle brackets 16 plus 2 comma fraction numerator space 20 over denominator 2 end fraction close angle brackets

Give your final answer in vector notation

open angle brackets 18 comma space 10 close angle brackets

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.