Quadratic approximation formula, part 1

TL;DR

This video explains how to extend the concept of local linearization to create a quadratic approximation, allowing for the use of terms like x squared, x times y, and y squared.

Transcript

• [Voiceover] So, our setup is that we have some kind of two variable function f of x, y, who has a scaler output, and the goal is to approximate it near a specific input point, and this is something I've already talked about in context of a local linearization, and I've written out the full local, the full local linearization, hard words to say, l... Read More

Key Insights

• 😥 Local linearization is a useful method for approximating a function near a specific input point.
• 🍉 The local linearization formula can be intimidating but can be simplified by breaking it down into smaller terms.
• 😥 The local linearization accurately approximates the value and partial derivatives of the original function at the input point.
• ❣️ Quadratic approximation involves extending the local linearization by adding terms like x squared, x times y, and y squared.
• 😥 The constant terms in the quadratic approximation are modified to ensure that the approximation matches the original function at the input point.
• 👻 The quadratic approximation allows for a more accurate representation of the function by incorporating additional terms.
• 🪡 The constants in the quadratic approximation need to be determined in order to create the most accurate approximation of the original function.

Questions & Answers

Q: What is local linearization?

Local linearization is a method used to approximate a function near a specific input point by linearizing it using the tangent line at that point.

Q: What are the terms allowed in a quadratic approximation?

A quadratic approximation allows for the use of terms like x squared, x times y, and y squared, which involve multiplying two variables together.

Q: How are constant terms handled in the quadratic approximation?

In order to ensure that the value and partial derivatives of the quadratic approximation match the original function at the input point, the constant terms in the approximation are modified by subtracting the corresponding input point values.

Q: How are the constants determined in the quadratic approximation?

The constants in the quadratic approximation are determined in a way that closely approximates the original function. The process of determining these constants is explained in a subsequent video.

Key Insights:

• Local linearization is a useful method for approximating a function near a specific input point.
• The local linearization formula can be intimidating but can be simplified by breaking it down into smaller terms.
• The local linearization accurately approximates the value and partial derivatives of the original function at the input point.
• Quadratic approximation involves extending the local linearization by adding terms like x squared, x times y, and y squared.
• The constant terms in the quadratic approximation are modified to ensure that the approximation matches the original function at the input point.
• The quadratic approximation allows for a more accurate representation of the function by incorporating additional terms.
• The constants in the quadratic approximation need to be determined in order to create the most accurate approximation of the original function.
• Further details on determining the constants in the quadratic approximation are provided in a subsequent video.

Summary & Key Takeaways

• The video introduces the concept of local linearization, which is used to approximate a two-variable function near a specific input point.

• The local linearization formula may seem complex, but it can be broken down into simpler terms.

• The video discusses the properties of the local linearization and how those properties are extended to create a quadratic approximation.