The proving process of Rolle's mean value theorem

The proving process of Rolle's mean value theorem

Rolle's mean value theorem
Rolle's mean value theorem:
If the function f (x) is continuous in the closed interval [a, b], has derivatives in the open interval (a, b), and has the same values at the end of the interval, that is, f (a) = f (b), then there is at least one point ξ (a) in (a, b)

Prove Rolle's mean value theorem with Lagrange's mean value theorem

Let f (x) satisfy: ① continuous on [a, b]; ② differentiable on (a, b); ③ f (a) = f (b) prove: there exists ξ ∈ (a, b), so that: F '(ξ) = 0 prove: from: F (x) satisfy: ① continuous on [a, b]; ② differentiable on (a, b); so according to Lagrange's mean value theorem, there exists ξ ∈ (a, b)

Rolle's mean value theorem
F (x) = (x-1) ^ 2 / 3, where x closed interval is [0,2], satisfies Rolle's theorem?

The values of 0 and 2 are equal. They are all 1 / 3

3 / 4 / [56 * (3 / 7-3 / 8)] simple calculation

3/4/[56*(3/7-3/8)]
=3/4÷(56*3/7-56*3/8)
=3/4÷(24-21)
=3/4÷3
=1/4

. I'm all in a daze
If the definition field of function y = f (x + 1) is [- 2,3], then it is - 2

You are right about the function y = f (x + 1) = f (T) t = x + 1. The domain of function is [- 2,3], which means that the domain of T is [- 2,3]. Naturally, the domain of X + 1 is also [- 2,3], and the domain of X can be known. If the domain of X in function y = f (x + 1) is [- 2,3], then - 2 = 0, x + 1 > = 0

If half of the oxygen in the oxygen bottle is used, will the volume change?

The volume does not change because the remaining oxygen can still fill the space in the bottle
The density is reduced to half of the original, because the mass is reduced by half and the volume is unchanged. From P = m / V, the density is reduced to half of the original
Don't confuse the concepts upstairs; what gas is liquid; gas is gas. Don't confuse the two

Define a kind of operation "*" in the range of rational number, whose rule is a * b = AB / A + B, then 2 * (- 3) * 4

2*(-3)*4=2.4
The method is as follows
2*（-3)=-6/-1=6
6*4=24/10==2.4

What are the geometric and physical meanings of integral, double integral and triple integral

The geometric meaning of definite integral is the directed area of trapezoid with curved edges, and the physical meaning is the work done by the distance or variable force of variable speed linear motion;
The geometric meaning of the double integral is the directed volume of the curved top cylinder, and the physical meaning is the pressure on the plane area (the pressure is variable);
The geometric and physical meaning of triple integral is considered as the mass of inhomogeneous space object

There is a 9 cm long ruler without scale. Now we need to use this ruler to measure 9 scales of 1-9 cm. What scales should be on the ruler at least?

Mark 1 scale, 4 scale and 7 scale
1 cm, 4 cm, 7 cm, 9 cm can be measured directly
9-7 = 2 cm
7-4 = 3cm
9-4 = 5cm
7-1 = 6cm
9-1 = 8 cm

For the first time, mathematician () of the Southern Dynasty accurate the value of pi to 7 decimal places

For the first time, mathematician Zu Chongzhi of the Southern Dynasty accurate the value of pi to 7 decimal places 3.1415926