3 at lower levels. Stupinnі chi show rіvnyannya. Powerful geometric progression
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For the beginning, we guess the basic formulas for the steps of that power.
Boot number a the same wine is called n times, we can write the whole viraz as a a ... a = a n
1. a 0 = 1 (a ≠ 0)
3. an m = an + m
4. (a n) m = a nm
5. a n b n = (ab) n
7. a n / a m \u003d a n - m
Steps or impressions - the ceremonies of some change are changed at the steps (chi pokazniki), and the basis is the number.
Apply ostentatious rivnyan:
For each butt, the number 6 is substantiated, for sure, stand at the bottom, and change x step chi pokanik.
Let's put some ostentatious rivnyan.
2 x *5 = 10
16x - 4x - 6 = 0
Now let's take a look, how are the displays of equivalence?
Let's take it easy:
2 x = 2 3
Such a butt can be found in the Duma. It can be seen that x = 3. Even if the left and right parts are equal, it is necessary to replace x by putting the number 3.
And now we wonder how it is necessary to issue a solution:
2 x = 2 3
x = 3
In order to win such equal, we were taken away however, substantiate(tobto two) and they wrote down those who were left out, the whole step. They took it away.
Now pіdіb'єmo pіdіb'єmo podbags of our solution.
Show alignment algorithm:
1. It is necessary to revise however, chi podstavi at rivnyannya right-handed and left-handed. As if they weren’t the same, we’re looking for options for perfecting this butt.
2. After that, yak imagine to become the same, comparable the step that virishuemo otrimane new equal.
Now we see a sprat of applications:
Let's start from a simple one.
Substitute in the left and right parts to equal the number 2, then we can substantiate that equate the steps.
x+2=4 It's easier to match.
x = 4 - 2
x=2
Suggestion: x=2
At the butt, it is clear that it was given a difference of 3 and 9.
3 3x - 9 x +8 = 0
For the cob, we can carry nine right-handed, we take:
Now it is necessary to work out the same substantiations. We know that 9 = 3 2 . Accelerated by the step formula (a n) m = a nm.
3 3x \u003d (3 2) x + 8
Subtract 9 x +8 = (32) x +8 = 3 2x +16
3 3x \u003d 3 2x + 16 now it is clear that in the left and right sides of the base there are the same equal trinity, so we can equate those steps.
3x=2x+16 took away the simplest equal
3x - 2x = 16
x=16
Suggestion: x = 16.
We marvel at the advancing butt:
2 2x+4 - 10 4 x = 2 4
We are marveling at the present, present two different chotiri. And we need to be the same. Let's rewrite the four by the formula (a n) m = a nm.
4 x = (2 2) x = 2 2x
And there is one more formula a n a m = an + m:
2 2x+4 = 2 2x 2 4
Dodaemo in equal:
2 2x 2 4 - 10 2 2x = 24
We brought the butt to the same bases. But we are honored with other numbers 10 and 24. What do you work with them? It’s amazing to see that in the left part we repeat 2 2x, axis and turn - 2 2x we can blame for the arms:
2 2x (2 4 - 10) = 24
Porahuemo viraz at the temples:
2 4 — 10 = 16 — 10 = 6
All equals are divisible by 6:
Visible 4 = 2 2:
2 2x \u003d 2 2 bases are the same, it seems that they are equal steps.
2x \u003d 2 was the simplest equal. Dilimo yoga for 2 is acceptable
x = 1
Verify: x = 1.
Rozv'yazhemo rivnyannya:
9 x - 12 * 3 x +27 = 0
Reworkable:
9 x = (3 2) x = 3 2x
We take equal:
3 2x - 12 3 x +27 = 0
Give us the same equal three. In this butt, it is clear that the first trio has two times (2x) larger feet, the lower one has the other (just x). In such a mood, you can win substitution method. The number with the smallest step is replaced:
Todi 3 2x \u003d (3 x) 2 \u003d t 2
Replace in equal all steps with ixes on t:
t 2 - 12t +27 = 0
Acceptable square alignment. Virishuemo through the discriminant, otrimuemo:
D=144-108=36
t1 = 9
t2 = 3
Let's turn to change x.
Beremo t 1:
t 1 \u003d 9 \u003d 3 x
Became booty
3 x = 9
3 x = 3 2
x 1 = 2
One root was known. Shukaєmo another, s t 2:
t 2 \u003d 3 \u003d 3 x
3 x = 3 1
x 2 = 1
Vidpovid: x 1 = 2; x 2 = 1.
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y (x) = e x, pokhіdna like a beautiful function itself.Exponent poznachayut so, or.
e number
Substitute step exponenti є e number. Tse irrational number. It's about as old as it gets
e ≈ 2,718281828459045...
The number e is determined through the intersequence. Tse so ranks another miraculous boundary:
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So the number e can be seen in the row:
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Exponent schedule
Exponential graph, y = e x.The graph shows the exhibitor, e at the step X.
y (x) = e x
The graph shows that the exponent is growing monotonously.
Formulas
Basic formulas are the same as for display functions with a substage e.
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Viraz display function with a sufficient basis step a through the exponent:
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Private values
Come on y (x) = e x. Todi
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Power exponenti
The exponent of the power of the display function with the basis of the stage e > 1 .
Designated area, anonymous value
Exponent y (x) = e x assigned to all x .
Її scope of assignment:
- ∞ < x + ∞
.
Її impersonal meaning:
0
< y < + ∞
.
Extreme, rising, falling
The exponent is a monotonically growing function, so there are no extremums. The main її authorities are presented in the tables.
Return function
Return for exhibitors is the natural logarithm.
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Pokhіdna exponenti
Pokhidna e at the step X dorivnyuє e at the step X
:
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Pokhіdna n-th order:
.
Visnovok formulas > > >
Integral
Complex numbers
Dії with complex numbers Euler formulas:
,
de є obvious loneliness:
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Virazi through hyperbolic functions
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Virazi through trigonometric functions
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Arrangement in stacked row
Wikoristan literature:
I.M. Bronstein, K.A. Semendyaev, Mathematics guide for engineers and university students, Lan, 2009.
A step function is called a function of the form y = x n (it is read as y is more expensive x at step n), where n is a given number. Private vipadami stack functionsє functions of the form y=x, y=x 2 , y=x 3 , y=1/x and many others. Let's talk about the skin of them.
Linear function y=x1 (y=x)
The graph is a straight line that passes through the point (0; 0) from the point 45 degrees to a positive straight line on the Ox axis.
The chart is shown below.
The main power of line functions:
- The function is growing and is assigned on the entire numerical axis.
- Do not have a maximum and a minimum value.
Quadratic function y=x2
Graph of the quadratic function of a parabola.
The main power of the quadratic function:
- 1. When x = 0, y = 0, i y> 0 at x0
- 2. The minimum value of the quadratic function reachable at its vertex. Ymin at x=0; Slid also means that the maximum value of the function is not used.
- 3. The function changes to intermittent (-∞;0] and increases to intermittent)
