Introduction to 3d graphs | Multivariable calculus | Khan Academy | Summary and Q&A

372.4K views
â€Ē
May 5, 2016
by
Khan Academy
YouTube video player
Introduction to 3d graphs | Multivariable calculus | Khan Academy

TL;DR

Three-dimensional graphs are a way to represent multi-variable functions using two inputs and one output.

Install to Summarize YouTube Videos and Get Transcripts

Key Insights

  • 📈 Multi-variable functions can be represented using three-dimensional graphs, which involve two inputs and one output.
  • ðŸĪŠ Inputs in three-dimensional graphs are pairs of points, and the natural way to visualize them is by using a triplet of x, y, and z coordinates.
  • 📈 The shape of the graph can be modified by altering the function, such as changing the height of the output for each input.
  • ✋ Three-dimensional graphs are limited to representing functions with a three-dimensional input and one-dimensional output, as visualizing higher dimensions becomes challenging.

Transcript

  • [Voiceover] Hello everyone. So what I'd like to do here is to describe how we think about three-dimensional graphs. Three-dimensional graphs are a way that we represent certain kind of multi-variable function that kind of has two inputs, or rather a two-dimensional input, and then one-dimensional of output of some kind. So the one that I have pic... Read More

Questions & Answers

Q: How are two-dimensional graphs different from three-dimensional graphs?

Two-dimensional graphs represent the relationship between a single input and its corresponding output, while three-dimensional graphs show the relationship between two inputs and the resulting output.

Q: How are input-output pairs visualized in two-dimensional graphs?

In two-dimensional graphs, input-output pairs are plotted as points on the graph. Each input corresponds to an x-coordinate, and the output is represented by the height of the graph above that point.

Q: How are input-output pairs visualized in three-dimensional graphs?

In three-dimensional graphs, input-output pairs are represented as points in three-dimensional space. The x and y coordinates determine the input, and the z coordinate represents the output.

Q: Is it possible to visualize all possible input-output pairs in a three-dimensional graph?

No, it is not feasible to plot all possible input-output pairs in a three-dimensional graph, as there are infinitely many combinations. Instead, a surface is drawn that represents the relationship between the inputs and outputs.

Summary & Key Takeaways

  • Three-dimensional graphs are a visual representation of multi-variable functions, showing the relationship between two inputs and one output.

  • In two-dimensional graphs, the inputs are plotted on the x-axis, the outputs are represented by the height of the graph above each point.

  • In three-dimensional graphs, inputs are pairs of points, and the natural way to visualize them is by using a triplet of x, y, and z coordinates.

Share This Summary 📚

Summarize YouTube Videos and Get Video Transcripts with 1-Click

Download browser extensions on:

Explore More Summaries from Khan Academy 📚

Summarize YouTube Videos and Get Video Transcripts with 1-Click

Download browser extensions on: