The tangent of the angle of inclination of the line is 0 25. Equation of the tangent to the graph of the function

Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. IN in this case The graph can be either a straight or curved line. That is, the derivative characterizes the rate of change of a function at a specific point in time. Remember general rules, by which derivatives are taken, and only then proceed to the next step.

• Read the article.
• How to take the simplest derivatives, for example, derivative exponential equation, described. The calculations presented in the following steps will be based on the methods described therein.

Learn to distinguish between tasks in which slope needs to be calculated through the derivative of the function. Problems do not always ask you to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x,y). You may also be asked to find the slope of the tangent at point A(x,y). In both cases it is necessary to take the derivative of the function.

The straight line y=f(x) will be tangent to the graph shown in the figure at point x0 if it passes through the point with coordinates (x0; f(x0)) and has an angular coefficient f"(x0). Find such a coefficient, Knowing the features of a tangent, it’s not difficult.

You will need

• - mathematical reference book;
• - a simple pencil;
• - notebook;
• - protractor;
• - compass;
• - pen.

Instructions

If the value f‘(x0) does not exist, then either there is no tangent, or it runs vertically. In view of this, the presence of a derivative of the function at the point x0 is due to the existence of a non-vertical tangent tangent to the graph of the function at the point (x0, f(x0)). In this case, the angular coefficient of the tangent will be equal to f "(x0). Thus, it becomes clear geometric meaning derivative – calculation of the slope of the tangent.

Draw additional tangents that would be in contact with the graph of the function at points x1, x2 and x3, and also mark the angles formed by these tangents with the x-axis (this angle is counted in the positive direction from the axis to the tangent line). For example, the angle, that is, α1, will be acute, the second (α2) will be obtuse, and the third (α3) equal to zero, since the tangent line is parallel to the OX axis. In this case, tangent obtuse angle– negative, the tangent of the acute angle is positive, and at tg0 the result is zero.

note

Correctly determine the angle formed by the tangent. To do this, use a protractor.

Helpful advice

Two inclined lines will be parallel if their angular coefficients are equal to each other; perpendicular if the product of the angular coefficients of these tangents is equal to -1.

Sources:

• Tangent to the graph of a function

Cosine, like sine, is classified as a “direct” trigonometric function. Tangent (together with cotangent) is classified as another pair called “derivatives”. There are several definitions of these functions that make it possible to find the tangent given by known value cosine of the same value.

Instructions

Subtract the quotient of one by the cosine given angle, and extract the square root from the result - this will be the tangent value of the angle, expressed by its cosine: tan(α)=√(1-1/(cos(α))²). Please note that in the formula the cosine is in the denominator of the fraction. The impossibility of dividing by zero precludes the use of this expression for angles equal to 90°, as well as those differing from this value by numbers that are multiples of 180° (270°, 450°, -90°, etc.).

There is an alternative way to calculate the tangent from a known cosine value. It can be used if there is no restriction on the use of others. To implement this method, first determine the angle value from a known cosine value - this can be done using the arc cosine function. Then simply calculate the tangent for the angle of the resulting value. IN general view this algorithm can be written as follows: tg(α)=tg(arccos(cos(α))).

There is also an exotic option using the definition of cosine and tangent through the acute angles of a right triangle. In this definition, cosine corresponds to the ratio of the length of the leg adjacent to the angle under consideration to the length of the hypotenuse. Knowing the value of the cosine, you can select the corresponding lengths of these two sides. For example, if cos(α) = 0.5, then the adjacent can be taken equal to 10 cm, and the hypotenuse - 20 cm. The specific numbers do not matter here - you will get the same and correct numbers with any values ​​that have the same . Then, using the Pythagorean theorem, determine the length of the missing side - opposite leg. It will be equal square root from the difference between the lengths of the squared hypotenuse and famous leg: √(20²-10²)=√300. By definition, the tangent corresponds to the ratio of the lengths of the opposite and adjacent legs (√300/10) - calculate it and get the tangent value found using classical definition cosine.

Sources:

• cosine through tangent formula

One of trigonometric functions, most often denoted by the letters tg, although the designations tan are also found. The easiest way to represent the tangent is as a sine ratio angle to its cosine. It is odd periodic and not continuous function, each cycle of which equal to the number Pi, and the break point corresponds to half this number.

In mathematics, one of the parameters describing the position of a line on Cartesian plane coordinates is the slope of this line. This parameter characterizes the slope of the straight line to the abscissa axis. To understand how to find the slope, first recall the general form of the equation of a straight line in the XY coordinate system.

In general, any straight line can be represented by the expression ax+by=c, where a, b and c are arbitrary real numbers, but necessarily a 2 + b 2 ≠ 0.

Using simple transformations, such an equation can be brought to the form y=kx+d, in which k and d are real numbers. The number k is the slope, and the equation of a line of this type is called an equation with a slope. It turns out that to find the angular coefficient, you just need to bring original equation to the above type. For a more complete understanding, consider a specific example:

Problem: Find the slope of the line given by the equation 36x - 18y = 108

Solution: Let's transform the original equation.

Answer: The required slope of this line is 2.

If, during the transformation of the equation, we received an expression like x = const and as a result we cannot represent y as a function of x, then we are dealing with a straight line parallel to the X axis. The angular coefficient of such a straight line is equal to infinity.

For lines expressed by an equation like y = const, the slope is zero. This is typical for straight lines parallel to the abscissa axis. For example:

Problem: Find the slope of the line given by the equation 24x + 12y - 4(3y + 7) = 4

Solution: Let's bring the original equation to its general form

24x + 12y - 12y + 28 = 4

It is impossible to express y from the resulting expression, therefore the angular coefficient of this line is equal to infinity, and the line itself will be parallel to the Y axis.

Geometric meaning

For a better understanding, let's look at the picture:

In the figure we see a graph of a function like y = kx. To simplify, let’s take the coefficient c = 0. In the triangle OAB, the ratio of side BA to AO will be equal to the angular coefficient k. At the same time, the ratio VA/AO is the tangent of the acute angle α in right triangle OAV. It turns out that the angular coefficient of the straight line is equal to the tangent of the angle that this straight line makes with the abscissa axis of the coordinate grid.

Solving the problem of how to find the angular coefficient of a straight line, we find the tangent of the angle between it and the X axis of the coordinate grid. Boundary cases, when the line in question is parallel to the coordinate axes, confirm the above. Indeed, for a straight line described by the equation y=const, the angle between it and the abscissa axis is zero. The tangent of the zero angle is also zero and the slope is also zero.

For straight lines perpendicular to the x-axis and described by the equation x=const, the angle between them and the X-axis is 90 degrees. Tangent right angle is equal to infinity, and the angular coefficient of similar straight lines is also equal to infinity, which confirms what was written above.

Tangent slope

A common task often encountered in practice is also to find the slope of a tangent to the graph of a function at a certain point. A tangent is a straight line, therefore the concept of slope is also applicable to it.

To figure out how to find the slope of a tangent, we will need to recall the concept of derivative. The derivative of any function at some point is a constant, numerically equal to tangent the angle formed between the tangent at a specified point to the graph of this function and the abscissa axis. It turns out that to determine the angular coefficient of the tangent at the point x 0, we need to calculate the value of the derivative of the original function at this point k = f"(x 0). Let's look at the example:

Problem: Find the slope of the line tangent to the function y = 12x 2 + 2xe x at x = 0.1.

Solution: Find the derivative of the original function in general form

y"(0.1) = 24. 0.1 + 2. 0.1. e 0.1 + 2. e 0.1

Answer: The required slope at point x = 0.1 is 4.831

The derivative of a function is one of difficult topics V school curriculum. Not every graduate will answer the question of what a derivative is.

This article explains in a simple and clear way what a derivative is and why it is needed.. We will not now strive for mathematical rigor in the presentation. The most important thing is to understand the meaning.

Let's remember the definition:

The derivative is the rate of change of a function.

The figure shows graphs of three functions. Which one do you think is growing faster?

The answer is obvious - the third. It has the highest rate of change, that is, the largest derivative.

Here's another example.

Kostya, Grisha and Matvey got jobs at the same time. Let's see how their income changed during the year:

The graph shows everything at once, isn’t it? Kostya’s income more than doubled in six months. And Grisha’s income also increased, but just a little. And Matvey’s income decreased to zero. The starting conditions are the same, but the rate of change of the function, that is derivative, - different. As for Matvey, his income derivative is generally negative.

Intuitively, we easily estimate the rate of change of a function. But how do we do this?

What we're really looking at is how steeply the graph of a function goes up (or down). In other words, how quickly does y change as x changes? Obviously, the same function in different points may have different meaning derivative - that is, it can change faster or slower.

The derivative of a function is denoted .

We'll show you how to find it using a graph.

A graph of some function has been drawn. Let's take a point with an abscissa on it. Let us draw a tangent to the graph of the function at this point. We want to estimate how steeply the graph of a function goes up. A convenient value for this is tangent of the tangent angle.

The derivative of a function at a point is equal to the tangent of the tangent angle drawn to the graph of the function at this point.

Please note that as the angle of inclination of the tangent we take the angle between the tangent and the positive direction of the axis.

Sometimes students ask what a tangent to the graph of a function is. This is a straight line that has only one common point with a graph, and as shown in our figure. It looks like a tangent to a circle.

Let's find it. We remember that the tangent of an acute angle in a right triangle equal to the ratio the opposite side to the adjacent one. From the triangle:

We found the derivative using a graph without even knowing the formula of the function. Such problems are often found in the Unified State Examination in mathematics under the number.

There is another important relationship. Recall that the straight line is given by the equation

The quantity in this equation is called slope of a straight line. It is equal to the tangent of the angle of inclination of the straight line to the axis.

.

We get that

Let's remember this formula. It expresses the geometric meaning of the derivative.

The derivative of a function at a point is equal to the slope of the tangent drawn to the graph of the function at that point.

In other words, the derivative is equal to the tangent of the tangent angle.

We have already said that the same function can have different derivatives at different points. Let's see how the derivative is related to the behavior of the function.

Let's draw a graph of some function. Let this function increase in some areas, and decrease in others, and with at different speeds. And let this function have maximum and minimum points.

At a point the function increases. The tangent to the graph drawn at the point forms sharp corner; with positive axis direction. This means that the derivative at the point is positive.

At the point our function decreases. The tangent at this point forms an obtuse angle; with positive axis direction. Since the tangent of an obtuse angle is negative, the derivative at the point is negative.

Here's what happens:

If a function is increasing, its derivative is positive.

If it decreases, its derivative is negative.

What will happen at the maximum and minimum points? We see that at the points (maximum point) and (minimum point) the tangent is horizontal. Therefore, the tangent of the tangent at these points is zero, and the derivative is also zero.

Point - maximum point. At this point, the increase in the function is replaced by a decrease. Consequently, the sign of the derivative changes at the point from “plus” to “minus”.

At the point - the minimum point - the derivative is also zero, but its sign changes from “minus” to “plus”.

Conclusion: using the derivative we can find out everything that interests us about the behavior of a function.

If the derivative is positive, then the function increases.

If the derivative is negative, then the function decreases.

At the maximum point, the derivative is zero and changes sign from “plus” to “minus”.

At the minimum point, the derivative is also zero and changes sign from “minus” to “plus”.

Let's write these conclusions in the form of a table:

Let's make two small clarifications. You will need one of them when solving the problem. Another - in the first year, with a more serious study of functions and derivatives.

It is possible that the derivative of a function at some point is equal to zero, but the function has neither a maximum nor a minimum at this point. This is the so-called :

At a point, the tangent to the graph is horizontal and the derivative is zero. However, before the point the function increased - and after the point it continues to increase. The sign of the derivative does not change - it remains positive as it was.

It also happens that at the point of maximum or minimum the derivative does not exist. On the graph, this corresponds to a sharp break, when it is impossible to draw a tangent at a given point.

How to find the derivative if the function is given not by a graph, but by a formula? In this case it applies

IN previous chapter it was shown that by choosing a certain coordinate system on the plane, we can geometric properties, which characterizes the points of the line under consideration, is expressed analytically by an equation between the current coordinates. Thus we get the equation of the line. This chapter will look at straight line equations.

To write the equation of a straight line in Cartesian coordinates, you need to somehow set the conditions that determine its position relative to the coordinate axes.

First, we will introduce the concept of the angular coefficient of a line, which is one of the quantities characterizing the position of a line on a plane.

Let's call the angle of inclination of the straight line to the Ox axis the angle by which the Ox axis needs to be rotated so that it coincides with the given line (or is parallel to it). As usual, we will consider the angle taking into account the sign (the sign is determined by the direction of rotation: counterclockwise or clockwise). Since an additional rotation of the Ox axis through an angle of 180° will again align it with the straight line, the angle of inclination of the straight line to the axis can not be chosen unambiguously (to within a term, a multiple of ).

The tangent of this angle is determined uniquely (since changing the angle does not change its tangent).

The tangent of the angle of inclination of the straight line to the Ox axis is called the angular coefficient of the straight line.

The angular coefficient characterizes the direction of the straight line (we do not distinguish between the two mutually opposite directions straight). If the slope of a line is zero, then the line is parallel to the x-axis. With a positive angular coefficient, the angle of inclination of the straight line to the Ox axis will be acute (we are considering here the smallest positive value tilt angle) (Fig. 39); Moreover, the larger the angular coefficient, the larger angle its inclination to the Ox axis. If the angular coefficient is negative, then the angle of inclination of the straight line to the Ox axis will be obtuse (Fig. 40). Note that a straight line perpendicular to the Ox axis does not have an angular coefficient (the tangent of the angle does not exist).