Degree exponential. Exponential function. What is an exponential function

Concentration of attention:

Definition. Function species is called exponential function .

Comment. Base exclusion a numbers 0; 1 and negative values a explained by the following circumstances:

The analytic expression itself a x in these cases, it retains its meaning and can be encountered in solving problems. For example, for the expression x y dot x = 1; y = 1 enters the range of acceptable values.

Construct graphs of functions: and .



Graph of an exponential function
y= a x, a > 1	y= a x , 0< a < 1

Properties of the exponential function

Properties of the exponential function	y= a x, a > 1	y= a x , 0< a < 1
Function scope
2. Range of function values
3. Intervals of comparison with the unit	at x> 0, a x > 1	at x > 0, 0< a x < 1
	at x < 0, 0< a x < 1	at x < 0, a x > 1
4. Even, odd.	The function is neither even nor odd (general function).
5. Monotony.	increases monotonically by R	decreases monotonically by R
6. Extremes.	The exponential function has no extrema.
7.Asymptote	Axis O x is a horizontal asymptote.
8. For any real values x and y;

When the table is filled, tasks are solved in parallel with the filling.

Task number 1. (To find the domain of the function).

What argument values are valid for functions:

Task number 2. (To find the range of the function).

The figure shows a graph of a function. Specify the scope and scope of the function:

Task number 3. (To indicate the intervals of comparison with the unit).

Compare each of the following powers with one:

Task number 4. (To study the function for monotonicity).

Compare real numbers by magnitude m and n if:

Task number 5. (To study the function for monotonicity).

Make a conclusion about the basis a, if:

y(x) = 10 x ; f(x) = 6 x ; z(x) - 4x

How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?

In one coordinate plane, graphs of functions are plotted:

y(x) = (0,1) x ; f(x) = (0.5) x ; z(x) = (0.8) x .

How are the graphs of exponential functions relative to each other for x > 0, x = 0, x< 0?

Number one of the most important constants in mathematics. By definition, it equal to the limit of the sequence with unlimited increasing n . Designation e introduced Leonard Euler in 1736. He calculated the first 23 digits of this number in decimal notation, and the number itself was named after Napier "the non-peer number."

Number e plays a special role in mathematical analysis. Exponential function with base e, called the exponent and denoted y = e x.

First signs numbers e easy to remember: two, a comma, seven, the year of Leo Tolstoy's birth - two times, forty-five, ninety, forty-five.

Homework:

Kolmogorov p. 35; No. 445-447; 451; 453.

Repeat the algorithm for constructing graphs of functions containing a variable under the module sign.

Lesson #2

Topic: An exponential function, its properties and graph.

Target: Check the quality of assimilation of the concept of "exponential function"; to form skills in recognizing an exponential function, in using its properties and graphs, to teach students to use the analytical and graphic forms of recording an exponential function; provide a working environment in the classroom.

Equipment: board, posters

Lesson Form: classroom

Type of lesson: practical lesson

Lesson type: skill training lesson

Lesson plan

1. Organizational moment

2. Independent work and checking homework

3. Problem solving

4. Summing up

5. Homework

During the classes.

1. Organizational moment :

Hello. Open notebooks, write down today's date and the topic of the lesson "Exponential function". Today we will continue to study the exponential function, its properties and graph.

2. Independent work and checking homework .

Target: check the quality of assimilation of the concept of "exponential function" and check the fulfillment of the theoretical part of the homework

Method: test task, frontal survey

As homework, you were given numbers from the problem book and a paragraph from the textbook. We will not check the execution of numbers from the textbook now, but you will hand over your notebooks at the end of the lesson. Now the theory will be tested in the form of a small test. The task is the same for everyone: you are given a list of functions, you must find out which of them are indicative (underline them). And next to the exponential function, you need to write whether it is increasing or decreasing.

Option 1

Answer

D) - exponential, decreasing

Option 2

Answer

D) - exponential, decreasing

D) - indicative, increasing

Option 3

Answer

BUT) - indicative, increasing

B) - exponential, decreasing

Option 4

Answer

BUT) - exponential, decreasing

AT) - indicative, increasing

Now let's remember together what function is called exponential?

A function of the form , where and , is called an exponential function.

What is the scope of this function?

All real numbers.

What is the range of the exponential function?

All positive real numbers.

Decreases if the base is greater than zero but less than one.

When does an exponential function decrease on its domain?

Increases if the base is greater than one.

3. Problem solving

Target: to form skills in recognizing an exponential function, in using its properties and graphs, to teach students to use the analytical and graphical forms of recording an exponential function

Method: demonstration by the teacher of solving typical problems, oral work, work at the blackboard, work in a notebook, teacher's conversation with students.

The properties of the exponential function can be used when comparing 2 or more numbers. For example: No. 000. Compare the values and if a) ..gif" width="37" height="20 src=">, then this is quite a tricky job: we would have to take the cube root of 3 and 9, and compare them. But we know that increases, this is in its own queue means that when the argument increases, the value of the function increases, that is, it is enough for us to compare the values of the argument with each other and, obviously, that (can be demonstrated on a poster with an increasing exponential function). And always when solving such examples, first determine the base of the exponential function, compare with 1, determine monotonicity and proceed to comparing the arguments. In the case of a decreasing function: as the argument increases, the value of the function decreases, therefore, the inequality sign is changed when moving from the inequality of arguments to the inequality of functions. Then we solve orally: b)

AT)

- No. 000. Compare the numbers: a) and

Therefore, the function is increasing, then

Why ?

Increasing function and

Therefore, the function is decreasing, then

Both functions increase over their entire domain of definition, since they are exponential with a base greater than one.

What is the meaning of it?

We build charts:

Which function grows faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25">

Which function decreases faster when striving https://pandia.ru/text/80/379/images/image062_0.gif" width="20 height=25" height="25">

On the interval, which of the functions has the greatest value at a particular point?

D), https://pandia.ru/text/80/379/images/image068_0.gif" width="69" height="57 src=">. First, let's find out the scope of these functions. Do they coincide?

Yes, the domain of these functions is all real numbers.

Name the scope of each of these functions.

The ranges of these functions coincide: all positive real numbers.

Determine the type of monotonicity of each of the functions.

All three functions decrease over their entire domain of definition, since they are exponential with a base less than one and greater than zero.

What is the singular point of the graph of an exponential function?

What is the meaning of it?

Whatever the base of the degree of an exponential function, if the exponent is 0, then the value of this function is 1.

We build charts:

Let's analyze the charts. How many intersection points do function graphs have?

Which function decreases faster when striving? https://pandia.ru/text/80/379/images/image070.gif

Which function grows faster when striving? https://pandia.ru/text/80/379/images/image070.gif

On the interval, which of the functions has the greatest value at a particular point?

Why do exponential functions with different bases have only one point of intersection?

The exponential functions are strictly monotonic over their entire domain of definition, so they can only intersect at one point.

The next task will focus on using this property. № 000. Find the largest and smallest value of a given function on a given interval a). Recall that a strictly monotonic function takes its minimum and maximum values at the ends of a given interval. And if the function is increasing, then its largest value will be at the right end of the segment, and the smallest at the left end of the segment (demonstration on the poster, using the exponential function as an example). If the function is decreasing, then its largest value will be at the left end of the segment, and the smallest at the right end of the segment (demonstration on the poster, using the exponential function as an example). The function is increasing, because, therefore, the smallest value of the function will be at the point https://pandia.ru/text/80/379/images/image075_0.gif" width="145" height="29">. Points b ) , in) d) solve notebooks on your own, we will check it orally.

Students solve the problem in their notebook

Decreasing function

the largest value of the function on the segment

the smallest value of the function on the interval

Increasing function

the smallest value of the function on the interval

the largest value of the function on the segment

- № 000. Find the largest and smallest value of a given function on a given interval a) . This task is almost the same as the previous one. But here is given not a segment, but a ray. We know that the function is increasing, and it has neither the largest nor the smallest value on the entire number line https://pandia.ru/text/80/379/images/image063_0.gif" width="68" height ="20">, and tends to at , i.e., on the ray, the function at tends to 0, but does not have its smallest value, but it has the largest value at the point . Points b) , in) , G) Solve your own notebooks, we will check it orally.

Exponential function is a generalization of the product of n numbers equal to a :
y (n) = a n = a a a a,
to the set of real numbers x :
y (x) = x.
Here a is a fixed real number, which is called the base of the exponential function.
An exponential function with base a is also called exponential to base a.

The generalization is carried out as follows.
For natural x = 1, 2, 3,... , the exponential function is the product of x factors:
.
Moreover, it has the properties (1.5-8) (), which follow from the rules for multiplying numbers. At zero and negative values of integers , the exponential function is determined by formulas (1.9-10). For fractional values x = m/n of rational numbers, , it is determined by formula (1.11). For real , the exponential function is defined as the limit of the sequence:
,
where is an arbitrary sequence of rational numbers converging to x : .
With this definition, the exponential function is defined for all , and satisfies the properties (1.5-8), as well as for natural x .

A rigorous mathematical formulation of the definition of an exponential function and a proof of its properties is given on the page "Definition and proof of the properties of an exponential function".

Properties of the exponential function

The exponential function y = a x has the following properties on the set of real numbers () :
(1.1) is defined and continuous, for , for all ;
(1.2) when a ≠ 1 has many meanings;
(1.3) strictly increases at , strictly decreases at ,
is constant at ;
(1.4) at ;
at ;
(1.5) ;
(1.6) ;
(1.7) ;
(1.8) ;
(1.9) ;
(1.10) ;
(1.11) , .

Other useful formulas
.
The formula for converting to an exponential function with a different power base:

For b = e , we get the expression of the exponential function in terms of the exponent:

Private values

, , , , .

The figure shows graphs of the exponential function
y (x) = x
for four values degree bases:a= 2 , a = 8 , a = 1/2 and a = 1/8 . It can be seen that for a > 1 exponential function is monotonically increasing. The larger the base of the degree a, the stronger the growth. At 0 < a < 1 exponential function is monotonically decreasing. The smaller the exponent a, the stronger the decrease.

Ascending, descending

The exponential function at is strictly monotonic, so it has no extrema. Its main properties are presented in the table.

	y = a x , a > 1	y = x, 0 < a < 1
Domain	- ∞ < x < + ∞	- ∞ < x < + ∞
Range of values	0 < y < + ∞	0 < y < + ∞
Monotone	increases monotonically	decreases monotonically
Zeros, y= 0	No	No
Points of intersection with the y-axis, x = 0	y= 1	y= 1
	+ ∞	0
	0	+ ∞

Inverse function

The reciprocal of an exponential function with a base of degree a is the logarithm to base a.

If , then
.
If , then
.

Differentiation of the exponential function

To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the rule for differentiating a complex function.

To do this, you need to use the property of logarithms
and the formula from the table of derivatives:
.

Let an exponential function be given:
.
We bring it to the base e:

We apply the rule of differentiation of a complex function. To do this, we introduce a variable

Then

From the table of derivatives we have (replace the variable x with z ):
.
Since is a constant, the derivative of z with respect to x is
.
According to the rule of differentiation of a complex function:
.

Derivative of exponential function

.
Derivative of the nth order:
.
Derivation of formulas > > >

An example of differentiating an exponential function

Find the derivative of a function
y= 35 x

Solution

We express the base of the exponential function in terms of the number e.
3 = e log 3
Then
.
We introduce a variable
.
Then

From the table of derivatives we find:
.
Because the 5ln 3 is a constant, then the derivative of z with respect to x is:
.
According to the rule of differentiation of a complex function, we have:
.

Answer

Integral

Expressions in terms of complex numbers

Consider the complex number function z:
f (z) = az
where z = x + iy ; i 2 = - 1 .
We express the complex constant a in terms of the modulus r and the argument φ :
a = r e i φ
Then

.
The argument φ is not uniquely defined. In general
φ = φ 0 + 2 pn,
where n is an integer. Therefore, the function f (z) is also ambiguous. Often considered its main importance
.