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

TL;DR

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

Key Insights

• 📈 Multi-variable functions can be represented using three-dimensional graphs, which involve two inputs and one output.
• 🤪 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

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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.