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

Learn how to find the tangent plane for multivariable functions using linear approximation, explained step-by-step by Dr. Trefor Bazett.

Key Takeaways

Tangent planes generalize tangent lines to functions of multiple variables.
The tangent plane at a point approximates the surface linearly near that point.
Partial derivatives of the function at the point provide the slope components for the tangent plane.
The normal vector to the tangent plane can be derived from the gradient of the function.
Understanding tangent planes aids in linear approximation and analysis of multivariable functions.

What the video covers

Introduction to tangent planes as a multivariable generalization of tangent lines from single variable calculus.
Definition and computation of tangent planes at a specific point on a function's graph.
Explanation of the tangent plane passing through a point (X₀, Y₀, Z₀) on the surface.
Discussion on the formula of a plane using a normal vector and a point on the plane.
Derivation of the tangent plane equation using partial derivatives of the function.
Interpretation of the tangent plane as a linear approximation to the nonlinear function near the point.
Use of the dot product to ensure the normal vector is perpendicular to vectors lying on the plane.
Simplifying assumption of the normal vector component related to Z to derive the final tangent plane formula.
Relationship between tangent lines and tangent planes, with tangent lines lying within tangent planes.
Worked example illustrating the application of the tangent plane formula to a specific function.

Chapters

What is a tangent plane in multivariable calculus?

A tangent plane is the plane that best approximates a surface at a given point, generalizing the concept of a tangent line from single variable calculus to functions of multiple variables.

How do you find the equation of a tangent plane to a function at a point?

You find the partial derivatives of the function with respect to each variable at the point, then use these as slope components in the plane equation along with the point coordinates to write the tangent plane equation.

Why is the normal vector important for finding the tangent plane?

The normal vector is perpendicular to the tangent plane and is derived from the gradient of the function; it is used in the plane equation to ensure the plane correctly approximates the surface near the point.

Full Transcript — Download SRT & Markdown

Speaker A

In this video, we're going to talk about tangent planes, which is the multivariable generalization of a tangent line that we saw in single variable calculus. We're going to grow how do we define a tangent plane, how do we compute a tangent plane.

Speaker A

so in this example I have the graph of a function and by specifying a particular point in red which is a spot on the graph of my function and X naught Y naught and it has a height Z naught and

Speaker A

So in this example, I have the graph of a function, and by specifying a particular point in red, which is a spot on the graph of my function at X naught, Y naught, and it has a height Z naught, and

Speaker A

approximation for the graphical function I can have a tangent plane in this spot but I could move it around up the fronton in a different location I'd get a different tangent plane to topple the tangent plane and the spot that's right

Speaker A

then the tangent plane is some plane that's going to go through that point. It just comes along. I'm sorry, this is the graph of this function at a particular point, and nearby it's a close value. That tangent plane is a good

Speaker A

this well let's begin with the goal a loose sense of my goal is I'm looking for some plane I'm asking two things I'm first asking that the tangent plane and the graph of my function that they meet at the

Speaker A

approximation for the graphical function. I can have a tangent plane in this spot, but I could move it around up the function in a different location. I'd get a different tangent plane to topple the tangent plane in the spot that's right

Speaker A

in it and we're going to come back to it later now tangent planes well it's a plane and let's talk about the formula for a plane in general we've seen this before in our multivariable presence course one way to

Speaker A

on the top here. At that top, the tangent plane is parallel to the XY plane, kind of analogously to what you would take a derivative equal to zero and have a horizontal tangent line in single variable calculus. So how do we find

Speaker A

zero when we came up with this as we had a picture we had a plane you had a normal vector n to the plane it came out of a particular point next on why not Zed not on the plane and then indeed we

Speaker A

this? Well, let's begin with the goal. A loose sense of my goal is I'm looking for some plane. I'm asking two things. I'm first asking that the tangent plane and the graph of my function meet at the

Speaker A

of the components a B and C then we're going to get the formula a times X minus X naught plus B times y minus y naught and C times n minus n naught and that to sum all these two things is 0 so if I'm

Speaker A

specific point, that's X naught, Y naught, Z naught, and then secondly, I'm asking that nearby that point, the tangent plane is close in some sense. It's close to the graph of the function, but what exactly I mean by that, let's put a pin

Speaker A

plug them into the formula the harder part is the a the B and the C how do we come up with this normal vector to the graph of the function which will give us the normal vector to the plane I'm going

Speaker A

in it, and we're going to come back to it later. Now tangent planes, well, it's a plane, and let's talk about the formula for a plane in general. We've seen this before in our multivariable calculus course. One way to

Speaker A

minus 1 this always works unless it's equal to 0 and I'm going to let you pause and think about why that is not relevant if I'm asking for a tangent plane to a differentiable function nevertheless if I make the simplifying

Speaker A

give the equation of the plane is as the dot product between the normal vector and a generic vector that lies in the plane, the vector that emanates from a point P naught. This dot product is zero.

Speaker A

function it's Z the height of this function depends on the epson Y it depends on X Y in a linear way the value of X is raised only to the power of one with multiplicative constants out at the

Speaker A

When we came up with this, we had a picture. We had a plane, you had a normal vector n to the plane. It came out of a particular point X naught, Y naught, Z naught on the plane, and then indeed we

Speaker A

it's much more simple I understand indeed this is the linear thing we're going to approximate we just to figure out the a and the B so how do we do that remember our conditions we want the a and the B to be close nearby in some

Speaker A

demanded that this normal vector was truly going to be normal. It had to be zero dot product with any vector that lay on the plane. From this particular equation that we have, we can expand it. If we give the normal vector

Speaker A

naught plus said not equal to Z this is like me taking my surface and I'm going to just restrict to the plane where Y is equal to Y not that's gonna cut through my surface and notice what this equation

Speaker A

the components a, b, and c, then we're going to get the formula a times X minus X naught plus b times Y minus Y naught and c times Z minus Z naught, and that sum of all these two things is zero. So if I'm

Speaker A

indeed in single variable calculus we asked and answered the problem of how you come up with a good approximation line a tangent line to the graph of a function of a single variable well this is the equation of a generic line the

Speaker A

asking the question, how do I find a tangent plane at a particular point? Well, the point is easy to figure out. The X naught, Y naught, and Z naught that you're asked to find the tangent plane at, just

Speaker A

equal to Y not and so we get the same answer that a if it's gonna be a linear approximation has to just be well the derivative of your function in this case is the only variable around that we're

Speaker A

plug them into the formula. The harder part is the a, the b, and the c. How do we come up with this normal vector to the graph of the function, which will give us the normal vector to the plane? I'm going

Speaker A

to this special single variable function f of X and why not play twice let's go and plug in c1 X naught that gives a different mirror equation B times y minus y naught plus that not equal to set and given a single variable function

Speaker A

to begin with the simplifying assumption that c is any nonzero number. If that was the case, like it was say 7, I could divide through so that the coefficient of the Z minus Z naught was just equal to the value of

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derivative in this case is the derivative Specter why now so the linear approximation will be a linear approximation when B is equal to the partial derivative of F now with respect to Y at the value of x not why not so

Speaker A

minus 1. This always works unless it's equal to zero, and I'm going to let you pause and think about why that is not relevant if I'm asking for a tangent plane to a differentiable function. Nevertheless, if I make the simplifying

Speaker A

partial derivatives back to X at our point times X minus X naught plus the partial derivatives back to Y at our point that's not why not times the Y minus y naught plus is said not is equal to Z now if I want to sort of interpret

Speaker A

assumption c equal to minus 1, then I can clean up my formula a little bit. I get the same a times X minus X naught and b times Y minus Y naught, but then I say plus Z minus Z naught equals zero. So what this is is a linear

Speaker A

on and I get this yellow curve here the tangent line to that yellow curve but the tangent line lives inside of the tangent plane if you come along and cut the function you come along and cut the plane as well you get a tangent line

Speaker A

function. It's Z, the height of this function, depends on the X and Y. It depends on X and Y in a linear way. The value of X is raised only to the power of one with multiplicative constants out at the

Speaker A

blue curve lives within the tangent plane at the particular point nevertheless we get this formula so let's see how it works out in this specific example the example I'm going to use is the one that I've been programming here is the function is 2

Speaker A

front. One way to think about tangent lines back in the day and tangent planes today is that what you're trying to come up with is an approximation for our nonlinear function. We approximate that nonlinear tricky thing with a linear

Speaker A

is the height of the function at this particular point well I just plug it in what is f 1/2 1/2 I plug it in then I get the value 3 house but I want to figure out the purpose but

Speaker A

one. It's much more simple. I understand. Indeed, this is the linear thing we're going to approximate. We just need to figure out the a and the b. So how do we do that? Remember our conditions. We want the a and the b to be close nearby in some

Speaker A

minus 1 likewise if I take the partial with respect to Y here this is minus 2y I evaluate at Y not equal to 1/2 and I get the value of minus 1 take those 3 points plug it in and finally I get the formula

Speaker A

sense that we haven't yet specified. Well, let's take that formula and let me plug in first the value y equal to Y naught. So plug the number in. The second term goes away under any times X minus X

Speaker A

algorithms so let's just help the you to bail you out by doing this video a like and finally if you want to watch more multivariable calculus videos this video is part of a larger playlist a multivariable calculus so you can check

Speaker A

naught plus Z minus Z naught equals zero. This is like me taking my surface and I'm going to just restrict to the plane where Y is equal to Y naught. That's going to cut through my surface, and notice what this equation

Topics:tangent planelinear approximationmultivariable calculuspartial derivativesnormal vectorgradienttangent linemultivariable functionsDr. Trefor Bazettcalculus

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