What is a tangent plane

TL;DR

Tangent planes of graphs are two-dimensional surfaces that barely kiss the graph of a function at various points.

Transcript

• [Voiceover] Hi everyone. Here and in the next few videos I'm gonna be talking about tangent planes of graphs, and I'll specify this is tangent planes of graphs and not of some other thing because in different context of multivariable calculus you might be taking a tangent plane of say a parametric surface or something like that but here I'm just ... Read More

Key Insights

• ✈️ Tangent planes of graphs are two-dimensional surfaces that approximate functions in multivariable calculus.
• 😥 The process of finding a tangent plane involves determining the input point, finding its corresponding output on the graph, and constructing a plane that is tangent to the graph at that point.
• 😥 The tangent plane allows for approximations of the function around the specified point and provides insights into its behavior in different directions.
• 🫥 Tangent planes in multivariable calculus are similar in concept to tangent lines in single variable calculus, but they operate in a higher dimension.

Questions & Answers

Q: What is the difference between tangent planes in multivariable calculus and tangent lines in single variable calculus?

Tangent planes are two-dimensional surfaces that approximate functions around a given point, while tangent lines are one-dimensional and provide information about functions in the single variable world.

Q: How do you determine the input point and its corresponding output on the graph?

In multivariable calculus, the input point is usually represented by coordinates (x, y), and you can find the output value by evaluating the function at that point.

Q: What is the purpose of finding the tangent plane?

The tangent plane allows us to approximate the behavior of the function near the specified point and understand how it changes in different directions.

Q: How does the process of finding the tangent plane relate to the original function?

The tangent plane is a new function, called L, that takes in x and y as inputs. It is dependent on the original function and the specified input point.

Summary & Key Takeaways

• Tangent planes of graphs are similar to tangent lines in the single variable world, but they are two-dimensional and give approximations of the function around a given point.

• To find the tangent plane, you first determine the input point and its corresponding output on the graph.

• The goal is to find a linear function that represents the tangent plane at the specified point.