Using the monotonicity of function, it is proved that when x > 0, there is x > arctan (x) I can read to understand the points

Let f (x) = arctan x, G (x) = x, X ＞ 0, f (0) = 0, G (0) = 0, f '(x) = 1 / (1 + X & sup2;) > 0, G' (x) = 1 ＞ 0, f '(x) - G' (x) = 1 / (1 + X & sup2;) - 1 = - X & sup2; / (1 + X & sup2;) ≤ 0, that is, f '(x) ≤ G' (x) because f (x) and G (x) on [0, + ∞) monotonically increase and f '(x) ≤ G' (x), so x > arctan (x)

Finding the inverse function of the function y = π + arctan X / 2

Note the domain and range of the original function: the domain x belongs to all real numbers, and the range y belongs to (π / 2,3 π / 2)
So the inverse function is obtained as follows:
Y = 2tan (x - π) = 2tanx, the domain of function definition is the range of original function (π / 2,3 π / 2)

On (- 1,1), an original function of 1 / (1 + x ^ 2) is () a, arctan (1 / x) B, - arccot (1 / x) C, - arctan ((1-x) / (1 + x))
D、arccot((1+x)/(1-x))

Trigonometric substitution: let x = Tan T, then DX = sec & # 178; t DT. Then ∫ 1 / (1 + X & # 178;) DX = ∫ 1 / (1 + Tan & # 178; t) * (1 / cos & # 178; t) DT = ∫ DT = t + C / / 1 + Tan & # 178; t = sec & # 178; t, SEC t = 1 / cos t = arctan x + C / / get t from the inverse solution t of x = Tan t

Let f (x) be a primitive function of F (x), f (1) = (√ 2) π) / 4. If x > 0, f (x) f (x) = (arctan √ x) / (√ x (1 + x)), try to find f (x)

According to the meaning of the question, there are f '(x) = f (x) f (x) f (x) = (arctan √ x) / [√ x (1 + x)] integral for X on both sides, there are ∫ f (x) f (x) DX = ∫ f (x) DF (x) = 1 / 2 * [f (x)] ^ 2 = ∫ (arctan √ x) / [√ x (1 + x)] DX = ∫ 2arctan √ x * D (arctan √ x) = (arctan √ x) ^ 2 + C, then [f (x)] ^ 2 = 2 (arctan √ x)

It takes 26 seconds for the train to pass through a 256 meter long tunnel, and the train takes 16 seconds to pass through a 96 meter long tunnel
It takes 26 seconds for the train to pass through a 256 meter long tunnel (from the front entrance to the rear exit). The train also takes 16 seconds to pass through a 96 meter long tunnel. Find the length of the train
And explain why this is done, what is the equivalent relationship

Let the length of the train be x meters
The formula (x + 256) △ 26 = (96 + x) △ 16
The solution is x = 160
The length of the train is 160 meters. Because the speed of the train is constant, so according to this series of equations, the length of the train is the length of the tunnel plus the length of the train itself

Triangle ABC is divided into two parts by De, be = 1 / 3AB, BD = 1 / 4bc

Taking the midpoint m of AE, connecting CE and cm, we can see that △ BCE = 1 / 3 △ ABC
Similarly, △ BDE = 1 / 4 △ BCE
So △ BDE = 1 / 12 △ ABC

At 300000 kilometers per second, how far is a light year?

Light year, a unit of length, refers to the distance light travels in a year, that is, about 9460 billion kilometers (or 5880 billion miles). It is more formally defined as: in a Julian year (365.25 days, equivalent to 86400 seconds per day), in free space and infinitely far away from any gravitational field or magnetic field, Because the speed of light in vacuum is 299792458 meters per second (accurate), a light year is equivalent to 9460730472580 meters (accurate), or about 9.46 × 1015 M = 9.46 beats meters. Light years are generally used to measure large distances, such as the distance between the solar system and another star. Light years are not a unit of time. In astronomy, the second gap is a very common unit, 26 light years. A light year is 63240 Au

Dad works for 5 days and has a rest for 1 day. Mom works for 6 days and has a rest for 1 day

The next time dad and mom rest together will be on the 42nd day

As shown in the figure, fold the square ABCD with a side length of 8cm, so that point d falls at the midpoint e of BC side, point a falls at F, and the crease is Mn, then the length of segment cn is ()
A. 3cmB. 4cmC. 5cmD. 6cm

Let CN = xcm, then DN = (8-x) cm. According to the properties of folding, en = DN = (8-x) cm, and EC = 12bc = 4cm. In RT △ ECN, according to Pythagorean theorem, en2 = EC2 + cn2, that is, (8-x) 2 = 16 + X2, we can get 16x = 48, so x = 3

When the density of water is 1.0 * 10 kg / m3, it is changed into g / cm3?
Wrong. It should be 1.0 * 10 cubic meter / cubic centimeter instead of? G / cubic centimeter

0 g / cm3
Read as 1.0 grams per cubic centimeter
The physical meaning is that the mass of water per cubic centimeter is 1 gram

Using the monotonicity of function, it is proved that when x > 0, there is x > arctan (x) I can read to understand the points