Linear Approximation and Tangent Planes: Calculus 3 Project by Abigail Powsner
Overview
This Project has been completed as part of a standard 10 weeks Calculus 3 asynhronous online course with optional office hours during Summer 2022 semester at MassBay Community College, Wellesley Hills, MA.
Summary
Author: Abigail Powsner
Instructor: Igor V Baryakhtar
Subject: Calculus 3
Course number: MA 202-700
Course type: Asynhronous online
Semester: Summer 2022, 10 weeks
College: MassBay Comminity College, MA
Tags: Calculus, Project Based learning, Active Learning
Language: English
Media Format: html
License: CC-BY 4.0
All project content created by Abigail Powsner
Content added to OER Commons by Igor V Baryakhtar
Linear Approximation and Tangent Planes
In earlier sections of Calculus, tangent lines were discussed. When you think of a function on a graph, there is a tangent line that touches the function. This is similar to what we are learning now, but instead, these tangent lines become tangent planes. When solving for tangent planes or lines, linear approximation can be used in this process.
Tangent planes are different compared to tangent lines because instead of thinking of a two-dimensional graph, tangent planes apply to three-dimensional graphs. Here is what a tangent line looks like: (the blue line)
(Dawkins)
There is usually only one tangent line, but now there is the possibility of having multiple tangent planes because it is being applied to three-dimensional graphs. Tangent planes are tangent to their graph and just touch the three-dimensional graph (Khan Academy). Here is what a tangent plane looks like: (gray plane)
(Khan Academy)
According to the Calculus Volume 3 textbook, the equation used to find tangent planes is: z = f(x0, y0) + fx(x0, y0)(x-x0) + fy(x0, y0)(y-y0) . This is specifically when S is a surface defined by z = f(x,y), which is a differentiable function. This equation gives the tangent plane to S at P0 when P0 = (x0, y0) is a point in the domain f (Gilbert & Herman, 2016). Here is one example of how to find the equation of a tangent plane at a point:
In step one, you find the partial derivative of f with respect to x, and then plug in the x and y values into the equation. Step two is similar, but instead, the partial derivative of f with respect to y is found, and the same two values as before are plugged in. In step 3, the function is evaluated at the point. From the equation used to find a tangent plane, values are plugged into the equation in step 4. You then distribute and simplify the equation to get z, which is the equation of the tangent plane.
Linear approximation is also known as the process of finding the tangent line. When looking at a graph, and this time it is a two-dimensional graph, linear approximation is used to approximate “values of f(x) as long as we stay ‘near’ x=a” (Dawkins). This means that linear approximation is a good technique to find a tangent line to a specific point. However, it is different when dealing with two-variable functions. In this case where there are two variables, a tangent plane is being approximated. Linear approximation is not going to give you an exact value or coordinate of a plane. Instead, it is going to give you the closest value possible to what you are solving for on the graph. The equation of linear approximation is: L(x, y) = f(x0, y0) + fx(x0, y0)(x-x0) + fy(x0, y0)(y-y0) . This equation is specifically used when a function, z = f(x, y), has continuous partial derivatives that exist at (x0, y0). L(x, y) finds the linear approximation of f at point (x0, y0). You use the tangent plane throughout the process of finding the linear approximation, here is a photo showing this:
Here is an example of using linear (tangent) plane approximation:
In step one, the partial derivative with respect to x was found. The values of x0 and y0 were then found. Step two was similar, but the partial derivative with respect to y was found. Then the same x0 and y0 values were plugged into the partial derivative and were calculated. This was the same process that was used in the previous example. Step 3 was evaluating what f was with the same values as before. In step 4, the equation of linear approximation was used, and all of the different values were plugged in. The equation for L(x, y) was then found. From the question, you were supposed to approximate the point by using the L(x, y) equation and see how similar it was to the function evaluated at the point. This was done by using x0 = 2 and y0 = 3 by plugging those numbers into the equation or function. The two values were very similar, so linear approximation shows that it is a valid way to approximate a function/plane.
Here are a few problems to try out yourself:
- Find the equation for the tangent plane to the surface at the point: z = - 9x2- 3y2 at point (2, 1, -39)
- Find the linear approximation of each function at the point: f(x, y) = x √ (y) at point (1, 4).
These are some videos to help further explain some topics (the last video is a video of more practice problems):
- What is a Tangent Plane
- Finding the Linearization of a Function using Tangent Line Approximations
- Tangent Planes and Linear Approximations
Overall, the topic of linear approximation and tangent planes brings together various Calculus concepts. We already knew about linear approximation and tangent lines, but now this was incorporating the fact that we are looking at three-dimensional figures/graphs. I hope you were able to use your previous knowledge on these topics and learn more about linear approximation and tangent planes.
References:
Dawkins, P. (n.d.). Section 4-11: Linear Approximations. Calculus I - linearapproximations. Retrieved July 8, 2022, from https://tutorial.math.lamar.edu/classes/calci/linearapproximations.aspx
Khan Academy. (n.d.). Tangent Planes (article). Khan Academy. Retrieved July 8, 2022, from https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/tangent-planes-and-local-linearization/a/tangent-planes
YouTube. (2016). What is a tangent plane. YouTube. Retrieved July 9, 2022, from https://www.youtube.com/watch?v=cHNT7_F8m1Y.
YouTube. (2018). Finding the linearization of a function using tangent line approximations. YouTube. Retrieved July 9, 2022, from https://www.youtube.com/watch?v=XQaCbFMnDo0.
YouTube. (2018). Tangent planes and linear approximation. YouTube. Retrieved July 9, 2022, from https://www.youtube.com/watch?v=0qDdUruJwFU.