Introduction to 3d graphs | Multivariable calculus | Khan Academy

Khan Academy
5 May 201607:06
EducationalLearning
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TLDRThis video script explores the concept of three-dimensional graphs as a method to represent multi-variable functions with two inputs and one output. It begins by drawing an analogy with two-dimensional graphs to facilitate understanding, then illustrates how to visualize 3D graphs with the example of f(x, y) = x^2 + y^2. The script discusses the visualization of input-output relationships in 3D space, the formation of surfaces from plotting numerous points, and the effects of modifying the function on the graph's shape. It also cautions against the limitations of visualizing higher-dimensional functions and introduces alternative methods such as contour maps and parametric functions for multi-variable analysis.

Takeaways
  • 📈 Three-dimensional graphs represent multi-variable functions with two-dimensional inputs and one-dimensional outputs.
  • 📚 Analogous to two-dimensional graphs, three-dimensional graphs help visualize the relationship between inputs and outputs.
  • 👀 The script uses the function f(x, y) = x^2 + y^2 to illustrate the concept of three-dimensional graphing.
  • 📊 Two-dimensional graphs are a foundation for understanding three-dimensional graphs, as they both involve plotting input-output pairs.
  • 📍 In two dimensions, inputs are plotted on the x-axis, and outputs are represented by the height of the graph.
  • 📌 The transition to three dimensions involves considering pairs of points as inputs and plotting them as triplets (x, y, z).
  • 🌐 Plotting multiple triplets results in a surface, which can be visualized as a three-dimensional parabola in the given example.
  • 🔍 Changing the function, such as multiplying by a fraction, alters the height of the graph, making it steeper or flatter.
  • 🚫 The script cautions against trying to visualize every multi-variable function as a three-dimensional graph due to limitations in human spatial visualization.
  • 🗺️ It introduces alternative methods for visualizing complex functions, such as contour maps, which look at the input space in a two-dimensional setting.
  • 🛰️ The script hints at other visualization techniques like parametric functions and vector spaces that will be covered in subsequent videos.
Q & A

  • What are three-dimensional graphs used for?

    -Three-dimensional graphs are used to represent multi-variable functions that have two-dimensional inputs and one-dimensional output.

  • How does the function f(x, y) = x^2 + y^2 differ from a two-dimensional graph?

    -The function f(x, y) = x^2 + y^2 is a three-dimensional representation of a function that would be a parabola in two dimensions, but in three dimensions, it forms a surface that resembles a three-dimensional parabola.

  • What is the purpose of plotting points in two-dimensional graphs?

    -Plotting points in two-dimensional graphs helps to visualize the relationship between inputs and outputs, showing all possible input-output pairs as a curve on the graph.

  • How does the height of a point in a three-dimensional graph relate to the function's output?

    -In a three-dimensional graph, the height of a point above the xy-plane corresponds to the output value of the function for a given pair of input values.

  • What happens to the graph of a function when the function is modified to output half of its original value?

    -When a function is modified to output half of its original value, the height of every point on the graph is reduced by half, causing the entire graph to 'sloop' downward.

  • Why is it challenging to visualize multi-variable functions with more than two input dimensions?

    -Visualizing multi-variable functions with more than two input dimensions is challenging because it would require a graph with more than three dimensions, which exceeds our ability to easily visualize in a standard three-dimensional space.

  • What is a contour map and how does it relate to three-dimensional graphs?

    -A contour map is a two-dimensional representation of a three-dimensional graph that shows the input space with lines connecting points of equal output value, providing a way to visualize 3-D graphs in a 2-D setting.

  • What is the significance of the xy-plane in the context of three-dimensional graphs?

    -The xy-plane in the context of three-dimensional graphs represents the space where the inputs live, and the outputs are represented by the height above this plane.

  • How can the shape of a three-dimensional graph be influenced by modifying the function's equation?

    -The shape of a three-dimensional graph can be influenced by modifying the function's equation, such as by scaling the output values, which can make the graph 'sloop' up or down, changing its overall form.

  • What are some alternative methods to visualize multi-variable functions besides three-dimensional graphs?

    -Some alternative methods to visualize multi-variable functions include contour maps, parametric functions, and vector space representations, each offering different perspectives on the input and output spaces.

Outlines
00:00
📊 Understanding 3D Graphs and Multivariable Functions

This paragraph introduces the concept of three-dimensional graphs as a method to represent multivariable functions with two-dimensional inputs and one-dimensional output. The analogy of two-dimensional graphs is used to explain the visualization process. The function f(x, y) = x² + y² is used as an example to illustrate how points are plotted in three dimensions, forming a surface that resembles a three-dimensional parabola. The importance of understanding the relationship between inputs and outputs is emphasized, and the process of plotting points in 3D space is described, highlighting how it differs from the two-dimensional approach.

05:02
🔍 Modifying 3D Graphs and Exploring Different Visualization Techniques

The second paragraph delves into how modifications to the function, such as scaling the output, affect the shape of the 3D graph. It uses the example of changing the function to one-half of x squared plus y squared, which results in the graph's height being reduced by half. The paragraph also cautions against the limitations of visualizing multivariable functions in three dimensions, especially when the input or output dimensions exceed what can be easily represented in a 3D space. It introduces alternative methods of visualization, such as contour maps, parametric functions, and vector spaces, suggesting that there are various ways to understand and represent multivariable functions beyond traditional 3D graphs.

Mindmap
Keywords
💡Three-dimensional graphs
Three-dimensional graphs are visual representations used to depict functions with two inputs and one output. They extend the concept of two-dimensional graphs into a spatial context, allowing for the visualization of multi-variable relationships. In the video, this concept is central as it explains how to represent the function f(x, y) = x^2 + y^2 in a 3D space, creating a surface that resembles a paraboloid.
💡Multi-variable function
A multi-variable function is a mathematical function that takes multiple inputs and produces a single output. The video script uses the example of f(x, y) = x^2 + y^2 to illustrate how such functions can be represented graphically in three dimensions, emphasizing the complexity of visualizing relationships involving more than one input variable.
💡Two-dimensional input
Two-dimensional input refers to the pair of values that serve as inputs for a multi-variable function. In the context of the video, the inputs are x and y, which together form the two-dimensional aspect of the function before the output is considered. This concept is foundational to understanding how three-dimensional graphs are constructed.
💡One-dimensional output
One-dimensional output is the single value that results from applying a function to a set of inputs. In the script, the output is the result of the function f(x, y), which is a single number representing the height of the graph above the xy-plane for a given pair of inputs (x, y).
💡Visualization
Visualization in the context of the video refers to the process of creating a mental or graphical representation of data or concepts to understand and analyze them better. The video discusses how three-dimensional graphs help visualize the relationship between inputs and outputs of multi-variable functions.
💡Input-output pairs
Input-output pairs are sets of values where the first value or set of values is the input to a function, and the resulting value is the output. The video script explains how plotting these pairs in two and three dimensions helps to understand the behavior of functions like f(x) = x^2 and f(x, y) = x^2 + y^2.
💡xy-plane
The xy-plane is the two-dimensional plane formed by the x and y axes in a three-dimensional coordinate system. In the video, the xy-plane is where the inputs of the multi-variable function are considered to reside, with the output represented by the height above this plane.
💡Surface
In the context of three-dimensional graphs, a surface is the shape formed when plotting the output of a multi-variable function for all possible input pairs. The video describes how the graph of f(x, y) = x^2 + y^2 forms a parabolic surface in three-dimensional space.
💡Contour map
A contour map is a type of two-dimensional representation used to depict a three-dimensional surface by showing lines (contours) of equal value. The video mentions contour maps as an alternative method for visualizing three-dimensional data in a two-dimensional space.
💡Parametric functions
Parametric functions are a way of representing a system of equations where the variables are expressed as functions of a single parameter. The video script hints at parametric functions as one of the methods to visualize multi-variable functions, focusing on the output space.
💡Vector space
Vector space, in the context of the video, refers to a mathematical structure that allows for the representation of all possible outputs of a function, considering the inputs as vectors. It is one of the alternative methods mentioned for visualizing and understanding multi-variable functions.
Highlights

Introduction to three-dimensional graphs as a way to represent multi-variable functions with two-dimensional inputs and one-dimensional output.

Analogous to two-dimensional graphs, three-dimensional graphs help visualize the relationship between inputs and outputs.

The example function f(x, y) = x^2 + y^2 is used to illustrate the concept of three-dimensional graphing.

Explanation of how two-dimensional graphs work by plotting input-output pairs on a graph.

The process of graphing involves plotting points for every possible input-output pair to form a curve.

In three-dimensional space, inputs are thought of as pairs of points, and outputs are the height above the xy-plane.

Visualization of a three-dimensional graph involves plotting triplets (x, y, z) to form a surface.

The shape of the graph for the function f(x, y) = (1/2)(x^2 + y^2) is discussed, showing a modification that reduces the height of the graph.

Further modification by reducing the output to one-twelfth results in a graph very close to the xy-plane, indicating very small outputs.

Caution against the temptation to visualize every multi-variable function as a three-dimensional graph due to limitations in human visualization capabilities.

Introduction to alternative methods for visualizing multi-variable functions beyond three-dimensional graphs.

Mention of contour maps as a two-dimensional representation of three-dimensional data by looking at the input space.

Discussion of parametric functions as a method to visualize output space for multi-variable functions.

Introduction to vector space as another method to look at input space and visualize all possible outputs.

Promise of covering these alternative visualization methods in the next few videos.

Emphasis on the importance of keeping an open mind to different methods for visualizing multi-variable functions.

Transcripts

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3D GraphsMulti-VariableFunction VisualizationInput-OutputAnalogiesTwo-DimensionalThree-DimensionalGraphing TechniquesEducational ContentMathematics Tutorial