Calculus knowledge supplement--concurrently bosom 102

Original

See this topic is not all kinds of wonderful ...
In fact, I really just want to write about calculus, and then (Shunbian) Miss a freshman year of life.
Think I 5 months ago, just entered the JSZX, but also just a little to know the limit, derivative, integral definition of the great evil, after a semester of diligent learning, although still do not understand anything, but there seems to be no relationship.
At the end of the paper again miss it ... Add some things first.
In the previous log, the function value is the limit when the argument approaches what value, but the real limit is: For any a>0, there is b>0 so that when X is inside the x0 (x0,b), there is |f (x)-p|<a, then P is f (x) The limit at the X0 place.
Then say some ways to limit it.
I'm not going to talk about the arithmetic law, huh?
First, the pinch theorem: if there is a x0 field, there is a constant g (x) ≤f (x) ≤h (x), and Lim (x->x0) g (x) =a,lim (x->x0) H (x) =a, there is Lim (x->x0) f (x) =a.
This can be used to prove Lim (x->0) sinx/x=1, only need to construct -1/|x|≤sinx/x≤1/|x|, set theorem is good.
Then there is the indeterminate form rule, which only needs to be remembered when Lim (X->x0) g (x) =0,lim (x->x0) H (x) =0, Lim (x->x0) g (x)/h (x) =lim (x->x0) G ' (x)/h ' (x), plus g ( x), H (x) is the infinite limit is also established, this law can be proved by Cauchy mean value theorem. It can also be used to prove Lim (X->0) sinx/x=1.
If Lim (a->0,b->0) A/b =∞, it is said that a is a high-order infinitesimal, equal to a non-0 constant, A/b is an equivalent infinitesimal, it is clear that it is the speed of approaching 0 to make a distinction. In the limit calculation, the substitution can often be substituted by the equivalent infinitesimal.
The limit probably complements so much ...
I found that I used to think of derivative as differential ... 233 ... In fact, the derivative is simply the change of the function value after the small change of the independent variable, in fact it is the micro quotient. Of course they have many similar algorithms ...
Derivative of inverse function: [f^ ( -1)] ' (x) =1/f ' (y), wit is the kingly ...
Implicit function derivation: the expression on both sides of the X derivative, and then the Y ' solution is good
Parametric equation derivation: F ' (x) =φ ' (t)/φ ' (t), i.e. dy/dx= (DY/DT)/(DX/DT)
Derivatives number: This can often write it a few orders and then find the law (playing table to find the law?) ), plus (UV) ^ (n) =σ (0≤i≤n) C (i,n) *[u^ (i) *v^ (n-i)].
Integral? Except for a little triangle substitution seems like nothing ... Wait, look at me, great and merciful. The increment step ladder method to calculate the product:

#include <stdio.h>#include<math.h>#include<iomanip>#defineE 2.7182818DoubleFDoublex) {    returnPow (e,-x*x);}DoubleCalc (DoubleADoubleBDoubleESP) {    intDone0); intn=1; Doubleh,tn,t2n,k,temp,x; H=b-A; Tn=h* (f (a) +f (b))/2.0;  while(!Done ) {Temp=0;  for(k=0; k<n;k++) {x=a+ (k +2.W)*h; Temp+=f (x); } t2n= (tn+h*temp)/2.0; if(Fabs (T2N-TN) <esp) done=1; Else{Tn=t2n; N=n*2; H=h/2; }    }    returnt2n;}intMain () {printf ("%.50lf\n", Calc (-10000,10000, 1e- -)); return 0;}

I've been told to write differential equations, I don't write, I don't write ...
The differential equation is actually the equation with the conduction function ... Just like a primary school, it's OK ...
Example: Dy/dx+f (x) y=0, then dy/y=f (x) dx, then y=ce^ (-∫f (x) dx). As for Dy/dx+f (x) y=g (x) is the expression after the substitution of the original formula, namely ce^ (-∫f (x) dx) * [∫g (x) e^ (∫f (x) dx) Dx+c].
It must be admitted that the differential equation only writes this point because ... Is...... The watermelon is so ... It's not quite familiar.
Then there is a second-order differential equation, the general solution structure y=c1y1+c2y2, wherein the Y1,Y2 is two special solutions, the special solution can be used y=e^ (RX), y ' =r e^ (rx), y ' =r^2 e^ (RX), the separation of e and then solve the characteristics of the equation, I am also more counseling ... Not much to say ... But you know, if Y is a multivariate function, then y ' +py ' +qy=f (x) is called second order constant coefficient nonhomogeneous linear partial differential equation, la la la ...
The median theorem:
Lowe's theorem:
If the function f (x) satisfies a continuous in the closed interval [a, b], in (A, a) can be guided, f (a) =f (a), then there is at least one ξ∈ (A, a), so that F ' (ξ) = 0.
It would be nice to prove it with the maximum minimum value.
LaGrand-Japanese value theorem:
If the function f (x) satisfies a continuous in the closed interval [A, a, b], in (A, a), there is at least one ξ∈ (A, a), which makes F ' (ξ) =[f (a)-f (a)]/(b-a).
constructor g (x) =f (x)-F (a)-(x-a) *[f (b)-F (a)]/(b-a), then use the Lowe theorem to prove it.
Cauchy median theorem:
If the function f (x) and F (x) satisfy the continuous on the closed interval [a, b], in the open interval (a, b) can be guided, to X∈ (A, A, b), F ' (x) ≠0, then at least a little ζ within (a, a), so that [F (c)-F (a)]/[f (a)-f (a)]=f ' (ζ)/F ' (ζ) Was founded.
Constructs g (x) =f (x)-F (a)-[f (b)-F (a)]/[g (b)-G (a)]*[g (x)-G (a)], the same set of Raul theorem.
Then mess up some definitions:
Break point:
The function f (x) is defined within a x0 neighborhood of the point. If the function f (x) has: there is no definition in x=x0 or although there is a definition in x=x0, but x→x0 LIMF (x) does not exist or is defined in x=x0, and x→x0 LIMF (x) exists, but x→x0 LIMF (x) ≠f (x0),
The function f (x) is x0 at the point, and the point x0 is called the break point of function f (x).
Several common types:
Can go to break point: the function at the point left limit, the right limit exists and is equal, but not equal to the point function value or function at that point is undefined.
Jumping breaks: functions exist at the left and right limits of the point, but are not equal.
Infinite discontinuity: The function can be undefined at this point, and at least one of the left and right limits is absent, and the function is at the limit of the point ∞.
Oscillation discontinuity: The function can be undefined at that point, and when the argument tends to that point, the function value changes infinitely multiple times between two constants.
The break point and the jumping break point are called the first kind of discontinuity, also called the limited discontinuity point. Other breaks are referred to as the second type of discontinuity point.

Asymptote
Horizontal progressive line: Lim (x->-∞) f (x) =a or Lim (x->+∞) f (x) =a, then y=a is the horizontal asymptotic line of f (x).
Vertical Progressive line: Lim (x->a+) f (x) =∞ or Lim (x->a-) f (x) =∞, then x=a to f (x) vertical asymptotic line.
Oblique Progressive line: Lim (x->+∞) f (x)-kx-b=0 or Lim (x->-∞) f (x)-kx-b=0, then y=kx+b is the oblique-asymptotic line of f (x).
———————— I'm a split line ————————————

Probably finished, I miss a ...
Describe my freshman year: F (x) =1/(|x|-3) (so bad a function dare to take it out?) ...... Because I'm a wuss. ）
OK, here it is ... r= (sin (t) *sqrt (ABS (cos (t)))/(sin (t) + (7/5)) -2*sin (t) +2
(It's not about calculus, it's about the polar coordinates ...) Blame me ...
Line ... dx/dt=a ( -2sint+2sin2t), dy/dx=-(cost-cos2t)/(SINT-SIN2T) Two equations from a solution ...
(Silly fork equation ...) Watermelon is a silly fork ...
This ... x>0, y ' +y=2[(1+x) sqrt (4-x^2)-(2x^2)/sqrt (4-x^2)], and it's even function ...
And this y=abs (x-a) -2*abs (x-a-1) +2*abs (x-a-2) -2*abs (x-a-3) +abs (x-a-4) +abs (x-b) -2*abs (x-b-1) +2*abs (x-b-2) -2*abs ( x-b-3) +abs (x-b-4) +lim (c->∞) [C/2 (ABS (X-D+3/C) -2*abs (x-d) +abs (x-d-1.5/c) +abs (x-d-2+1.5/c) -2*abs (x-d-2) +abs ( X-D-2-3/C)]
Which a=-7,b=-2,d=3.
If some of my classmates would like to use a little bit of their toes to see what the above functions mean, they probably know what's in the watermelon (.?).
Finally send one to 102 students: x=0,y∈[-1,1]; r=|sint|+1; [sqrt (1-x^2)-y] [|x-1-y|+ (X∈ ( -1,1)? 0:1)] [|y+2|+ (X∈ ( -1,1)? 0:1)]=0

Calculus knowledge supplement--concurrently bosom 102