How to find the TANGENT PLANE | Linear approximation of multi-variable functions

Dr. Trefor Bazett
19 Nov 201909:23
EducationalLearning
32 Likes 10 Comments

TLDRIn this video, we explore tangent planes, which generalize tangent lines from single-variable calculus to multivariable functions. The tangent plane at a specific point on a surface approximates the surface near that point. The video explains how to find the tangent plane by deriving the formula involving partial derivatives. An example using the function 2 - x^2 - y^2 at the point (1/2, 1/2) demonstrates the process. By calculating partial derivatives and substituting values, we find the equation of the tangent plane. This tutorial is part of a multivariable calculus series.

Takeaways
  • 📚 The video discusses the concept of tangent planes, which is a multivariable extension of the tangent line concept from single-variable calculus.
  • 📐 A tangent plane is a plane that closely approximates the graph of a function at a specific point and is parallel to the XY plane at the top of the graph.
  • 🔍 The goal is to find a plane that meets the graph of the function at a given point and is a close approximation of the function's graph near that point.
  • 📈 The general equation of a plane is given by the dot product between a normal vector and a generic vector in the plane, which equals zero.
  • 📝 By assuming the normal vector's z-component to be -1, the equation simplifies to a linear function, which is an approximation of the nonlinear function at that point.
  • 🔑 The coefficients 'a' and 'b' in the tangent plane equation are determined by the partial derivatives of the function with respect to 'x' and 'y' at the given point.
  • 🧭 The partial derivative with respect to 'x' gives the slope of the tangent line when 'y' is held constant, and vice versa for 'y'.
  • 📉 The tangent plane equation is derived by setting the partial derivatives equal to the differences in 'x' and 'y' coordinates from the point of tangency, plus the z-value at that point.
  • 📌 Geometrically, the tangent plane contains the tangent lines obtained by slicing the function at fixed 'x' or 'y' values.
  • 📚 The example given in the video uses the function f(x, y) = 2 - x^2 - y^2, with a specific point (1/2, 1/2) to illustrate the calculation of the tangent plane.
  • 📝 The final equation of the tangent plane for the example is Z = 3/2 - x + 1/2 - y + 1/2, which is derived by plugging in the values of the function and its partial derivatives at the specified point.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is the concept of tangent planes in multivariable calculus, which is a generalization of the tangent line concept from single-variable calculus.

  • What is the purpose of a tangent plane in the context of the video?

    -The purpose of a tangent plane is to provide a good approximation for the graphical function of a multivariable function at a specific point and in the vicinity of that point.

  • What is the general formula for the equation of a plane?

    -The general formula for the equation of a plane is given by the dot product between the normal vector to the plane and a generic vector that lies in the plane, which equals zero.

  • How does the video simplify the formula for the tangent plane?

    -The video simplifies the formula for the tangent plane by assuming that the coefficient of the z-component of the normal vector is -1, which allows for a cleaner linear representation of the plane.

  • What is the role of the normal vector in defining a tangent plane?

    -The normal vector is crucial in defining a tangent plane as it provides the orientation of the plane, and it must be perpendicular to the surface of the function at the point of tangency.

  • How does the video relate the concept of a tangent plane to single-variable calculus?

    -The video relates the concept of a tangent plane to single-variable calculus by drawing an analogy between the linear approximation of a tangent line in single-variable functions and the linear approximation of a tangent plane in multivariable functions.

  • What are the conditions for the coefficients a and b in the tangent plane formula?

    -The coefficients a and b in the tangent plane formula are the partial derivatives of the function with respect to x and y, respectively, evaluated at the point of tangency (x_not, y_not).

  • How does the video explain the process of finding the tangent plane for a given function?

    -The video explains the process by first identifying the point of tangency, then finding the partial derivatives of the function at that point, and finally plugging these values into the simplified tangent plane formula.

  • What is the example function used in the video to illustrate the concept of a tangent plane?

    -The example function used in the video is f(x, y) = 2 - x^2 - y^2, and a specific point (1/2, 1/2) is chosen to demonstrate the calculation of the tangent plane.

  • What is the final equation of the tangent plane obtained in the example provided in the video?

    -The final equation of the tangent plane obtained in the example is Z = 3/2 - (X - 1/2) - (Y - 1/2), which represents the tangent plane at the point (1/2, 1/2, 3/2) for the given function.

Outlines
00:00
📚 Introduction to Tangent Planes in Multivariable Calculus

This paragraph introduces the concept of tangent planes as the multivariable equivalent of tangent lines in single-variable calculus. The speaker explains the purpose of tangent planes, which is to approximate the graph of a function near a specific point. The point in question is marked in red on the graph, with coordinates (x₀, y₀, z₀), and the tangent plane is a plane that passes through this point and closely approximates the function's graph nearby. The speaker also discusses the general formula for a plane, which involves a normal vector and a dot product, and sets up the goal of finding the tangent plane's equation by satisfying two conditions: meeting the function's graph at the specific point and approximating the function's graph nearby.

05:00
🔍 Deriving the Equation of a Tangent Plane

In this paragraph, the speaker delves into the process of deriving the equation of a tangent plane. They begin by simplifying the general plane equation by assuming the normal vector's z-component is -1, which simplifies the equation to a linear form. The speaker then explains that to find the tangent plane, one must determine the coefficients a and b, which are related to the partial derivatives of the function with respect to x and y at the point (x₀, y₀). By setting y to y₀ and solving for the tangent line in the x-direction, the speaker shows that the coefficient a is the partial derivative of the function with respect to x at the point (x₀, y₀). Similarly, by setting x to x₀ and solving for the tangent line in the y-direction, the coefficient b is found to be the partial derivative of the function with respect to y at the point (x₀, y₀). The paragraph concludes with the formula for the tangent plane's equation, which is a linear combination of the differences in x and y coordinates from the point (x₀, y₀), plus the z-coordinate of the function at that point.

Mindmap
Keywords
💡Tangent Plane
A tangent plane is a concept in multivariable calculus that generalizes the idea of a tangent line from single-variable calculus. It is a plane that touches a surface at a single point and provides a linear approximation of the surface near that point. In the video, the tangent plane is described as a plane that passes through a specified point on the graph of a function and closely approximates the function's graph near that point, similar to how a tangent line touches a curve at a single point in single-variable calculus.
💡Multivariable Generalization
The term 'multivariable generalization' refers to the extension of mathematical concepts from single-variable to multiple variables. In the context of the video, it is used to describe how the concept of a tangent line is extended to a tangent plane when dealing with functions of more than one variable. The video script explains that while a tangent line is a flat line that touches a curve at a point, a tangent plane is a flat surface that touches a three-dimensional graph at a point.
💡Normal Vector
A normal vector is a vector that is perpendicular to a surface, such as a plane or a curve. In the video, the normal vector is crucial for defining the tangent plane because it provides the direction that is perpendicular to the plane. The script discusses how the equation of a plane can be expressed using the dot product involving a normal vector and a position vector on the plane, emphasizing the importance of the normal vector in determining the orientation of the tangent plane.
💡Dot Product
The dot product is an algebraic operation that takes two equal-length vectors and returns a single number. In the video, the dot product is used to express the equation of a plane, where the dot product between the normal vector to the plane and a vector from a point on the plane to any other point on the plane equals zero. This is a key concept in defining the tangent plane, as it ensures that any point on the plane satisfies the plane's equation.
💡Partial Derivative
A partial derivative is the derivative of a function of multiple variables with respect to one of its variables, while treating the other variables as constants. In the video, partial derivatives are used to find the coefficients of the tangent plane's equation. The script explains that the partial derivatives of a function at a specific point give the rates of change of the function in the x and y directions, which are essential for constructing the linear approximation of the function's surface near that point.
💡Linear Approximation
Linear approximation is a method of approximating a function or a surface with a linear function, which is simpler to work with. In the video, the tangent plane is described as a linear approximation of a nonlinear function near a specific point. The script illustrates that by using the partial derivatives of the function at a point, one can create a plane that closely approximates the function's surface in the vicinity of that point.
💡Differentiable Function
A differentiable function is a function that has a derivative at every point in its domain. In the video, the concept of a tangent plane is discussed in the context of differentiable functions, as the existence of a tangent plane at a point on a function's graph implies that the function is differentiable at that point. The script mentions that the tangent plane is used to approximate a differentiable function near a specific point.
💡Graph of a Function
The graph of a function is a visual representation of the function's values, typically plotted on a coordinate plane. In the video, the graph of a function is used to illustrate the concept of a tangent plane. The script describes how the tangent plane touches the graph at a specific point and how it can be visualized as the surface that closely follows the function's graph near that point.
💡Slicing a Surface
Slicing a surface refers to the process of intersecting a three-dimensional surface with a plane to create a two-dimensional curve. In the video, the concept is used to explain how the tangent plane can be understood by considering the tangent lines to the curves obtained by slicing the surface at fixed values of one of the variables. The script provides an example where the surface is sliced at a fixed y value, resulting in a curve with a tangent line that lies within the tangent plane.
💡Equation of a Plane
The equation of a plane is a mathematical formula that describes all the points that lie on the plane. In the video, the equation of a plane is discussed in the context of the tangent plane, where the coefficients of the plane's equation are derived from the partial derivatives of the function at a specific point. The script explains that by knowing the normal vector to the plane and a point through which the plane passes, one can write the equation of the tangent plane.
Highlights

Introduction to tangent planes as a multivariable generalization of tangent lines in single-variable calculus.

Explanation of how to define and compute a tangent plane at a specific point on a function's graph.

The tangent plane as an approximation for the graph of a function near a given point.

The formula for a plane using the dot product between a normal vector and a generic vector in the plane.

Assumption that the coefficient of the z-component in the normal vector is -1 for simplification.

Derivation of the tangent plane formula involving partial derivatives with respect to x and y.

Geometric interpretation of the tangent plane and its relationship to tangent lines in single-variable functions.

How to find the normal vector to the graph of a function to determine the tangent plane.

The process of finding the partial derivatives of a function to approximate the tangent plane.

Using the partial derivatives to construct the linear approximation of the function at a point.

The importance of the normal vector in determining the orientation of the tangent plane.

Illustration of how the tangent plane is derived from the graph of a function at a specific point.

The role of the tangent plane in approximating the function's graph near the point of tangency.

Practical example of finding the tangent plane for the function f(x, y) = 2 - x^2 - y^2 at the point (1/2, 1/2).

Calculation of the function's value and partial derivatives at the specified point for the tangent plane equation.

Final equation of the tangent plane for the given function and point.

Invitation for viewers to leave questions in the comments for further discussion.

Encouragement for viewers to like the video and subscribe for more multivariable calculus content.

Transcripts

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