Mastering Domain & Range of Multivariable Functions | Calculus 3 Lesson 39 📘

How to Find the Domain & Range of Multivariable Functions (Calculus 3 Lesson 39)

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In this video you will learn about multivariable functions, or functions of several independent variables. You’ll be introduced to their notation, what defines a multivariable function, and what their graphs look like. Additionally, we talk about combinations of multivariable functions and what a composite function may look like. Finally, you’ll learn how to find the domain and range of multivariable function as well as graph the domain. We also briefly talk about how to graph multivariable functions by recognizing special surfaces, but the actual techniques have been learned in previous lessons (planes, cylinders, quadric surfaces), so we will not look at any examples.

This video series is designed to help students understand the concepts of Calculus 3 at a grounded level. No long, boring, and unnecessary explanations, just what you need to know at a reasonable and digestible pace, with the goal of each video being shorter than the average school lecture!

Calculus 3 requires a solid understanding of concepts from calculus 2, calculus 1, precalculus, and algebra. This includes limits, differentiation, advanced integration techniques, applications of derivatives and integrals, polar coordinates, parametric equations, trigonometric functions, logarithms, factoring, equation manipulation, graphing, and much more. If you are not familiar with these prerequisite topics, be sure to learn them first!


Video Chapters:
0:00 What is a Multivariable Function?
5:51 Single Variable vs Multivariable Functions
12:49 Definition of Multivariable Functions
17:09 Example: Evaluating Multivariable Functions
19:36 Combinations & Composite Multivariable Functions
23:05 How to Find Domain & Range
30:02 Example: Domain & Range of 2xy/(x-y)
32:28 Example: Domain & Range of √(9-x²-y²)
36:32 Graphing Domain of Multivariable Functions
42:13 Example: Graph Domain of ln(y-x)/√(x-y+2)
46:56 Example: Graph Domain of √(25-x²-y²-z²)
50:18 Graphing Multivariable Functions

Corrections:
0:48 I mistakenly wrote the formula for the volume of a right circular cylinder, not a cone. If you divide the height h by 3, then the formula would be correct. This ultimately does not change the point I was making at this point in the video. Both formulas are examples of multivariable functions.


📝 Examples Video: https://youtu.be/deuQbizie0I

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#calculus

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