Square inequalities. How to solve cubic equations of expression through trigonometric functions
Number e. It is an important mathematical constant, which is the basis of a natural logarithm. Number e. approximately 2.71828 with the limit (1 + 1/n.)n. for n. seeking infinity.
Enter the value x to find the value of the exponential function eX.
To calculate numbers with the letter E. Use a calculator for converting an exponential number into an integer
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Calculator algebra calculation
The number E is an important mathematical constant underlying a natural logarithm.
0.3 with power X multiplied by 3 by power X, the same
The number E is approximately 2.71828 with the limit (1 + 1 / n) n for n, which tends to infinity.
This number is also called the number of Euler or the number of faeces.
Exponential - exponential function f (x) \u003d exp (x) \u003d Ex, where E is the number of Euler.
Enter the value of x to find the value of the exponential function ex
Calculation of the value of the exponential function on the network.
When the Euler number (E) rises to zero, the answer is 1.
When you raise more than one level, the answer will be greater than the original. If the speed is greater than zero, but less than 1 (for example, 0.5), the answer will be greater than 1, but less than the original (e). When the indicator increases to negative power, 1 must be divided into a given power to the specified power, but with a "plus" sign.
Definitions
exhibitor This is the exponential function y (x) \u003d e x, the derivative of which coincides with the function itself.
The indicator is marked as, or.
E. number
The basis of the exponent is the number E.
This is an irrational number. It's about the same
e. ≈ 2,718281828459045 …
The number E is defined abroad of the sequence. This is the so-called other exceptional limit:
.
The number E can also be represented as a series:
.
Schedule Exhibitor
The graph shows an indicator of the degree e. In the Stage h..
y (x) \u003d ex
The schedule shows that it increases monotonously exponentially.
formula
The basic formulas are the same as for the exponential function with the level E base.
An expression of exponential functions with an arbitrary basis A in the sense of exhibitors:
.
also, the "Exponential Function" department "\u003e\u003e\u003e
Private values
Let y (x) \u003d e x.
5 to power X and equal to 0
Exponential properties
The indicator has the properties of the exponential function with the basis of the degree e. \u003e First
Definition field, set of values
For X, the indicator y (x) \u003d E x is defined.
Its volume:
— ∞ < x + ∞.
Its value:
0 < Y < + ∞.
Extreme, increase, reduction
The exponent is a monotonous increasing function, so it does not have extremes.
Its basic properties are shown in the table.
Reverse function
The inverse indicator is a natural logarithm.
;
.
Derived indicators
derivative e. In the Stage h. it e. In the Stage h.
:
.
Derivative N-order:
.
Formulas \u003e\u003e\u003e
integral
also, the "table of uncertain integrals" \u003e\u003e\u003e
Comprehensive rooms
Operations with complex numbers are performed using Formula Euler:
,
where the imaginary unit:
.
Expressions through hyperbolic functions
Expressions through trigonometric functions
Expansion of power rows
When x is zero?
Normal or online calculator
Normal calculator
The standard calculator gives you simple operations in a calculator, such as adding, subtraction, multiplication and division.
You can use fast mathematical calculator
The scientific calculator allows you to perform more complex operations, as well as a calculator, such as sinus, cosine, inverse sinus, reverse cosine, which concerns tangent, exponent indicator, indicator, logarithm, interest, and business in a web-based memory calculator.
You can enter directly from the keyboard, first click on the area using the calculator.
It performs simple operations with numbers, as well as more complex, such as
mathematical calculator online.
0 + 1 = 2.
Here are two calculators:
- Calculate the first as usual
- Another calculates it as engineering
Rules apply to the calculator calculated on the server
Terms of Input Terms and Functions
Why do I need this online calculator?
Online calculator - how does it differ from the usual calculator?
First, the standard calculator is not suitable for transport, and secondly - now the Internet is practically everywhere, it does not mean that there are problems, go to our site and use a web calculator.
Online Calculator - How does it differ from the Java Calculator, as well as from other calculators for operating systems?
- Again - mobility. If you are on another computer, you do not need to reinstall it.
So use this site!
Expressions may consist of functions (in alphabetical order):
absolute (x) Absolute value h.
(module h. or | X |) arcCOS (X) Function - Arkoxin from h.arccosh (x) Arsosin is hyperbolic from h.arcsin (X) Private Son. h.arcsinh (x) Hyperx hyperbolic h.arctg (x) Function - Arctangent from h.arctgh (x) Arctangent is hyperbolic h.e.e. number - about 2.7 exp (x) Function - indicator h. (as e.^h.) log (x) or lN (X) Natural logarithm h.
(Yes log7 (x), You must enter log (x) / log (7) (or, for example, for log10 (x)\u003d log (x) / log (10)) p. The number "PI", which is about 3.14 sIN (X) Function - sinus h.cOS (X) Function - cone from h.sINH (X) Function - sinus hyperbolic h.cOSH (X) Function - cosine-hyperbolic h.sQRT (X) The function is a square root of h.sQR (X) or x ^ 2. Function - Square h.tG (X) Function - Tangent from h.tGH (X) Function - tangency hyperbolic from h.cBRT (X) Function is a cubic root h.soil (x) Rounding function h. On the underside (sample soil (4.5) \u003d\u003d 4.0) symbol (x) Function - symbol h.eRF (X) Error function (Laplace or Integral Probability)
The following operations can be used in terms:
Real numbers Enter in the form 7,5 , not 7,5 2 * X. - Multiplication 3 / X. - separation x ^ 3. - Eksponentiacija. x + 7. - Moreover, x - 6. - countdown
Download PDF.
Indicative equations are the equations of the form
x--Nexuality indicator,
a. and b.- Some numbers.
Examples of the indicative equation:
And equations:
no longer will be indicative.
Consider examples of solving indicative equations:
Example 1.
Find the root of the equation:
Let's give degrees to the same basis to take advantage of the degree property with the actual indicator
Then it will be possible to remove the foundation of the degree and go to the equality of indicators.
We transform the left part of the equation:
We transform the right side of the equation:
Use the degree property
Answer: 4.5.
Example 2.
Solve inequality:
We divide both parts of the equation on
Reverse replacement:
Answer: x \u003d 0.
Solve the equation and find the roots at the specified interval:
We give all the components to the same base:
Replacement:
We are looking for the roots of the equation, by selecting a multiple free member:
- Suitable, because
equality is performed.
- Suitable, because
How to solve? E ^ (x-3) \u003d 0 e to degree x-3
equality is performed.
- Suitable, because Equality is performed.
- not suitable, because Equality is not performed.
Reverse replacement:
The number refers to 1 if its indicator is 0
Not suitable, because
The right side is 1, because
From here:
Solve the equation:
Replacement:, then
Reverse replacement:
1 Equation:
if the bases of numbers are equal, then their indicators will be equal, then
2 Equation:
Log froze both parts based on 2:
The indicator of the degree gets up before expression, because
The left side is 2x, because
From here:
Solve the equation:
We transform the left side:
Reduce degrees by the formula:
We simplify: by the formula:
Imagine in the form:
Replacement:
Transfer the fraction in the wrong:
a2 - it is suitable, because
Reverse replacement:
Conduct to a general basis:
If a
Answer: X \u003d 20.
Solve the equation:
OD
We transform the left side by the formula:
Replacement:
Calculate the root from the discriminant:
a2 is not suitable, because
and does not take negative values
Conduct to a general basis:
If a
We will be erected both parts:
Article editors: Gavrilina Anna Viktorovna, Ageeva Lyubov Aleksandrovna
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Large article "An Intuitive Guide to Exponential Functions & E"
The number E has always worried me - not as a letter, but as a mathematical constant.
What does the number really mean?
Different mathematical books and even my hot beloved Wikipedia describes this majestic constant with a completely stupid scientific jargon:
The mathematical constant E is the basis of a natural logarithm.
If you are interested in what is natural logarithm, you will find such a definition:
The natural logarithm, previously known as hyperbolic logarithm, is a logarithm with a base of E, where E is an irrational constant, approximately equal to 2.718281828459.
Definitions of course correct.
But it is extremely difficult to understand them. Of course, Wikipedia is not to blame for this: usually mathematical explanations of dry and formal, are compiled throughout the rigor of science. Because of this, newcomers are difficult to master the subject (and once every newcomer).
I'm over it! Today I share my highly intelligent considerations about what is the number eAnd what it is so cool! Set your thick, leaving the fear of mathematical books to the side!
Number e is not just a number
Describe e as a "constant, approximately equal to 2,71828 ..." is it all equal to calling the number Pi "an irrational number, approximately equal to 3,1415 ...".
Undoubtedly, it is, but the essence still eludes us.
The number Pi is the ratio of the circumference of the circle to the diameter, the same for all circles. This is a fundamental proportion peculiar to all circles, and therefore it is involved in calculating the length of the circle, area, volume and surface area for circles, spheres, cylinders, etc.
Pi shows that all circles are connected, not to mention the trigonometric functions derived from the circles (sinus, cosine, tangent).
The number E is a basic growth ratio for all continuously growing processes. The number E allows you to take a simple growth rate (where the difference is visible only at the end of the year) and calculate the components of this indicator, normal growth, in which with each nanosecond (or even faster) everything grows on a bit.
The number E is participating both in systems with exponential and constant growth: population, radioactive decay, counting interest, and many, many others.
Even step-in systems that do not grow evenly, can be approximated by the number E.
Also, as any number can be viewed in the form of "scaled" version 1 (base unit), any circumference can be considered as a "scaled" version of the unit circle (with a radius 1).
The equation is given: e to the degree x \u003d 0. What is equal to x?
And any growth rate can be considered in the form of a "scaled" version E ("single" growth coefficient).
So the number E is not a random, taken at random. The number E embodies the idea that all continuously growing systems are scaled versions of the same indicator.
The concept of exponential growth
Let's start with the consideration of the basic system that doubles for a certain period of time.
For example:
- Bacteria share and "double" in quantity every 24 hours
- We get twice as many lapshoks if we smoke them in half
- Your money is doubled every year if you get 100% profit (lucky!)
And it looks like this:
Delivery into two or doubles is a very simple progression. Of course, we can triple or set up, but doubles more conveniently for explanation.
Mathematically, if we have x separation, we get 2 ^ x times more good than it was first.
If only 1 partition is done, we get 2 ^ 1 times more. If partition 4, we will have 2 ^ 4 \u003d 16 parts. The general formula looks like this:
In other words, doubling is 100% growth.
We can rewrite this formula like this:
height \u003d (1 + 100%) x
This is the same equality, we only divided "2" into composite parts, which are essentially this number: the initial value (1) plus 100%. Cleverly, yes?
Of course, we can substitute any other number (50%, 25%, 200%) instead of 100% and get a growth formula for this new coefficient.
The general formula for x periods of the time series will look at:
height \u003d (1 + increase) x
It simply means that we use the refund rate, (1 + increase), "x" in a row.
Close closer
Our formula suggests that the increase occurs with discrete steps. Our bacteria are waiting, waiting, and then Batz!, And at the last minute they double in quantity. Our profit on interest from the deposit is magically appear exactly after 1 year.
Based on the formula written above, the profit grows stepped. Green dots appear suddenly.
But the world is not always like that.
If we increase the picture, we will see that our bacteria friends are constantly divided:
Green small does not arise from nothing: it slowly grows out of a blue parent. After 1 period of time (24 hours in our case), the green friend is completely ripe. Having matured, he becomes a full-fledged blue member of herd and can create new green cells himself.
This information will somehow change our equation?
In the case of bacteria, half-defined green cells can still do anything until they grow up and do not get off from their blue parents at all. So the equation is fair.
In the next article we will look at an example of the exponential growth of your money.
Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")
What "Square inequality"? Not a question!) If you take anyone Square equation and replace in it sign "=" (equal) to any inequality icon ( > ≥ < ≤ ≠ ), It will be square inequality. For example:
1. x 2 -8x + 12 ≥ 0
2. -X 2 + 3X > 0
3. x 2 ≤ 4
Well, you understood ...)
I am not in vain here tied equations and inequalities. The fact is that the first step is in solving anyone square inequality - solve the equation from which this inequality is done. For this reason, the inability to solve square equations automatically leads to complete failure and in inequalities. A hint is clear?) If that, see how to solve any square equations. Everything is described in detail. And in this lesson we will deal with inequalities.
Ready for solving inequality is: left - Square Three aX 2 + BX + C, right - zero. The sign of inequality can be absolutely any. The first two examples here already ready to solve. The third example should still be prepared.
If you like this site ...
By the way, I have another couple of interesting sites for you.)
It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)
You can get acquainted with features and derivatives.
In the cubic equation, the highest indicator of the degree is 3, in such an equation 3 root (solutions) and it has the form. Some cubic equations are not so easy to solve, but if you apply the correct method (with good theoretical preparation), you can find the roots of even the most complex cubic equation - to do this, use the formula to solve the square equation, find whole roots or calculate the discriminant.
Steps
How to solve a cubic equation without a free member
- In our example, substitute the values \u200b\u200bof the coefficients A (\\ DisplayStyle a), B (\\ DisplayStyle B), C (\\ DisplayStyle C) ( 3 (\\ DisplayStyle 3), - 2 (\\ displayStyle -2), 14 (\\ DisplayStyle 14)) in the formula: - B ± B 2 - 4 A C 2 A (\\ DisplayStyle (\\ FRAC (-b \\ pm (\\ sqrt (b ^ (2) -4ac))) (2a))) - (- 2) ± ((- 2) 2 - 4 (3) (14) 2 (3) (\\ displayStyle (\\ FRAC (- (- 2) \\ pm (\\ sqrt (((-2) ^ (2 ) -4 (3) (14)))) (2 (3)))) 2 ± 4 - (12) (14) 6 (\\ displayStyle (\\ FRAC (2 \\ PM (\\ SQRT (4- (12) (14)))) (6))) 2 ± (4 - 168 6 (\\ DisplayStyle (\\ FRAC (2 \\ PM (\\ SQRT ((4-168))) (6))) 2 ± - 164 6 (\\ DisplayStyle (\\ FRAC (2 \\ PM (\\ SQRT (-164))) (6)))
- First root: 2 + - 164 6 (\\ DisplayStyle (\\ FRAC (2 + (\\ SQRT (-164))) (6))) 2 + 12, 8 i 6 (\\ DisplayStyle (\\ FRAC (2 + 12.8I) (6)))
- Second root: 2 - 12, 8 i 6 (\\ DisplayStyle (\\ FRAC (2-12.8) (6)))
-
Use zero and roots of the square equation as solutions of a cubic equation. The square equations have two roots, and in cubic - three. Two solutions you have already found - these are the roots of the square equation. If you have taken out "x" for brackets, the third solution will be.
How to find whole roots using multipliers
-
Make sure that there is a free dick in the cubic equation D. (\\ DisplayStyle D) . If in the view equation a x 3 + b x 2 + c x + d \u003d 0 (\\ displaystyle ax ^ (3) + bx ^ (2) + cx + d \u003d 0) There is a free dick D (\\ DisplayStyle D) (which is not zero), to make "x" for brackets will not work. In this case, use the method set out in this section.
Remove the factors of the coefficient A. (\\ DisplayStyle A) and free member D. (\\ DisplayStyle D) . That is, find the multipliers of the number when x 3 (\\ displaystyle x ^ (3)) and the number before the sign of equality. Recall that the number of numbers are numbers, when multiplying which this number is obtained.
Divide each factor A. (\\ DisplayStyle A) for every multiplier D. (\\ DisplayStyle D) . As a result, a lot of fractions and several integers are obtained; The roots of the cubic equation will be one of the integers or the negative value of one of the integers.
- In our example, divide multipliers A (\\ DisplayStyle a) (1 and 2 ) on multipliers D (\\ DisplayStyle D) (1 , 2 , 3 and 6 ). You'll get: 1 (\\ DISPLAYSTYLE 1), , , , 2 (\\ DISPLAYSTYLE 2) and. Now add the negative values \u200b\u200bof the fractions obtained and numbers to this list: 1 (\\ DISPLAYSTYLE 1), - 1 (\\ displayStyle -1), 1 2 (\\ DisplayStyle (\\ FRAC (1) (2))), - 1 2 (\\ displayStyle - (\\ FRAC (1) (2))), 1 3 (\\ displayStyle (\\ FRAC (1) (3))), - 1 3 (\\ displayStyle - (\\ FRAC (1) (3))), 1 6 (\\ displayStyle (\\ FRAC (1) (6))), - 1 6 (\\ displayStyle - (\\ FRAC (1) (6))), 2 (\\ DISPLAYSTYLE 2), - 2 (\\ displayStyle -2), 2 3 (\\ DisplayStyle (\\ FRAC (2) (3))) and - 2 3 (\\ DisplayStyle - (\\ FRAC (2) (3))). Whole roots of the cubic equation are some numbers from this list.
-
Submold integers into the cubic equation. If at the same time the equality is observed, the substituted number is the root of the equation. For example, substitute in the equation 1 (\\ DISPLAYSTYLE 1):
Take advantage of the division of polynomials by gorner scheme To faster find the roots of the equation. Do it if you do not want to manually substitute the numbers into the equation. In the Gorner scheme, the integers are divided into the values \u200b\u200bof the equation coefficients A (\\ DisplayStyle a), B (\\ DisplayStyle B), C (\\ DisplayStyle C) and D (\\ DisplayStyle D). If the numbers are divided by a focus (that is, the residue is equal), an integer is the root of the equation.
-
Find out whether there is a free member in the cubic equation D. (\\ DisplayStyle D) . The cubic equation has a view a x 3 + b x 2 + c x + d \u003d 0 (\\ displaystyle ax ^ (3) + bx ^ (2) + cx + d \u003d 0). To the equation is considered cubic, it is enough for only a member in it. x 3 (\\ displaystyle x ^ (3)) (That is, other members may not be at all).
Take out for braces X. (\\ DisplayStyle X) . Since there is no free member in the equation, each member of the equation includes a variable X (\\ DisplayStyle X). This means that one X (\\ DisplayStyle X) You can take out the brackets to simplify the equation. Thus, the equation will be recorded like this: x (a x 2 + b x + c) (\\ displaystyle x (AX ^ (2) + BX + C)).
Spread on multipliers (on the work of two benomes) a square equation (if possible). Many square equations of the form a x 2 + b x + c \u003d 0 (\\ displaystyle ax ^ (2) + bx + c \u003d 0) You can decompose on multipliers. Such an equation will succeed if you make X (\\ DisplayStyle X) for brackets. In our example:
Decide the square equation with the help of a special formula. Do it if the square equation cannot be decomposed on multipliers. To find two root equations, the values \u200b\u200bof the coefficients A (\\ DisplayStyle a), B (\\ DisplayStyle B), C (\\ DisplayStyle C) Substitute in the formula.
