How to find the equation of the tangent plane. Tangent plane. How to write the equations of the tangent plane and the normal at a point, if the surface is given by an explicit function

The topic "The angular coefficient of the tangent as the tangent of the angle of inclination" in the certification exam is given several tasks at once. Depending on their condition, the graduate may be required to provide both a full answer and a short answer. When preparing for the exam in mathematics, the student should definitely repeat the tasks in which it is required to calculate the slope of the tangent.

The Shkolkovo educational portal will help you do this. Our experts have prepared and presented theoretical and practical material as accessible as possible. Having become acquainted with it, graduates with any level of training will be able to successfully solve problems related to derivatives, in which it is required to find the tangent of the slope of the tangent.

Basic moments

To find the correct and rational solution to such tasks in the USE, it is necessary to recall the basic definition: the derivative is the rate of change of the function; it is equal to the tangent of the slope of the tangent drawn to the graph of the function at a certain point. It is equally important to complete the drawing. It will allow you to find the correct solution to the USE problems on the derivative, in which it is required to calculate the tangent of the slope of the tangent. For clarity, it is best to plot a graph on the OXY plane.

If you have already familiarized yourself with the basic material on the topic of the derivative and are ready to start solving problems for calculating the tangent of the slope of the tangent, similar to the USE tasks, you can do this online. For each task, for example, tasks on the topic "Relationship of the derivative with the speed and acceleration of the body", we wrote down the correct answer and the solution algorithm. In this case, students can practice performing tasks of various levels of complexity. If necessary, the exercise can be saved in the "Favorites" section, so that later you can discuss the decision with the teacher.

Let a function f be given, which at some point x 0 has a finite derivative f (x 0). Then the line passing through the point (x 0; f (x 0)), which has a slope f '(x 0), is called a tangent.

But what happens if the derivative at the point x 0 does not exist? There are two options:

1. The tangent to the graph also does not exist. The classic example is the function y = |x | at the point (0; 0).
2. The tangent becomes vertical. This is true, for example, for the function y = arcsin x at the point (1; π /2).

Tangent equation

Any non-vertical straight line is given by an equation of the form y = kx + b, where k is the slope. The tangent is no exception, and in order to compose its equation at some point x 0, it is enough to know the value of the function and the derivative at this point.

So, let a function be given y \u003d f (x), which has a derivative y \u003d f '(x) on the segment. Then at any point x 0 ∈ (a; b) a tangent can be drawn to the graph of this function, which is given by the equation:

y \u003d f '(x 0) (x - x 0) + f (x 0)

Here f ’(x 0) is the value of the derivative at the point x 0, and f (x 0) is the value of the function itself.

Task. Given a function y = x 3 . Write an equation for the tangent to the graph of this function at the point x 0 = 2.

Tangent equation: y \u003d f '(x 0) (x - x 0) + f (x 0). The point x 0 = 2 is given to us, but the values ​​f (x 0) and f '(x 0) will have to be calculated.

First, let's find the value of the function. Everything is easy here: f (x 0) = f (2) = 2 3 = 8;
Now let's find the derivative: f '(x) \u003d (x 3) ' \u003d 3x 2;
Substitute in the derivative x 0 = 2: f '(x 0) = f '(2) = 3 2 2 = 12;
So we get: y = 12 (x - 2) + 8 = 12x - 24 + 8 = 12x - 16.
This is the tangent equation.

Task. Compose the equation of the tangent to the graph of the function f (x) \u003d 2sin x + 5 at the point x 0 \u003d π / 2.

This time we will not describe in detail each action - we will only indicate the key steps. We have:

f (x 0) \u003d f (π / 2) \u003d 2sin (π / 2) + 5 \u003d 2 + 5 \u003d 7;
f '(x) \u003d (2sin x + 5) ' \u003d 2cos x;
f '(x 0) \u003d f '(π / 2) \u003d 2cos (π / 2) \u003d 0;

Tangent equation:

y = 0 (x − π /2) + 7 ⇒ y = 7

In the latter case, the line turned out to be horizontal, because its slope k = 0. There is nothing wrong with that - we just stumbled upon an extremum point.

A surface is defined as a set points , coordinates which satisfy a certain type of equations:

If the function F (x , y , z) (\displaystyle F(x,\,y,\,z)) continuous at some point and has continuous partial derivatives at it, at least one of which does not vanish, then in the neighborhood of this point the surface given by equation (1) will be correct surface.

In addition to the above implicit way of setting, the surface can be defined clearly, if one of the variables, for example, z, can be expressed in terms of the others:

More strictly, plain surface called image homeomorphic mapping (that is, a one-to-one and mutually continuous mapping) of the interior of the unit square. This definition can be given analytical expression.

Let on a plane with a rectangular coordinate system u and v given square, whose interior points coordinates satisfy the inequalities 0< u < 1, 0 < v < 1. Гомеоморфный образ квадрата в пространстве с rectangular coordinate system x, y, z is given by the formulas x = x(u, v), y = y(u, v), z = z(u, v) ( parametric definition of the surface). In this case, the functions x(u, v), y(u, v) and z(u, v) are required to be continuous and that for different points (u, v) and (u", v") the corresponding points (x, y, z) and (x", y", z") be different.

An example simple surface is a hemisphere. All the same sphere is not plain surface. This necessitates a further generalization of the concept of a surface.

A subset of space in which each point has a neighborhood that is plain surface, is called correct surface .

Surface in differential geometry

Helicoid

catenoid

The metric does not uniquely determine the shape of the surface. For example, metrics helicoid and catenoid, parameterized accordingly, coincide, that is, there is a correspondence between their regions that preserves all lengths ( isometry). Properties that are preserved under isometric transformations are called internal geometry surfaces. The internal geometry does not depend on the position of the surface in space and does not change when it is bent without tension and compression (for example, when bending cylinder in cone) .

Metric coefficients E , F , G (\displaystyle E,\ F,\ G) determine not only the lengths of all curves, but in general the results of all measurements inside the surface (angles, areas, curvature and etc.). Therefore, everything that depends only on the metric refers to the internal geometry.

Normal and normal section

Normal vectors at surface points

One of the main characteristics of a surface is its normal - unit vector perpendicular to the tangent plane at a given point:

The sign of the normal depends on the choice of coordinates.

The section of the surface by a plane containing the normal of the surface at a given point forms a certain curve, which is called normal section surfaces. Main normal for a normal section coincides with the normal to the surface (up to a sign).

If the curve on the surface is not a normal section, then its principal normal forms an angle with the surface normal θ (\displaystyle \theta ). Then the curvature k (\displaystyle k) curve is related to curvature k n (\displaystyle k_(n)) normal section (with the same tangent) Meunier formula :

The coordinates of the normal vector for different ways of specifying the surface are given in the table:

Here D (y , z) D (u , v) = | y u ′ y v ′ z u ′ z v ′ | , D (z , x) D (u , v) = | z u ′ z v ′ x u ′ x v ′ | , D (x, y) D (u, v) = | x u ′ x v ′ y u ′ y v ′ | (\displaystyle (\frac (D(y,z))(D(u,v)))=(\begin(vmatrix)y"_(u)&y"_(v)\\z"_(u) &z"_(v)\end(vmatrix)),\quad (\frac (D(z,x))(D(u,v)))=(\begin(vmatrix)z"_(u)&z" _(v)\\x"_(u)&x"_(v)\end(vmatrix)),\quad (\frac (D(x,y))(D(u,v)))=(\ begin(vmatrix)x"_(u)&x"_(v)\\y"_(u)&y"_(v)\end(vmatrix))).

All derivatives are taken at the point (x 0 , y 0 , z 0) (\displaystyle (x_(0),y_(0),z_(0))).

Curvature

For different directions at a given point on the surface, a different curvature of the normal section is obtained, which is called normal curvature; it is assigned a plus sign if the main normal of the curve goes in the same direction as the normal to the surface, or a minus sign if the directions of the normals are opposite.

Generally speaking, at every point on the surface there are two perpendicular directions e 1 (\displaystyle e_(1)) and e 2 (\displaystyle e_(2)), in which the normal curvature takes the minimum and maximum values; these directions are called main. An exception is the case when the normal curvature is the same in all directions (for example, at the sphere or at the end ellipsoid rotation), then all directions at the point are principal.

Surfaces with negative (left), zero (center), and positive (right) curvature.

Normal curvatures in principal directions are called principal curvatures; let's denote them κ 1 (\displaystyle \kappa _(1)) and κ 2 (\displaystyle \kappa _(2)). Size:

called Gaussian curvature , full curvature or just the curvature of the surface. There is also the term curvature scalar, which means the result convolutions curvature tensor; in this case, the curvature scalar is twice as large as the Gaussian curvature.

The Gaussian curvature can be calculated in terms of the metric, and therefore it is an object of the intrinsic geometry of surfaces (note that the principal curvatures do not belong to the intrinsic geometry). By the sign of curvature, you can classify the points of the surface (see figure). The curvature of the plane is zero. The curvature of a sphere of radius R is everywhere equal to 1 R 2 (\displaystyle (\frac (1)(R^(2)))). There is also a surface of constant negative curvature -

Namely, about what you see in the title. In essence, this is a "spatial analog" problems of finding a tangent and normals to the graph of a function of one variable, and therefore no difficulties should arise.

Let's start with basic questions: WHAT IS a tangent plane and WHAT IS a normal? Many are aware of these concepts at the level of intuition. The simplest model that comes to mind is a ball on which lies a thin flat cardboard. The cardboard is located as close as possible to the sphere and touches it at a single point. In addition, at the point of contact, it is fixed with a needle sticking straight up.

In theory, there is a rather witty definition of a tangent plane. Imagine an arbitrary surface and the point that belongs to it. It is obvious that a lot passes through the point. spatial lines that belong to this surface. Who has what associations? =) …I personally introduced the octopus. Suppose that each such line has spatial tangent at point .

Definition 1: tangent plane to the surface at a point is plane, containing the tangents to all curves that belong to the given surface and pass through the point .

Definition 2: normal to the surface at a point is straight passing through the given point perpendicular to the tangent plane.

Simple and elegant. By the way, so that you do not die of boredom from the simplicity of the material, a little later I will share with you one elegant secret that allows you to forget about cramming various definitions ONCE AND FOR ALL.

We will get acquainted with the working formulas and the solution algorithm directly on a specific example. In the vast majority of problems, it is required to compose both the equation of the tangent plane and the equation of the normal:

Example 1

Decision:if the surface is given by the equation (i.e. implicitly), then the equation of the tangent plane to a given surface at a point can be found by the following formula:

I pay special attention to unusual partial derivatives - their should not be confused with partial derivatives of an implicitly defined function (even though the surface is implicitly defined). When finding these derivatives, one should be guided by rules for differentiating a function of three variables, that is, when differentiating with respect to any variable, the other two letters are considered constants:

Without departing from the cash register, we find the partial derivative at the point:

Similarly:

This was the most unpleasant moment of the decision, in which an error, if not allowed, is constantly imagining. However, there is an effective verification technique here, which I talked about in the lesson. Directional derivative and gradient.

All the “ingredients” have been found, and now it’s up to careful substitution with further simplifications:

– general equation desired tangent plane.

I strongly recommend checking this stage of the decision. First you need to make sure that the coordinates of the touch point really satisfy the found equation:

- true equality.

Now we “remove” the coefficients of the general equation of the plane and check them for coincidence or proportionality with the corresponding values. In this case they are proportional. As you remember from analytic geometry course, - This normal vector tangent plane, and he - guide vector normal straight line. Let's compose canonical equations normals by point and direction vector:

In principle, the denominators can be reduced by a "two", but there is no particular need for this.

Answer:

It is not forbidden to designate the equations with some letters, however, again - why? Here and so it is very clear what's what.

The following two examples are for independent solution. A small "mathematical tongue twister":

Example 2

Find the equations of the tangent plane and the normal to the surface at the point .

And a task interesting from a technical point of view:

Example 3

Compose the equations of the tangent plane and the normal to the surface at a point

At the point.

There is every chance not only to get confused, but also to face difficulties when writing. canonical equations of the line. And the normal equations, as you probably understood, are usually written in this form. Although, due to forgetfulness or ignorance of some nuances, a parametric form is more than acceptable.

Examples of finishing solutions at the end of the lesson.

Is there a tangent plane at any point on the surface? In general, of course not. The classic example is conical surface and point - the tangents at this point directly form a conical surface, and, of course, do not lie in the same plane. It is easy to verify the discord and analytically: .

Another source of problems is the fact non-existence some partial derivative at a point. However, this does not mean that there is no single tangent plane at a given point.

But it was rather popular science than practically significant information, and we return to pressing matters:

How to write the equations of the tangent plane and the normal at a point,
if the surface is given by an explicit function?

Let's rewrite it implicitly:

And by the same principles we find partial derivatives:

Thus, the tangent plane formula is transformed into the following equation:

And accordingly, the canonical equations of the normal:

As it is easy to guess - it's "real" partial derivatives of a function of two variables at the point , which we used to designate with the letter "Z" and found 100500 times.

Note that in this article it is enough to remember the very first formula, from which, if necessary, it is easy to derive everything else. (obviously, having a basic level of training). It is this approach that should be used in the course of studying the exact sciences, i.e. from a minimum of information, one should strive to “pull out” a maximum of conclusions and consequences. "Soobrazhalovka" and already existing knowledge to help! This principle is also useful because it is very likely to save you in a critical situation when you know very little.

Let's work out the "modified" formulas with a couple of examples:

Example 4

Compose the equations of the tangent plane and the normal to the surface at point .

A small overlay here turned out with symbols - now the letter denotes a point of the plane, but what can you do - such a popular letter ....

Decision: we will compose the equation of the desired tangent plane according to the formula:

Let's calculate the value of the function at the point :

Compute partial derivatives of the 1st order at this point:

Thus:

carefully, do not rush:

Let us write the canonical equations of the normal at the point :

Answer:

And a final example for a do-it-yourself solution:

Example 5

Compose the equations of the tangent plane and the normal to the surface at the point.

The final one is because, in fact, I explained all the technical points and there is nothing special to add. Even the functions themselves offered in this task are dull and monotonous - in practice you are almost guaranteed to come across a "polynomial", and in this sense, Example No. 2 with the exponent looks like a "black sheep". By the way, it is much more likely to meet a surface given by an equation, and this is another reason why the function was included in the article as the “second number”.

And finally, the promised secret: so how to avoid cramming definitions? (of course, I don’t mean the situation when a student is feverishly cramming something before the exam)

The definition of any concept/phenomenon/object, first of all, gives an answer to the following question: WHAT IS IT? (who/such/such/such). Consciously In answering this question, you should try to reflect significant signs, definitely identifying this or that concept/phenomenon/object. Yes, at first it turns out to be somewhat tongue-tied, inaccurate and redundant (the teacher will correct =)), but over time, quite a worthy scientific speech develops.

Practice on the most abstract objects, for example, answer the question: who is Cheburashka? It's not so simple ;-) Is it a "fairytale character with big ears, eyes and brown hair"? Far and very far from the definition - you never know there are characters with such characteristics .... But this is much closer to the definition: "Cheburashka is a character invented by the writer Eduard Uspensky in 1966, which ... (listing the main distinguishing features)". Pay attention to how well started