What is exponential What is the exponential formula?

There are many kinds of differential equations, there are separable variables, there are homogeneous equations, there are homogeneous differential equations with first-order constant coefficients, there are non-homogeneous constants of the first order, homogeneous coefficients of the second order constant coefficients, and Bernoulli's equation ......

These are all specific types, and the main categories are first-order linearity, first-order nonlinearity, second-order linearity, etc., and the following are examples of common first-order linear differential equations.

The standard form of a first-order linear differential equation is .

dy/dx + yp(x) = q(x)

Differential equations like the above equation are called first-order linear differential equations, and vice versa.

If q(x)=0, then the above equation is called a first-order linear homogeneous differential equation, and vice versa.

For example: dy dx = y + x 2

dy dt = x * sint + t 2 they are both linear differential equations that conform to the above equation.

y * y' -2*xy = 3

y' - cosy = 1

They do not conform to the standard form of first-order linear differential equations, so they are not the standard form of Bernoulli's equations.

Dy dx + p(x)*y = q(x) *y n is called Bernoulli's equation if it conforms to the above form.

Read the definitions in the textbook carefully, and don't look at many examples to distinguish between my definitions.

The explanations are clear and easy to understand!!

In the case of linear algebra, the order refers to the number of rows and series of determinants. Because a determinant is a set of numbers enclosed in curly braces. The upper determinant has a total of 4 rows and 4 columns, so it is called a 4th-order determinant.

Enclosed in parentheses, 3 rows * 3 columns is called a 3rd order determinant, that is, the number of rows = column = order of the determinant, understand?

Linear relationships are embodied in matrices, as well as in space. It is a numerical relationship between them. It is reflected in the fact that there is a certain number of spatial relationships between them, which can be expressed uniformly by a mathematical expression or a spatial vector.

Linearity can also refer to linear operations, such as:

5a + 43b - 4c + 21f = n + f - e, the above equation only contains the multiplication and addition and subtraction of numbers, so it is called a linear expression, and its operation can be called a linear operation.

Operations other than multiplication and addition and subtraction cannot be linear!

Indices are statistics designed and calculated based on certain samples** or bonds to measure the volatility of ** markets or bond markets.

a b=n is exponential.

logan=b logarithmic.

And they are inverse functions of each other.

in the power of the square. A n, where a is called the base.

n is called exponential, and the result is called power.

The index formula is as follows:1. y=c (c is constant) y'=0

2、y=x^n y'=nx^(n-1)

3、y=a^x y'=a^xlna y=e^x y'=e^x4、y=logax y'=logae/x y=lnx y'=1/x5、y=sinx y'=cosx

6、y=cosx y'=-sinx

7、y=tanx y'=1/cos^2x

8、y=cotx y'=-1/sin^2x<>

Explanation of terms: exponential function.

is an important basic elementary function.

One. In general, the y=ax function (a is a constant and a>0,a≠1) is called an exponential function, and the domain of the function is r. Note the definition expression in the exponential function.

, the coefficient before ax must be the number 1, the independent variable.

x must be in the position of the exponent, and cannot be any other expression of x, otherwise, it is not an exponential function.

Exponential functionsRefers to functions that are important in mathematics. The formula for the hail index is:

1. Same base.

The powers are multiplied, the base is unchanged, and the exponents are added; (a^m)*（a^n）=a^(m+n)。

2. Divide by the power of the same base, the base number remains unchanged, and the number of defeats is subtracted; (a^m)÷（a^n）=a^(m-n)。

3. The power of the power, the base is unchanged, and the exponents are multiplied; (a^m)^n=a^(mn)。

4. The power of the product is equal to each factor.

Powers of Each; (ab)^n=(a^n)(b^n)。

The properties of the exponential function operation are as follows:

1) The domain in which the exponential function is defined.

is r, where the premise of the collision is that a is greater than 0 and not equal to 1. If a is not greater than 0, it will inevitably make the definition domain of the function discontinuous, so we do not consider it, and the function that a is equal to 0 is meaningless is generally not considered.

2) The range of the exponential function.

is (0, +

3) The function graphs are all concave.

4) a>1, the exponential function increases monotonically; If 0

The exponent is calculated as y=a x(a>0 and not 1).

Exponential functionsThe general form of y=a x(a>0 and not 1), the graph of the number of dead numbers in the function is concave, and a is greater than 1, then the exponential function increases monotonically; A less than 1 and greater than 0 is a monotonically decreasing function. The exponential function is neither an odd function.

Nor is it an even function.

In order to make x possible to take the entire set of real numbers as a defined domain.

Then there is only one case where the different magnitude of a affects the graph of the function.

Logarithmic arithmetic formula:

If a>0, a≠1, m>0, n>0, then:

1、loga(mn)=logam+logan。

2、logamn=logam-logan。

3、logamn=nlogam(n∈r)。

Indices are exponential calculations.

a (a≠0), a is the base, n is the exponent, the exponent is located in the upper right corner of the base, and the early with the power operation means that the exponential bases are multiplied. When n is a positive integer.

a denotes n multiplications of A. When n=0, a 1.

Index, or statistical index, is an important statistical method for analyzing the quantitative changes in socio-economic phenomena. Broadly speaking, the relative numbers that reflect the overall quantitative changes in the phenomenon are all exponential. The index in the narrow sense reflects the change in the overall number of complex phenomena.

The compilation of the index is derived from the movement of prices.

Generalization of indicesThe index is a relative number that indicates the dynamics of socio-economic phenomena, and the index can be used to determine the total dynamics of socio-economic phenomena that cannot be directly added and directly compared, to analyze the influence of the changes of various factors in the total changes of socio-economic phenomena, and to study the role of the changes in the level of each group of markers and the overall structure of the changes in the changes of the total average indicators.

Indices are divided into individual indices and aggregate indices according to the range of phenomena reflected. The general relative number is the ratio of two related indicators, which can quantitatively reflect the contrast between two interrelated phenomena. The quality index is an index that explains changes in the quality of economic activities, such as the product cost index and the labor productivity index.