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

The topic “The angular coefficient of a tangent as the tangent of the angle of inclination” is given several tasks in the certification exam. Depending on their condition, the graduate may be required to provide either a full answer or a short answer. In preparation for passing the Unified State Exam in mathematics, the student should definitely repeat problems in which it is required to calculate slope tangent.

It will help you do this educational portal"Shkolkovo". Our specialists prepared and presented theoretical and practical material in the most accessible way possible. Having become familiar with it, graduates with any level of training will be able to successfully solve problems related to derivatives in which it is necessary to find the tangent of the tangent angle.

Basic moments

To find the correct and rational decision For similar tasks in the Unified State Exam, you need to remember the basic definition: the derivative represents the rate of change of a function; it is equal to the tangent of the tangent angle 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 correct solution Unified State Examination problems on the derivative, in which it is necessary to calculate the tangent of the angle of inclination of the tangent. For clarity, it is best to plot the graph on the OXY plane.

If you have already familiarized yourself with the basic material on the topic of derivatives and are ready to start solving problems on calculating the tangent of the tangent angle, such as Unified State Exam assignments, you can do this online. For each task, for example, problems on the topic “Relationship of a derivative with the speed and acceleration of a body,” we wrote down the correct answer and solution algorithm. At the same time, students can practice performing tasks of varying levels of complexity. If necessary, the exercise can be saved in the “Favorites” section so that you can discuss the solution with the teacher later.

Let a function f be given, which at some point x 0 has a finite derivative f (x 0). Then the straight line passing through the point (x 0 ; f (x 0)), having an angular coefficient f ’(x 0), is called a tangent.

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

1. There is no tangent to the graph either. A classic example is the function y = |x | at 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 create 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 y = f (x) be given, which has a derivative y = 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 = f ’(x 0) (x − x 0) + f (x 0)

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

Task. Given the 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 = 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) = (x 3)’ = 3x 2;
We substitute x 0 = 2 into the derivative: f ’(x 0) = f ’(2) = 3 2 2 = 12;
In total we get: y = 12 · (x − 2) + 8 = 12x − 24 + 8 = 12x − 16.
This is the tangent equation.

Task. Write an equation for the tangent to the graph of the function f (x) = 2sin x + 5 at point x 0 = π /2.

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

f (x 0) = f (π /2) = 2sin (π /2) + 5 = 2 + 5 = 7;
f ’(x) = (2sin x + 5)’ = 2cos x;
f ’(x 0) = f ’(π /2) = 2cos (π /2) = 0;

Tangent equation:

y = 0 · (x − π /2) + 7 ⇒ y = 7

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

A surface is defined as a set points , coordinates which are satisfied a certain type 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 defined by equation (1) will be the right surface.

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

More strictly simple 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 the plane with rectangular system u and v coordinates are given square, coordinates internal points which satisfy the inequalities 0< u < 1, 0 < v < 1. Гомеоморфный образ квадрата в пространстве с rectangular coordinate system x, y, z is specified using the formulas x = x(u, v), y = y(u, v), z = z(u, v) ( parametric surface definition). In this case, the functions x(u, v), y(u, v) and z(u, v) are required to be continuous and so that for different points (u, v) and (u", v") the corresponding points (x, y, z) and (x", y", z") are different.

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

A subset of space, each point of which has a neighborhood that is simple surface, called the right 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, between their areas there is a correspondence 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 stretching or compressing (for example, when bending cylinder V cone) .

Metric coefficients E , F , G (\displaystyle E,\ F,\ G) determine not only the lengths of all curves, but also 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 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.

A section of a surface by a plane containing the surface normal at a given point forms a certain curve called normal section surfaces. Home normal for a normal section coincides with the normal to the surface (up to sign).

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

Coordinates of the normal unit vector for different ways surface assignments 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, 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 a surface there are two perpendicular directions e 1 (\displaystyle e_(1)) And e 2 (\displaystyle e_(2)), in which the normal curvature takes minimum and maximum values; these directions are called main. The exception is the case when the normal curvature is the same in all directions (for example, near a 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 the principal directions are called main curvatures; let's designate them κ 1 (\displaystyle \kappa _(1)) And κ 2 (\displaystyle \kappa _(2)). Size:

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

Gaussian curvature can be calculated via a metric, and is therefore an object of the intrinsic geometry of surfaces (note that the principal curvatures do not belong to the intrinsic geometry). You can classify surface points based on the sign of curvature (see figure). The curvature of the plane is zero. The curvature of a sphere of radius R is equal everywhere 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. Essentially, this is a “spatial analogue” tangent finding problems And normals to the graph of a function of one variable, and therefore no difficulties should arise.

Let's start with the basic questions: WHAT IS a tangent plane and WHAT IS a normal? Many people understand these concepts at the level of intuition. The most simple model The one that comes to mind is a ball on which lies a thin flat piece of 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 secured with a needle sticking straight up.

In theory, there is a rather ingenious definition of a tangent plane. Imagine a free surface and the point belonging to it. Obviously, a lot passes through the point spatial lines, which belong to this surface. Who has what associations? =) ...personally, I imagined an octopus. Let us assume that each such line has spatial tangent at point .

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

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

Simple and elegant. By the way, so that you don’t 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’ll get acquainted with the working formulas and solution algorithm directly at specific example. In the vast majority of problems, it is necessary to construct both the tangent plane equation and the normal equation:

Example 1

Solution: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 using the following formula:

Special attention I draw attention to unusual partial derivatives - their should not be confused With partial derivatives of an implicitly specified function (although the surface is specified implicitly). When finding these derivatives, one must 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 leaving the cash register, we find the partial derivative at the point:

Likewise:

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

All the “ingredients” have been found and now it’s a matter of careful substitution with further simplifications:

– general equation the desired tangent plane.

I strongly recommend checking this stage of the solution as well. First you need to make sure that the coordinates of the tangent point really satisfy the found equation:

- true equality.

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

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

Answer:

It is not forbidden to designate the equations with some letters, but, again, why? Here it’s already extremely clear what’s what.

The following two examples are for independent decision. A little “mathematical tongue twister”:

Example 2

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

And a task that is interesting from a technical point of view:

Example 3

Write equations for the tangent plane and normal to the surface at a point

At the point.

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

Sample samples finalization of solutions at the end of the lesson.

Is there a tangent plane at any point on the surface? IN general case, 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 that something is wrong analytically: .

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

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

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

Let's rewrite it implicitly:

And using the same principles we find partial derivatives:

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

And accordingly, the canonical normal equations:

As you might guess, - these are already “real” partial derivatives of a function of two variables at the point, which we used to denote by the letter “z” and were found 100500 times.

Please 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 (of course, having a basic level of training). This is the approach that should be used when studying exact sciences, i.e. from a minimum of information we must strive to “draw” a maximum of conclusions and consequences. “Consideration” and existing knowledge will help! This principle is also useful because it will most likely 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

Write equations for the tangent plane and normal to the surface at point .

There is a slight overlay here with the notations - now the letter denotes a point on the plane, but what can you do - such a popular letter...

Solution: let’s compose the equation of the desired tangent plane using the formula:

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

Let's calculate 1st order partial derivatives at this point:

Thus:

carefully, don't rush:

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

Answer:

And a final example for your own solution:

Example 5

Write down equations for the tangent plane and the normal to the surface at the point.

Final - because I have explained virtually all the technical points and there is nothing special to add. Even the functions themselves proposed 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 an exponent looks like a “black sheep”. By the way, it is much more likely to encounter a surface defined by an equation, and this is another reason why the function was included in the article as number two.

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

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

Practice on the most abstract objects, for example, answer the question: who is Cheburashka? It's not that simple ;-) This is " fairy tale character With big ears, eyes and brown fur"? Far and very far from 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, who ... (list of main distinctive features)» . Notice how well it started