Ask questions about higher numbers 3 (about differential mean value theorem) Let B not be equal to a, and prove that | arctanb arctana | is less than or equal to | b-a|

Ask questions about higher numbers 3 (about differential mean value theorem) Let B not be equal to a, and prove that | arctanb arctana | is less than or equal to | b-a|

Let b > A, then the original proposition is arctanb arctana ≤ b-a
F (x) = arctanx is continuous in [a, b] and differentiable in (a, b),
According to Lagrange's mean value theorem, there is a point ξ (a)

A problem of differential mean value theorem of higher numbers
Description: Let f (x) be continuous on [- A, a] and continuous on (- A, a)
It is proved that f (- a) = f (a), a > 0-
a. There is at least one point θ in a) such that F
'(θ) = 2 θ f (θ). ('guidance number)

Do auxiliary function
F(x) = lnf(x) - x^2,
Then f (x) is continuous on [- A, a], differentiable in (- A, a), and
F(-a) = F(a),
According to Rolle's theorem, there is at least one point θ in (- A, a) such that
F‘(θ) = 0,
Namely
f'(θ)=2θf(θ),
It has been proved

Differential mean value theorem of higher numbers
Let f (x) have the third derivative on [0,1], and f (0) = 0, f (1) = 1 / 2, f '(1 / 2) = 0, then we prove that there is a ᦉ 8364; belonging to (0,1), such that | f' '' (ᦉ 8364;) | > = 12

Let f (0) and f (1) do Taylor expansion at x = 0.5
0=f(0)=f(0.5)+0.5f‘’(0.5)*(0.5)^2-f'''(c)/48；
0.5=f(1)=f(0.5)+0.5f''(0.5)*(0.5)^2+f'''(d)/48;
It's absolutely worthwhile to reduce the two formulas
24=12.
So the conclusion is valid

The number of girls is four times that of boys. There are 18 more girls than boys. How many boys and girls are there in the chorus

The number of female students is X
The number of boys is y
There are
x=4y
x-y=18
The solution is x = 24
y=6

If the algebraic formula 6mx & sup2; + 4nxy + 2x + 2xy-x & sup2; + y-4 does not contain X & sup2; term and XY term, find the value of 6m-2n + 2

6m-1=0
4n+2=0
∴m=1/6
n=-1/2
∴6m-2n+2
=1+1+2
=4

Xiao Hua read a 200 page story book. On the first day, he read one eighth of the whole book. On the second day, which page did he start to read?
The original price of a computer was 6000 yuan. First, the price was reduced by 1 / 12. Because it was too popular, the price was increased by 1 / 10. Now how much is the price of this computer?
No one answered the question just now. Now I'm going to talk about practical questions

6000 × 1 / 12 = 4500, 4500 × 11 / 10 = 4950
Xiaohua's question is: 200 × 0.125 = 25

When Wang Ming calculated the difference of a polynomial minus 2 (3a + a-5), he forgot to use "2" and the
When Wang Ming calculates a polynomial minus the difference of 2 (3a + a-5), he forgot to multiply the two terms after "2" and "3A's square + a-5" in the bracket due to his negligence. The result is the square of the errand a + 3a-1. Based on this, can you work out the polynomial? And please work out the correct result

3A & # 178; + a-5 followed by a and - 5
This polynomial is:
a²+3a-1+2×3a²
=a²+3a-1+6a²
=7a²+3a-1
The correct results are as follows
(7a²+3a-1)-2(3a²+a-5)
=7a²+3a-1-6a²-2a+10
=a²+a+9

A large truck and a small car drive from place a to place B at their respective speeds at the same time. When they arrive at place B, they immediately return, and their respective speeds increase by 20%. 1.5 hours after starting, the car meets the large truck on the way back. When the large truck arrives at place B, the car still has 15% of the distance between place a and place B How many hours does a round trip between B and B take?

1: (1 + 23) = 3:53: [5 × (1 + 20%)] = 3:6 = 1:2 after the meeting, the car only has 1.5 hours to go, so the time for the car is 1.5 △ 2 = 0.75 hours

How to draw the number axis

A straight line, determine the origin, divided into several segments, and then add an arrow

From city a to city B, it takes 8 hours for passenger cars and 12 hours for freight cars. The two cars come from two places at the same time, and the freight cars are 40 kilometers away from the midpoint when they meet
How many kilometers are there between the two cities?

The distance ratio of bus to truck is 1 / 8:1 / 12 = 3:2
The distance between city a and city B: 40 × 2 ÷ (3-2) × (3 + 2) = 400 km