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 slope f"(x0). Finding such a coefficient, knowing the features of the tangent, is 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.
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.
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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 function graph 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 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 learn 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:
| increases | maximum point | decreases | minimum point | increases | |
| + | 0 | - | 0 | + |
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
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 necessary to calculate the angular coefficient of a tangent.
It will help you do this educational portal"Shkolkovo". Our experts prepared and presented theoretical and practical material as accessible as 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 similar tasks in the Unified State Exam must be remembered 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 Exam problems on the derivative, in which it is necessary to calculate the tangent of the tangent angle. For clarity, it is best to plot the graph on the OXY plane.
If you have already read base material on the topic of derivatives and are ready to begin solving problems on calculating the tangent of the tangent angle, similar 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 completing tasks various levels difficulties. If necessary, the exercise can be saved in the “Favorites” section so that you can discuss the solution with the teacher later.
