Provides reference data on the exponential function - basic properties, graphs and formulas. The following issues are considered: domain of definition, set of values, monotonicity, inverse function, derivative, integral, expansion in power series and representation using complex numbers.
Definition
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) = ax.
Here a is a fixed real number, which is called basis of the exponential function.
An exponential function with base a is also called exponent 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 properties (1.5-8) (), which follow from the rules for multiplying numbers. For zero and negative values of integers, the exponential function is determined using formulas (1.9-10). For fractional values x = m/n rational numbers, , it is determined by formula (1.11). For real ones, the exponential function is defined as sequence limit:
,
where is an arbitrary sequence of rational numbers converging to x: .
With this definition, the exponential function is defined for all , and satisfies properties (1.5-8), as for natural x.
A rigorous mathematical formulation of the definition of an exponential function and the 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)
defined and continuous, for , for all ;
(1.2)
for 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.
.
Formula for converting to an exponential function with a different exponent base:
When b = e, we obtain the expression of the exponential function through the exponential:
Private values
, , , , .
The figure shows graphs of the exponential function
y (x) = ax
for four values degree bases: a = 2
, a = 8
, a = 1/2
and a = 1/8
. It can be seen that for a > 1
the exponential function increases monotonically. The larger the base of the degree a, the more strong growth. At 0
< a < 1
the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.
Ascending, descending
The exponential function is strictly monotonic and therefore has no extrema. Its main properties are presented in the table.
| y = a x , a > 1 | y = ax, 0 < a < 1 | |
| Domain | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | 0 < y < + ∞ | 0 < y < + ∞ |
| Monotone | monotonically increases | monotonically decreases |
| Zeros, y = 0 | No | No |
| Intercept points with the ordinate axis, x = 0 | y= 1 | y= 1 |
| + ∞ | 0 | |
| 0 | + ∞ |
Inverse function
The inverse of an exponential function with base a is the logarithm to base a.
If , then
.
If , then
.
Differentiation of an exponential function
To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the differentiation rule complex function.
To do this you need to use the property of logarithms
and the formula from the derivatives table:
.
Let an exponential function be given:
.
We bring it to the base e:
Let's apply the rule of differentiation of complex functions. To do this, introduce the 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 equal to
.
According to the rule of differentiation of a complex function:
.
Derivative of an exponential function
.
Derivative of nth order:
.
Deriving formulas > > >
An example of differentiating an exponential function
Find the derivative of a function
y= 3 5 x
Solution
Let's express the base of the exponential function through the number e.
3 = e ln 3
Then
.
Enter 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 equal to:
.
According to the rule of differentiation of a complex function, we have:
.
Answer
Integral
Expressions using complex numbers
Consider the complex number function z:
f (z) = a z
where z = x + iy; i 2 = - 1
.
Let us express the complex constant a in terms of modulus r and argument φ:
a = r e i φ
Then
.
The argument φ is not uniquely defined. IN general view
φ = φ 0 + 2 πn,
where n is an integer. Therefore the function f (z) is also not clear. Its main significance is often considered
.
Series expansion
.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
EXPONENTARY AND LOGARITHMIC FUNCTIONS VIII
§ 179 Basic properties of the exponential function
In this section we will study the basic properties of the exponential function
y = a x (1)
Let us remember that under A in formula (1) we mean any fixed positive number other than 1.
Property 1. The domain of an exponential function is the set of all real numbers.
In fact, with a positive A expression A x defined for any real number X .
Property 2. The exponential function accepts only positive values.
Indeed, if X > 0, then, as was proven in § 176,
A x > 0.
If X <. 0, то
A x =
Where - X already more than zero. That's why A - x > 0. But then
A x = > 0.
Finally, when X = 0
A x = 1.
The 2nd property of the exponential function has a simple graphical interpretation. It lies in the fact that the graph of this function (see Fig. 246 and 247) is located entirely above the abscissa axis.
Property 3. If A >1, then when X > 0 A x > 1, and when X < 0 A x < 1. If A < 1, тoh, on the contrary, when X > 0 A x < 1, and when X < 0 A x > 1.
This property of the exponential function also allows for a simple geometric interpretation. At A > 1 (Fig. 246) curves y = a x located above the straight line at = 1 at X > 0 and below straight line at = 1 at X < 0.
If A < 1 (рис. 247), то, наоборот, кривые y = a x located below the straight line at = 1 at X > 0 and above this line at X < 0.
Let us give a rigorous proof of the 3rd property. Let A > 1 and X - an arbitrary positive number. Let's show that
A x > 1.
If the number X rational ( X = m / n ) , That A x = A m/ n = n √a m .
Because the A > 1, then A m > 1, But the root of a number greater than one is obviously also greater than 1.
If X is irrational, then there are positive rational numbers X" And X" , which serve as decimal approximations of a number x :
X"< х < х" .
But then, by definition of a degree with an irrational exponent
A x" < A x < A x"" .
As shown above, the number A x" more than one. Therefore the number A x , greater than A x" , must also be greater than 1,
So, we have shown that when a >1 and arbitrary positive X
A x > 1.
If the number X was negative, then we would have
A x =
where the number is X would already be positive. That's why A - x > 1. Therefore,
A x = < 1.
Thus, when A > 1 and arbitrary negative x
A x < 1.
The case when 0< A < 1, легко сводится к уже рассмотренному случаю. Учащимся предлагается убедиться в этом самостоятельно.
Property 4. If x = 0, then regardless of a A x =1.
This follows from the definition of degree zero; the zero power of any number other than zero is equal to 1. Graphically, this property is expressed in the fact that for any A curve at = A x (see Fig. 246 and 247) intersects the axis at at a point with ordinate 1.
Property 5. At A >1 exponential function = A x is monotonically increasing, and for a < 1 - monotonically decreasing.
This property also allows for a simple geometric interpretation.
At A > 1 (Fig. 246) curve at = A x with growth X rises higher and higher, and when A < 1 (рис. 247) - опускается все ниже и ниже.
Let us give a rigorous proof of the 5th property.
Let A > 1 and X 2 > X 1 . Let's show that
A x 2 > A x 1
Because the X 2 > X 1 ., then X 2 = X 1 + d , Where d - some positive number. That's why
A x 2 - A x 1 = A x 1 + d - A x 1 = A x 1 (A d - 1)
By the 2nd property of the exponential function A x 1 > 0. Since d > 0, then by the 3rd property of the exponential function A d > 1. Both factors in the product A x 1 (A d - 1) are positive, therefore this product itself is positive. Means, A x 2 - A x 1 > 0, or A x 2 > A x 1, which is what needed to be proven.
So, when a > 1 function at = A x is monotonically increasing. Similarly, it is proved that when A < 1 функция at = A x is monotonically decreasing.
Consequence. If two powers of the same positive number other than 1 are equal, then their exponents are equal.
In other words, if
A b = A c (A > 0 and A =/= 1),
b = c .
Indeed, if the numbers b And With were not equal, then due to the monotonicity of the function at = A x the greater of them would correspond to A >1 greater, and when A < 1 меньшее значение этой функции. Таким образом, было бы или A b > A c , or A b < A c . Both contradict the condition A b = A c . It remains to admit that b = c .
Property 6. If a > 1, then with an unlimited increase in the argument X (X -> ∞ ) function values at = A x also grow indefinitely (at -> ∞ ). When the argument decreases without limit X (X -> -∞ ) the values of this function tend to zero while remaining positive (at->0; at > 0).
Taking into account the monotonicity of the function proved above at = A x , we can say that in the case under consideration the function at = A x monotonically increases from 0 to ∞ .
If 0 <A < 1, then with an unlimited increase in the argument x (x -> ∞), the values of the function y = a x tend to zero, while remaining positive (at->0; at > 0). When the argument x decreases without limit (X -> -∞ ) the values of this function grow unlimitedly (at -> ∞ ).
Due to the monotonicity of the function y = a x we can say that in this case the function at = A x monotonically decreases from ∞ to 0.
The 6th property of the exponential function is clearly reflected in Figures 246 and 247. We will not strictly prove it.
All we have to do is establish the range of variation of the exponential function y = a x (A > 0, A =/= 1).
Above we proved that the function y = a x takes only positive values and either increases monotonically from 0 to ∞ (at A > 1), or decreases monotonically from ∞ to 0 (at 0< A <. 1). Однако остался невыясненным следующий вопрос: не претерпевает ли функция y = a x Are there any jumps when you change? Does it take any positive values? This issue is resolved positively. If A > 0 and A =/= 1, then whatever the positive number is at 0 will definitely be found X 0 , such that
A x 0 = at 0 .
(Due to the monotonicity of the function y = a x specified value X 0 will, of course, be the only one.)
Proving this fact is beyond the scope of our program. Its geometric interpretation is that for any positive value at 0 function graph y = a x will definitely intersect with a straight line at = at 0 and, moreover, only at one point (Fig. 248).
From this we can draw the following conclusion, which we formulate as property 7.
Property 7. The area of change of the exponential function y = a x (A > 0, A =/= 1)is the set of all positive numbers.
Exercises
1368. Find the domains of definition of the following functions:
1369. Which of these numbers is greater than 1 and which is less than 1:
1370. Based on what property of the exponential function can it be stated that
a) (5 / 7) 2.6 > (5 / 7) 2.5; b) (4 / 3) 1.3 > (4 / 3) 1.2
1371. Which number is greater:
A) π - √3 or (1/ π ) - √3 ; c) (2 / 3) 1 + √6 or (2 / 3) √2 + √5 ;
b) ( π / 4) 1 + √3 or ( π / 4) 2; d) (√3) √2 - √5 or (√3) √3 - 2 ?
1372. Are the inequalities equivalent:
1373. What can be said about numbers X And at , If a x = and y , Where A - a given positive number?
1374. 1) Is it possible among all the values of the function at = 2x highlight:
2) Is it possible among all the values of the function at = 2 | x| highlight:
A) highest value; b) the smallest value?
Concentration of attention:
Definition. Function species is called exponential function .
Comment. Exclusion from base values a numbers 0; 1 and negative values a is explained by the following circumstances:
The analytical expression itself a x in these cases, it retains its meaning and can be used in solving problems. For example, for the expression x y dot x = 1; y = 1 is within 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 |
|
||
| 2. Function range | ||
| 3. Intervals of comparison with 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 (a function of general form). | |
| 5.Monotony. | monotonically increases by R | decreases monotonically by R |
| 6. Extremes. | The exponential function has no extrema. | |
| 7.Asymptote | O-axis x is a horizontal asymptote. | |
| 8. For any real values x And y; |  |
|
When the table is filled out, tasks are solved in parallel with the filling.
Task No. 1. (To find the domain of definition of a function).
What argument values are valid for functions:
Task No. 2. (To find the range of values of a function).
The figure shows the graph of the function. Specify the domain of definition and range of values of the function:
 |
 |
 |
 |
Task No. 3. (To indicate the intervals of comparison with one).
Compare each of the following powers with one:
Task No. 4. (To study the function for monotonicity).
Compare real numbers by size m And n If:
Task No. 5. (To study the function for monotonicity).
Draw a conclusion regarding 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?
One coordinate plane graphs of functions were constructed:
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 entered Leonard Euler
in 1736. He calculated the first 23 digits of this number in decimal notation, and the number itself was named in honor of Napier the “non-Pierre number.”
Number e plays special role in mathematical analysis. Exponential function with base e, called exponent and is designated y = e x. First signs numbers e easy to remember: two, comma, seven, year of birth of Leo Tolstoy - two times, forty-five, ninety, forty-five. |
 |
Homework:
Kolmogorov paragraph 35; No. 445-447; 451; 453.
Repeat the algorithm for constructing graphs of functions containing a variable under the modulus sign.
1. An exponential function is a function of the form y(x) = 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).
Let's 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 рациональную степень с нечетным показателем.
a = -2
If a = 0, the function y = is defined and has a constant value of 0
c) a =1
If a = 1, the function y = is defined and has a constant value of 1
Function Domain (DOF) Range of permissible function values (APV) 3. Zeros of the function (y = 0) 4. Points of intersection with the ordinate axis oy (x = 0) 5. Increasing, decreasing functions If , then the function f(x) increases 6. Even, odd function The function y = is not symmetrical with respect to the 0y axis and with respect to the origin, therefore it is neither even nor odd. (General function) 7. The function y = has no extrema 8. Properties of a degree with a real exponent: Let a > 0; a≠1 Then for xϵR; yϵR: Properties of degree monotonicity: if , then If a> 0, then . 9. Relative position of the function The larger the base a, the closer to the axes x and oy a > 1, a = 20 Example 1. Let's find the value of the expression for various rational values of the variable x=2; 0; -3; - Note that no matter what number we substitute for the variable x, we can always find the value of this expression. This means that we are considering an exponential function (E is equal to three to the power of x), defined on the set of rational numbers: . Let's build a graph of this function by compiling a table of its values. Let's draw a smooth line passing through these points (Figure 1) Using the graph of this function, let’s consider its properties: 3.Increases throughout the entire area of definition. 8. The function is convex downward. If we construct graphs of functions in one coordinate system; y=(y is equal to two to the power of x, y is equal to five to the power of x, y is equal to seven to the power of x), then you can see that they have the same properties as y=(y is equal to three to the power of x) (Fig. .2), that is, all functions of the form y = (a is equal to a to the x power, for a greater than one) will have such properties Let's plot the function: 1. Compiling a table of its values. Let us mark the obtained points on the coordinate plane. Let's draw a smooth line passing through these points (Figure 3). Using the graph of this function, we indicate its properties: 1. The domain of definition is the set of all real numbers. 2. Is neither even nor odd. 3.Decreases throughout the entire domain of definition. 4. Has neither the largest nor the smallest values. 5.Limited below, but not limited above. 6.Continuous throughout the entire domain of definition. 7. range of values from zero to plus infinity. 8. The function is convex downward. Similarly, if we construct graphs of functions in one coordinate system; y = (y is equal to one-half to the power of x, y is equal to one-fifth to the power of x, y is equal to one-seventh to the power of x), then you can notice that they have the same properties as y = (y is equal to one-third to the power x (Fig. 4), that is, all functions of the form y = (the y is equal to one divided by a to the x power, with a greater than zero but less than one) will have such properties. Let us construct graphs of functions in one coordinate system This means that the graphs of the functions y=y= will also be symmetrical (y is equal to a to the x power and y is equal to one divided by a to the x power) for the same value of a. Let us summarize what has been said by defining the exponential function and indicating its main properties: Definition: A function of the form y=, where (a is equal to a to the power x, where a is positive and different from one), is called an exponential function. It is necessary to remember the differences between the exponential function y= and the power function y=, a=2,3,4,…. both audibly and visually. The exponential function X is a degree, and power function X is the basis. Example1: Solve the equation (three to the power x equals nine) (Y is equal to three to the power of X and Y is equal to nine) Fig. 7 Note that they have one common point M (2;9) (em with coordinates two; nine), which means that the abscissa of the point will be the root given equation. That is, the equation has a single root x = 2. Example 2: Solve the equation In one coordinate system, we will construct two graphs of the function y= (the y is equal to five to the power of x and the y is equal to one twenty-fifth) Fig. 8. The graphs intersect at one point T (-2; (te with coordinates minus two; one twenty-fifth). This means that the root of the equation is x = -2 (the number minus two). Example 3: Solve the inequality In one coordinate system we will construct two graphs of the function y= (Y is equal to three to the power of X and Y is equal to twenty-seven). Fig.9 The graph of the function is located above the graph of the function y=at x Therefore, the solution to the inequality is the interval (from minus infinity to three) Example 4: Solve the inequality In one coordinate system, we will construct two graphs of the function y= (the y is equal to one fourth to the power of x and the y is equal to sixteen). (Fig. 10). The graphs intersect at one point K (-2;16). This means that the solution to the inequality is the interval (-2; (from minus two to plus infinity), since the graph of the function y= is located below the graph of the function at x Our reasoning allows us to verify the validity of the following theorems: Theme 1: If true if and only if m=n. Theorem 2: If is true if and only if, inequality is true if and only if (Fig. *) Theorem 4: If true if and only if (Fig.**), the inequality is true if and only if. Theorem 3: If true if and only if m=n. Example 5: Graph the function y= Let's modify the function by applying the property of degree y= Let us construct an additional coordinate system and in new system coordinates, we will construct a graph of the function y = (the y is equal to two to the x power) Fig. 11. Example 6: Solve the equation In one coordinate system we will construct two graphs of the function y= (Y is equal to seven to the power of X and Y is equal to eight minus X) Fig. 12. The graphs intersect at one point E (1; (e with coordinates one; seven). This means that the root of the equation is x = 1 (x equal to one). Example 7: Solve the inequality In one coordinate system we will construct two graphs of the function y= (Y is equal to one-fourth to the power of X and Y is equal to X plus five). The graph of the function y=is located below the graph of the function y=x+5 when the solution to the inequality is the interval x (from minus one to plus infinity).
2. Let's take a closer look at the exponential function:
0
If , then the function f(x) decreases
Function y= , at 0
This follows from the properties of monotonicity of a power 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 ox and oy axes and the graph takes a form close to the function y = 1.
Construct a graph of y =
