The definition range of y = arcsin √ X and y = arcsin (1 / √ x) The definition range of y = arcsin √ X and y = arcsin (1 / √ x)

The definition range of y = arcsin √ X and y = arcsin (1 / √ x) The definition range of y = arcsin √ X and y = arcsin (1 / √ x)

1. Domain: [0,1], range [0, π / 2];
2. Domain: [1, + ∞), range (0, π / 2]

How can TI-89 calculate the differential when x = 2? For example, when x = 2, 3x ^ 2 + 2x is the derivative~
I'm in a hurry! I'll take the AP exam soon!

Enter home
First click F3 to find D (differential)
On the calculator is d (3x ^ 2 + 2x, x) | x = 2. The answer is 14
D is differential, 3x ^ 2 + 2x is your equation, X is the derivative of X | x = 2 is when x = 2 (| just below the equal sign)

Given the function f (x) = ax2-2x + LNX (I) if f (x) has no extreme point, but its derivative f '(x) has zero point, find the value of a; (II) if f (x) has two extreme points, find the value range of a, and prove that the minimum value of F (x) is less than - 32

Solution & nbsp; (I) first, X ＞ 0f / (x) = 2aX − 2 + 1x = 2ax2 − 2x + 1xf '(x) has zero point and f (x) has no extremum point, which indicates that the zero point has the same sign about f' (x), so a ≠ 0, and △ of 2ax2-2x + 1 = 0 = 0. Thus, we can get a = 12 (II) from the meaning of the problem, 2ax2-2x + 1 = 0 has two different positive roots, so △ 0, a > 0, Because in the interval (0, x1), (X2, + ∞), f ′ (x) > 0, and in the interval (x1, x2), f ′ (x) < 0, so X2 is the minimum point of F (x). Because f (x) in the interval (X1, x2) is a decreasing function, if we can prove that f (x1 + X22) < 32, then f (x2) < 32 is more important. According to Weida's theorem, X1 + X22 = 12a, f (12a) = a (12a) 2 − 2 (12a) + ln12a = ln12a − 32 · 12a Let 12a = t, where g (T) = LNT − 32t + 32, it is easy to prove that G (T) decreases monotonically when t ＞ 1, and G (1) = 0, G (T) = lnt-32, t + 32 ＜ 0, so f (12a) ＜ - 32, so f (x2) ＜ - 32 is the minimum of F (x)

Finding the decreasing interval of function y = 2x - INX

y'=2-1/x=(2x-1)/x (x>0)
y'0 ==>（2x-1)/x 0

On the image of function y = INX, the abscissa from the nearest point to the line 2x-y + 2 = 0 is

y=lnx
y'=1/x
The closest point to the line 2x-y + 2 = 0 is the tangent point parallel to the line. Then the slope of the tangent is k = 2
That is: 1 / x = 2, get x = 1 / 2
The abscissa of the point is x = 1 / 2

What's the difference between "=" and "≡", it seems that they are similar? For example, "f (x) = 0" and "f (x) ≡ 0"?

Identical to
It's definitely equal to
Let's put it this way
That is, if f (x) ≡ 0, the solutions of X are infinite and arbitrary
If f (x) = 0, it means that the solution of X is some or no solution

High number sign D
I read advanced mathematics books at home, and the derivative of composite function is dy / DX = dy / Du * Du / DX
What does the symbol D mean? Does it mean Delta?

Yes
Equivalent to unit trace

What's the sign in front of the partial differential,
How do you read it,
What kind of symbol is it,

∂ in the school, the teachers study English in class

What is integral sign internal derivative?
What is the mathematical tool that Charlie Feynman often mentions in his autobiography

Taking differential in integral sign: when the integrand is a continuous differentiable function of two variables, the order of integration and derivation can be exchanged
This mathematical tool is calculus

Solution of differential sign D
For example, the indefinite integral f 3xdx (F is the symbol of indefinite integral, I can't type it to replace it), does this DX mean to multiply by 3x, or is it just a symbol? In addition, does the meaning of D mean △ or just infinitesimal?

DX is multiplied by 3x
This point, you can recall from the concept of definite integral at the beginning. 3x is the height of the rectangle, DX is the length of the rectangle, and their multiplication represents the area of the rectangle. Then D is infinitesimal, and the area of the curve edge figure is obtained by integrating again
D means △ which means more infinitesimals
Welcome to ask!