National Research University
Department of Applied Geology
Abstract on higher mathematics
On the topic: “Basic elementary functions,
their properties and graphs"
Completed:
Checked:
teacher
Definition. The function given by the formula y=a x (where a>0, a≠1) is called an exponential function with base a.
Let us formulate the main properties of the exponential function:
1. The domain of definition is the set (R) of all real numbers.
2. Range - the set (R+) of all positive real numbers.
3. For a > 1, the function increases along the entire number line; at 0<а<1 функция убывает.
4. Is a function of general form.
, on the interval xО [-3;3]
, on the interval xО [-3;3] A function of the form y(x)=x n, where n is the number ОR, is called a power function. The number n can take on different values: both integer and fractional, both even and odd. Depending on this, the power function will have a different form. Let's consider special cases that are power functions and reflect the basic properties of this type of curve in the following order: power function y=x² (function with an even exponent - a parabola), power function y=x³ (function with an odd exponent - cubic parabola) and function y=√x (x to the power of ½) (function with a fractional exponent), function with a negative integer exponent (hyperbola).
Power function y=x²
1. D(x)=R – the function is defined on the entire numerical axis;
2. E(y)= and increases on the interval
Power function y=x³
1. The graph of the function y=x³ is called a cubic parabola. The power function y=x³ has the following properties:
2. D(x)=R – the function is defined on the entire numerical axis;
3. E(y)=(-∞;∞) – the function takes all values in its domain of definition;
4. When x=0 y=0 – the function passes through the origin of coordinates O(0;0).
5. The function increases over the entire domain of definition.
6. The function is odd (symmetrical about the origin).
, on the interval xО [-3;3] Depending on the numerical factor in front of x³, the function can be steep/flat and increasing/decreasing.
Power function with negative integer exponent:
If the exponent n is odd, then the graph of such a power function is called a hyperbola. A power function with an integer negative exponent has the following properties:
1. D(x)=(-∞;0)U(0;∞) for any n;
2. E(y)=(-∞;0)U(0;∞), if n is an odd number; E(y)=(0;∞), if n is an even number;
3. The function decreases over the entire domain of definition if n is an odd number; the function increases on the interval (-∞;0) and decreases on the interval (0;∞) if n is an even number.
4. The function is odd (symmetrical about the origin) if n is an odd number; a function is even if n is an even number.
5. The function passes through the points (1;1) and (-1;-1) if n is an odd number and through the points (1;1) and (-1;1) if n is an even number.
, on the interval xО [-3;3] Power function with fractional exponent
A power function with a fractional exponent (picture) has a graph of the function shown in the figure. A power function with a fractional exponent has the following properties: (picture)
1. D(x) ОR, if n is an odd number and D(x)= 
, on the interval xО 
, on the interval xО [-3;3]
The logarithmic function y = log a x has the following properties:
1. Domain of definition D(x)О (0; + ∞).
2. Range of values E(y) О (- ∞; + ∞)
3. The function is neither even nor odd (of general form).
4. The function increases on the interval (0; + ∞) for a > 1, decreases on (0; + ∞) for 0< а < 1.
The graph of the function y = log a x can be obtained from the graph of the function y = a x using a symmetry transformation about the straight line y = x. Figure 9 shows a graph of the logarithmic function for a > 1, and Figure 10 for 0< a < 1.
; on the interval xО 
; on the interval xО The functions y = sin x, y = cos x, y = tan x, y = ctg x are called trigonometric functions.
The functions y = sin x, y = tan x, y = ctg x are odd, and the function y = cos x is even.
Function y = sin(x).
1. Domain of definition D(x) ОR.
2. Range of values E(y) О [ - 1; 1].
3. The function is periodic; the main period is 2π.
4. The function is odd.
5. The function increases on intervals [ -π/2 + 2πn; π/2 + 2πn] and decreases on the intervals [π/2 + 2πn; 3π/2 + 2πn], n О Z.
The graph of the function y = sin (x) is shown in Figure 11.
Provides reference data on the exponential function - basic properties, graphs and formulas. The following topics are considered: domain of definition, set of values, monotonicity, inverse function, derivative, integral, power series expansion 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 , 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 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 stronger the growth. At 0
< a < 1
the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.
Ascending, descending
The exponential function for 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 rule for differentiating a 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
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.
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