The exponential function y. Exponential function
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 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:
.
Insofar as 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 function complex number 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 view
φ = φ 0 + 2 pn,
where n is an integer. Therefore, the function f (z) is also ambiguous. Often considered its main importance
.
Expansion in series
.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.
1. An exponential function is a function of the form y(x) \u003d a x, depending on the exponent x, with a constant value of the base of the degree a, where a > 0, a ≠ 0, xϵR (R is the set of real numbers).
Consider graph of the function if the base does not satisfy the condition: a>0
a) a< 0
If a< 0 – возможно возведение в целую степень или в rational degree with an odd score.
a = -2
If a = 0, the function y = is defined and has constant value 0
c) a \u003d 1
If a = 1 - the function y = is defined and has a constant value of 1
Function domain (OOF) Area of allowable function values (ODZ) 3. Zeros of the function (y = 0) 4. Points of intersection with the y-axis (x = 0) 5. Increasing, decreasing function If , then the function f(x) increases 6. Even, odd functions The function y = is not symmetrical about the 0y axis and about the origin, therefore it is neither even nor odd. (general function) 7. The function y \u003d has no extremums 8. Properties of a degree with a real exponent: Let a > 0; a≠1 Then for xϵR; yϵR: Degree monotonicity properties: if , then If a> 0, then . 9. Relative location of the function The larger the base a, the closer to the x and y axes a > 1, a = 20 Example 1 Majority Decision math problems somehow connected with the transformation of numerical, algebraic or functional expressions. This applies especially to the solution. In the USE variants in mathematics, this type of task includes, in particular, task C3. Learning how to solve C3 tasks is important not only for the purpose of successful passing the exam, but also for the reason that this skill is useful when studying a mathematics course in higher education. Performing tasks C3, you have to decide different kinds equations and inequalities. Among them are rational, irrational, exponential, logarithmic, trigonometric, containing modules (absolute values), as well as combined ones. This article discusses the main types of exponential equations and inequalities, as well as various methods their decisions. Read about solving other types of equations and inequalities under the heading "" in articles devoted to methods for solving C3 problems from USE options mathematics. Before proceeding to the analysis of specific exponential equations and inequalities, as a math tutor, I suggest you brush up on some of the theoretical material that we will need. View function y = a x, where a> 0 and a≠ 1, called exponential function. Main exponential function properties y = a x: The graph of the exponential function is exhibitor: Graphs of exponential functions (exponents) indicative called equations in which the unknown variable is found only in exponents of any powers. For solutions exponential equations you need to know and be able to use the following simple theorem: Theorem 1. exponential equation a f(x) = a g(x) (where a > 0, a≠ 1) is equivalent to the equation f(x) = g(x). In addition, it is useful to remember the basic formulas and actions with degrees: Title="(!LANG:Rendered by QuickLaTeX.com">!} Example 1 Solve the equation: Solution: use the above formulas and substitution: The equation then becomes: Received discriminant quadratic equation positive: Title="(!LANG:Rendered by QuickLaTeX.com">!} This means that this equation has two roots. We find them: Going back to substitution, we get: The second equation has no roots, since the exponential function is strictly positive over the entire domain of definition. Let's solve the second one: Taking into account what was said in Theorem 1, we pass to the equivalent equation: x= 3. This will be the answer to the task. Answer: x = 3. Example 2 Solve the equation: Solution: the equation has no restrictions on the area of admissible values, since the radical expression makes sense for any value x(exponential function y = 9 4 -x positive and not equal to zero). We solve the equation by equivalent transformations using the rules of multiplication and division of powers: The last transition was carried out in accordance with Theorem 1. Answer:x= 6.
Example 3 Solve the equation: Solution: both sides of the original equation can be divided by 0.2 x. This transition will be equivalent, since this expression is greater than zero for any value x(the exponential function is strictly positive on its domain). Then the equation takes the form: Answer: x = 0. Example 4 Solve the equation: Solution: we simplify the equation to an elementary one by equivalent transformations using the rules of division and multiplication of powers given at the beginning of the article: Dividing both sides of the equation by 4 x, as in the previous example, is an equivalent transformation, since this expression is not equal to zero for any values x. Answer: x = 0. Example 5 Solve the equation: Solution: function y = 3x, standing on the left side of the equation, is increasing. Function y = —x-2/3, standing on the right side of the equation, is decreasing. This means that if the graphs of these functions intersect, then at most at one point. In this case, it is easy to guess that the graphs intersect at the point x= -1. There will be no other roots. Answer: x = -1. Example 6 Solve the equation: Solution: we simplify the equation by equivalent transformations, bearing in mind everywhere that the exponential function is strictly greater than zero for any value x and using the rules for calculating the product and partial powers given at the beginning of the article: Answer: x = 2. indicative called inequalities in which the unknown variable is contained only in the exponents of some powers. For solutions exponential inequalities knowledge of the following theorem is required: Theorem 2. If a> 1, then the inequality a f(x) > a g(x) is equivalent to an inequality of the same meaning: f(x) > g(x). If 0< a < 1, то показательное неравенство a f(x) > a g(x) is equivalent to an inequality of the opposite meaning: f(x) < g(x). Example 7 Solve the inequality: Solution: represent the original inequality in the form: Divide both parts of this inequality by 3 2 x, and (due to the positiveness of the function y= 3 2x) the inequality sign will not change: Let's use a substitution: Then the inequality takes the form: So, the solution to the inequality is the interval: passing to the reverse substitution, we get: The left inequality, due to the positiveness of the exponential function, is fulfilled automatically. Using the well-known property of the logarithm, we pass to the equivalent inequality: Since the base of the degree is a number greater than one, equivalent (by Theorem 2) will be the transition to the following inequality: So we finally get answer: Example 8 Solve the inequality: Solution: using the properties of multiplication and division of powers, we rewrite the inequality in the form: Let's introduce a new variable: With this substitution, the inequality takes the form: Multiply the numerator and denominator of the fraction by 7, we get the following equivalent inequality: So, the inequality is satisfied by the following values of the variable t: Then, going back to substitution, we get: Since the base of the degree here is greater than one, it is equivalent (by Theorem 2) to pass to the inequality: Finally we get answer: Example 9 Solve the inequality: Solution: We divide both sides of the inequality by the expression: It is always greater than zero (because the exponential function is positive), so the inequality sign does not need to be changed. We get: t , which are in the interval: Passing to the reverse substitution, we find that the original inequality splits into two cases: The first inequality has no solutions due to the positivity of the exponential function. Let's solve the second one: Example 10 Solve the inequality: Solution: Parabola branches y = 2x+2-x 2 are directed downwards, hence it is bounded from above by the value it reaches at its apex: Parabola branches y = x 2 -2x+2, which is in the indicator, are directed upwards, which means it is limited from below by the value that it reaches at its top: At the same time, the function turns out to be bounded from below y = 3 x 2 -2x+2 on the right side of the equation. It reaches its smallest value at the same point as the parabola in the index, and this value is equal to 3 1 = 3. So, the original inequality can only be true if the function on the left and the function on the right take the value , equal to 3 (the intersection of the ranges of these functions is only this number). This condition is satisfied at a single point x = 1. Answer: x= 1. To learn how to solve exponential equations and inequalities you need to constantly train in their solution. In this difficult matter, various teaching aids, problem books in elementary mathematics, collections of competitive problems, mathematics classes at school, as well as individual lessons with a professional tutor. I sincerely wish you success in your preparation and brilliant results in the exam. P.S. Dear guests! Please do not write requests for solving your equations in the comments. Unfortunately, I don't have time for this at all. Such messages will be deleted. Please read the article. Perhaps in it you will find answers to questions that did not allow you to solve your task on your own.
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 verification 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.
2. Consider the exponential function in more detail:
0
If , then the function f(x) decreases
Function y= , at 0
This follows from the monotonicity properties of a degree with a real exponent.
b > 0; b≠1
For example:
The exponential function is continuous at any point ϵ R.
If a0, then the exponential function takes a form close to y = 0.
If a1, then further from the axes x and y and the graph takes the form close to the function y \u003d 1.
Plot y=
Exponential function
What is an exponential function?
Graph of an exponential function
Solution of exponential equations
Solving exponential inequalities
Sergey Valerievich
Option 1 Answer B) 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 IN) - 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) 
- 
IN) 
- 
G) 
- 
- 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 greater value at a specific 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?
On the interval, which of the functions has the greatest value at a particular point?
Why exponential functions with different grounds 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. No. 000. Find the largest and smallest value 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
|
Decreasing function
|
Increasing function
|
the largest value of the function on the segment
the smallest value of the function on the interval
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.