tanx-1≥0 What is the value range of X?

tanx-1≥0 What is the value range of X?

tanx≥√3/3=tan(π/6)=tan(kπ+π/6)
TaNx is an increasing function at (K π - π / 2, K π + π / 2)
So K π + π / 6 ≤ X

Let TaNx = 3 / 2, tany = 1 / 2 /, find X -- y =? The value is not required. It can be expressed by other trigonometric functions

tan(X－Y)＝(tanX－tanY)/(1＋tanXtanY)＝(3/2－1/2)/(1＋3/2×1/2)＝4/5

Why "the derivative of inverse function is equal to the reciprocal of derivative of direct function" does not conform to the derivative of arctanx! (arctanx) '= 1 / (1 + x ^ 2) instead of (cosx) ^ 2

x=tany dx/dy=(secy)^2 (arctanx)'=dy/dx=1/dx/dy =(cosy)^2=1/(1+x^2)
You've confused the symbols X and y

How to draw the inverse function image of y = SiNx Our teacher said to draw a 45 degree line to find the symmetry point

The inverse function of the function y = SiNx, X ∈ [- π / 2, π / 2] is called arcsine function
Because of the definition of function, we only consider one interval [- π / 2, π / 2], which is a monotone interval near the origin of sine function, which is called principal value interval of sine function. Otherwise, if the value of function is one to many, it is not a function
Two images which are inverse functions of each other are symmetric about y = X

How to draw the image of inverse function?

The original function and the inverse function are symmetrical about the straight line y = x, so it is relatively simple to draw a picture according to this. For example, if the original function has a point (x1, Y1), then the inverse function is a bit (Y1, x1)

Parabola area formula The parabola y = ax ^ 2 + BX + C, how to calculate the area of the curved polygon (if it can be enclosed) with the X axis?

Let f (x) = ax ^ 2 + BX + C = 0 be p, Q
Let f (x) = (A / 3) x ^ 3 + (B / 2) * x ^ 2 + C * X
Then the area s = [f (q) - f (P)]
[] represents absolute value

What is the area formula enclosed by parabola and straight line?

The area of the tangent line s + 1 / S + is equal to the area of the tangent line of the parabola =4/3*S

Parabolic equation expression

Parabola equation refers to the trajectory equation of parabola, which is a method to express parabola by equation. In geometric plane, parabola can be drawn according to parabola equation. The specific expression of equation is y = a * x * x + b * x + C (a ≠ 0) (2) if a > 0, the opening of parabola is upward; if a < 0, the opening of parabola is downward; 3) extreme point: (- B / 2a, (4ac-b * b) / 4A); 4) Δ = b * b-4ac, Δ > 0, The image intersects the x-axis at two points: ([- B - √Δ] / 2a, 0) and ([- B + √ Δ] / 2a, 0); Δ = 0, the image intersects the x-axis at one point: (- B / 2a, 0); Δ < 0, there is no intersection between the image and the x-axis; if the parabola intersects the y-axis, then c > 0; if the parabola intersects the y-axis, then c > 0

The parabola y = x ^ - 2 is translated up one unit to get a new parabola

y=x²-1

What is the difference between integral and differential?

There are three kinds of integral calculus and indefinite integral
1.0 indefinite integral
Let f (x) be a primitive function of function f (x). We call all primitive functions f (x) + C (C is any constant) of function f (x) as indefinite integral of function f (x)
DX
Where ∫ is called integral sign, f (x) is called integrand, X is called integral variable, f (x) DX is called integrand, C is called integral constant. The process of finding indefinite integral of known function is called integrating this function
According to the definition:
To find the indefinite integral of function f (x) is to obtain all the original functions of F (x). From the properties of the original function, we can get the indefinite integral of function f (x) as long as we find one primitive function of function f (x) and add any constant C
It can also be expressed that integral is the inverse operation of differential
2.0 definite integral
As we all know, the two parts of calculus are differential and integral. Differential is actually to find the derivative of a function, while integral is to find the derivative of a function. Therefore, differential and integral are inverse operations
In fact, the integral can be divided into two parts. The first is simple integration, that is, the derivative of F (x) + C (C is a constant) is also f (x) if the derivative of F (x) is f (x), that is, the derivative of F (x) + C (C is a constant) is also f (x). In other words, the derivative of F (x) + C is also f (x), and C is an infinite constant, so the results of F (x) integration are numerous and uncertain, We always use f (x) + C instead, which is called indefinite integral
And relative to indefinite integral, it is definite integral
The so-called definite integral is in the form of ∫ f (x) DX (the upper limit a is written above ∫ and the lower limit B is written below ∫). The reason why it is called a definite integral is that the value obtained after integration is certain and is a number, not a function
The formal name of definite integral is Riemann integral. See Riemann integral for details. In my own words, the image of function in rectangular coordinate system is divided into innumerable rectangles by straight lines parallel to y axis, and then the rectangles on a certain interval [a, b] are accumulated to get the area of the image of this function in the interval [a, b], The upper and lower bounds of definite integral are the two endpoints of interval a and B
We can see that the essence of definite integral is to infinitely subdivide the image and then add it up. The essence of integral is to find the original function of a function. They seem to have no connection. So why is the definite integral written in the form of integral?
Definite integral and integral seem to be different from each other, but because of the support of an important mathematical theory, they have an essential close relationship. It seems impossible to subdivide and accumulate a graph infinitely, but because of this theory, it can be transformed into computational integral. This important theory is the famous Newton Leibniz formula
If f '(x) = f (x)
Then ∫ f (x) DX (upper limit a, lower limit b) = f (a) - f (b)
Newton Leibniz formula is expressed in words, that is, the value of a definite integral formula is the difference between the upper limit of the original function and the lower limit of the original function
Because of this theory, it reveals the essential relationship between integral and Riemann integral, which shows its important position in calculus and even higher mathematics. Therefore, Newton Leibniz formula is also called the basic theorem of calculus
3.0 calculus
Integral is the inverse operation of differential, that is to know the derivative function of the function and reverse the original function. In application, the function of integration is not only that, but also is widely used in summation. Generally speaking, it is to calculate the area of curved triangle. This ingenious solution is determined by the special properties of integral
Another family of indefinite functions is also called an integral function
Where: [f (x) + C] '= f (x)
The definite integral of a real variable function on the interval [a, b] is a real number. It is equal to the value of a minus the value of a of an original function of the function
The indefinite integral is proposed to solve the inverse operation of derivation and differential. For example, given the function f (x) defined on the interval I, find a curve y = f (x), X ∈ I, If f (x) is a primitive function of F (x), then C is an arbitrary constant. For example, the definite integral is derived from the area problem of plane graph. Y = f (x) is a function defined on [a, b], in order to obtain x = a, X = B, The area s of the figure enclosed by y = 0 and y = f (x) is obtained by using the exhaustion method of ancient Greek. First, the curve is replaced by a straight line in a small range, and then the area s is obtained by taking the limit. Therefore, we divide [a, B] into N equal parts: a = x0 < x1 When n → + ∞, the limit of PN can be regarded as the area s. abstract the thinking method of this kind of problem, then we can get the concept of definite integral: for the function y = f (x) defined on [a, b], make a partition a = x0 < x1 ＜ If there is a constant I which is independent of partition and the choice of ζ I ∈ [XI-1, Xi], then I is called the definite integral of F (x) on [a, b], which is called [a, b] as the integral interval, f (x) as the integrand function, a and B as the upper limit and lower limit of the integral respectively, The calculation of definite integral can be transformed into the indefinite integral of F (x): This is C Newton Leibniz formula
differential
One variable differential
definition:
Let y = f (x) be defined in the neighborhood of X. let x0 and x0 + Δ X be defined in this interval. If the increment of function Δ y = f (x0 + Δ x) − f (x0) can be expressed as Δ y = a Δ x + O (Δ x) (where a is a constant independent of Δ x), and O (Δ x0) is infinitely smaller than Δ x, then f (x) is said to be differentiable at point x0, and a Δ x is called the differential of function at point x0 corresponding to the increment of independent variable Δ x, denoted as dy, That is, Dy = a Δ X
In general, the increment Δ X of the independent variable x is called the differential of the independent variable DX, that is, DX = Δ X. therefore, the differential of the function y = f (x) can be written as dy = f '(x) DX. The quotient of the differential of the function and the differential of the independent variable is equal to the derivative of the function. Therefore, the derivative is also called the derivative
When the independent variable x is changed to x + △ x, the value of the function is changed from F (x) to f (x + △ x). If there is a constant a independent of △ x such that the difference between F (x + △ x) - f (x) and a ·△ x is an infinitesimal of higher order with respect to △ x → 0, then a ·△ x is called the differential of F (x) in X, denoted as Dy, and f (x) is differentiable at X. the function is differentiable, and vice versa. In this case, a = f '(x), For example, D (SiNx) = cosxdx
Geometric meaning:
Let Δ X be the increment of point m on the abscissa of the curve y = f (x), Δ y be the increment of Δ x corresponding to Δ x at point m, and Dy be the increment of Δ x corresponding to the tangent of the curve at point M. when | Δ x | is very small, | Δ y-dy | is much smaller than | Δ y | (infinitesimal of higher order), so near point m, we can use tangent line segment to approximately replace curve segment
Multivariate differential
Similarly, when the independent variables are multiple, we can get the definition of multivariate differential
Algorithm:
dy=f'(x)dx
d(u+v)=du+dv
d(u-v)=du-dv
d(uv)=du·v+dv·u
d(u/v)=(du·v-dv·u)/v^2