(1) parabola

y = ax^2 + bx + c (a≠0)

That is, y equals a times x plus the square of b times x plus C.

Placed in a plane rectangular coordinate system

When a> is 0, the opening is upward.

When a< is 0, the opening is downward.

(a=0 is a linear function of one variable)

C>0, the function image intersects with the positive direction of Y axis.

C<0, the function image intersects with the negative direction of Y axis.

When c = 0, the parabola passes through the origin.

When b = 0, the axis of symmetry of parabola is the Y axis.

(Of course, this function is linear when a=0 and b≠0).

There are also vertex formulas y = a (x+h) * 2+k, (h, k) = (-b/(2a), (4ac-b 2)/(4a)).

That is, y equals a times the square of (x+h)+K.

-h is x of vertex coordinates.

K is y of vertex coordinates.

Generally used to find the maximum, minimum and symmetry axis.

Parabolic standard equation: y 2 = 2px (p > 0)

It means that the focus of parabola is on the positive semi-axis of X, the focal coordinate is (p/2,0), and the directrix equation is x=-p/2.

Since the focus of parabola can be on any semi-axis, * * has the standard equation y 2 = 2px y 2 =-2px x 2 = 2py x 2 =-2py.

(2) circle

Sphere volume = (4/3) π (r 3)

Area = π (r 2)

Perimeter =2πr =πd

The standard equation of a circle (X-A) 2+(Y-B) 2 = R 2 Note: (A, B) is the center coordinate.

General equation of circle x2+y2+Dx+Ey+F=0 Note: D2+E2-4f >; 0

(A) ellipse circumference calculation formula

According to the standard elliptic equation, the major axis A and minor axis B are λ=(a-b)/(a+b).

Ellipse circumference l = π (a+b) (1+λ 2/4+λ 4/64+λ 6/256+25λ 8/16384+.

......)

Simplification: L≈π[ 1.5(a+b)- sqrt(ab)]

Or l ≈ π (a+b) (64-3λ 4)/(64-16λ 2).

(2) Calculation formula of ellipse area

Elliptic area formula: S=πab

Ellipse area theorem: the area of an ellipse is equal to π times the product of the major semi-axis length (a) and the minor semi-axis length (b) of an ellipse.

Although there is no ellipse πT in the above formula of ellipse circumference sum area, these two formulas are all derived from ellipse πT .. Constant is the body, so it can be used.

Volume calculation formula of ellipsoid long radius * short radius * ellipse π height

(3) trigonometric function

Sum and difference angle formula

sin(A+B)= Sina cosb+cosa sinb； sin(A-B)=sinAcosB - sinBcosA

cos(A+B)= cosa cosb-Sina sinb； cos(A-B)=cosAcosB + sinAsinB

tan(A+B)=(tanA+tanB)/( 1-tanA tanB)； tan(A-B)=(tanA-tanB)/( 1+tanA tanB)

cot(A+B)=(cotA cotB- 1)/(cot B+cotA)

; cot(A-B)=(cotA cotB+ 1)/(cot b-cotA)

Double angle formula

tan2a=2tana/( 1-tan^2a)； cot2A=(cot^2A- 1)/2cota

cos2a=cos^2a-sin^2a=2cos^2a- 1= 1-2sin^2a

sin2A=2sinAcosA=2/(tanA+cotA)

In addition: sin α+sin (α+2π/n)+sin (α+2π * 2/n)+sin (α+2π * 3/n)+...+sin [α+2π * (n-1)/n] = 0.

cosα+cos(α+2π/n)+cos(α+2π* 2/n)+cos(α+2π* 3/n)+……+cos[α+2π*(n- 1)/n]= 0

and

sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2

tanAtanBtan(A+B)+tanA+tan B- tan(A+B)= 0

Quadruple angle formula:

sin4a=-4*(cosa*sina*(2*sina^2- 1))

cos4a= 1+(-8*cosa^2+8*cosa^4)

tan4a=(4*tana-4*tana^3)/( 1-6*tana^2+tana^4)

Quintuple angle formula:

sin5a= 16sina^5-20sina^3+5sina

cos5a= 16cosa^5-20cosa^3+5cosa

tan5a=tana*(5- 10*tana^2+tana^4)/( 1- 10*tana^2+5*tana^4)

Hexagon formula:

sin6a=2*(cosa*sina)*(2*sina+ 1)*(2*sina- 1)*(-3+4*sina^2))

cos6a=((- 1+2*cosa^2)*( 16*cosa^4- 16*cosa^2+ 1))

tan6a=(-6*tana+20*tana^3-6*tana^5)/(- 1+ 15*tana^2- 15*tana^4+tana^6)

Seven-fold angle formula:

sin7a=-(sina*(56*sina^2- 1 12*sina^4-7+64*sina^6))

cos7a=(cosa*(56*cosa^2- 1 12*cosa^4+64*cosa^6-7))

tan7a=tana*(-7+35*tana^2-2 1*tana^4+tana^6)/(- 1+2 1*tana^2-35*tana^4+7*tana^6)

Octagonal angle formula:

sin8a=-8*(cosa*sina*(2*sina^2- 1)*(-8*sina^2+8*sina^4+ 1))

cos8a= 1+( 160*cosa^4-256*cosa^6+ 128*cosa^8-32*cosa^2)

tan8a=-8*tana*(- 1+7*tana^2-7*tana^4+tana^6)/( 1-28*tana^2+70*tana^4-28*tana^6+tana^8)

Nine-fold angle formula:

sin9a=(sina*(-3+4*sina^2)*(64*sina^6-96*sina^4+36*sina^2-3))

cos9a=(cosa*(-3+4*cosa^2)*(64*cosa^6-96*cosa^4+36*cosa^2-3))

tan9a=tana*(9-84*tana^2+ 126*tana^4-36*tana^6+tana^8)/( 1-36*tana^2+ 126*tana^4-84*tana^6+9*tana^8)

Ten times angle formula:

sin 10a=2*(cosa*sina*(4*sina^2+2*sina- 1)*(4*sina^2-2*sina- 1)*(-20*sina^2+5+ 16*sina^4))

cos 10a=((- 1+2*cosa^2)*(256*cosa^8-5 12*cosa^6+304*cosa^4-48*cosa^2+ 1))

tan 10a=-2*tana*(5-60*tana^2+ 126*tana^4-60*tana^6+5*tana^8)/(- 1+45*tana^2-2 10*tana^4+2 10*tana^6-45*tana^8+tana^ 10)

General formula:

sinα=2tan(α/2)/[ 1+tan^2(α/2)]

cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]

tanα=2tan(α/2)/[ 1-tan^2(α/2)]

half-angle formula

sin(A/2)=√(( 1-cosA)/2)sin(A/2)=-√(( 1-cosA)/2)

cos(A/2)=√(( 1+cosA)/2)cos(A/2)=-√(( 1+cosA)/2)

tan(A/2)=√(( 1-cosA)/(( 1+cosA))tan(A/2)=-√(( 1-cosA)/(( 1+cosA))

cot(A/2)=√(( 1+cosA)/(( 1-cosA))cot(A/2)=-√(( 1+cosA)/(( 1-cosA))

Sum difference product

2 Sina cosb = sin(A+B)+sin(A-B)； 2cosAsinB=sin(A+B)-sin(A-B)

2 cosa cosb = cos(A+B)+cos(A-B)； -2sinAsinB=cos(A+B)-cos(A-B)

sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2

; cosA+cosB = 2cos((A+B)/2)sin((A-B)/2)

tanA+tanB = sin(A+B)/cosa cosb； tanA-tanB=sin(A-B)/cosAcosB

cotA+cotB = sin(A+B)/Sina sinb； -cotA+cotB=sin(A+B)/sinAsinB

Reduced power formula

Sin? (A)=( 1-cos(2A))/2 = versin(2A)/2

Because? (α)=( 1+cos(2A))/2 = covers(2A)/2

Tan? (α)=( 1-cos(2A))/( 1+cos(2A))

Sine theorem a/sinA=b/sinB=c/sinC=2R Note: where r represents the radius of the circumscribed circle of a triangle.

Cosine Theorem B 2 = A 2+C 2-2 ACCOSB Note: Angle B is the included angle between side A and side C.

Inductive formula

Formula 1:

Representation of angles in arc system;

sin(2kπ+α)=sinα (k∈Z)

cos(2kπ+α)=cosα (k∈Z)

tan(2kπ+α)=tanα (k∈Z)

cot(2kπ+α)=cotα (k∈Z)

sec(2kπ+α)=secα (k∈Z)

csc(2kπ+α)=cscα (k∈Z)

Representation of angle in angle system;

sin (α+k 360 )=sinα(k∈Z)

cos(α+k 360 )=cosα(k∈Z)

tan (α+k 360 )=tanα(k∈Z)

cot(α+k 360 )=cotα (k∈Z)

sec(α+k 360 )=secα (k∈Z)

csc(α+k 360 )=cscα (k∈Z)

Equation 2:

Representation of angles in arc system;

sin(π+α)=-sinα (k∈Z)

cos(π+α)=-cosα(k∈Z)

tan(π+α)=tanα(k∈Z)

cot(π+α)=cotα(k∈Z)

sec(π+α)=-secα(k∈Z)

csc(π+α)=-cscα(k∈Z)

Representation of angle in angle system;

sin( 180 +α)=-sinα(k∈Z)

cos( 180 +α)=-cosα(k∈Z)

tan( 180 +α)=tanα(k∈Z)

cot( 180 +α)=cotα(k∈Z)

sec( 180 +α)=-secα(k∈Z)

csc( 180 +α)=-cscα(k∈Z)

Formula 3:

sin(-α)=-sinα(k∈Z)

cos(-α)=cosα(k∈Z)

tan(-α)=-tanα(k∈Z)

cot(-α)=-cotα(k∈Z)

sec(-α)=secα(k∈Z)

csc-α)=-cscα(k∈Z)

Equation 4:

Representation of angles in arc system;

sin(π-α)=sinα(k∈Z)

cos(π-α)=-cosα(k∈Z)

tan(π-α)=-tanα(k∈Z)

cot(π-α)=-cotα(k∈Z)

sec(π-α)=-secα(k∈Z)

cot(π-α)= csα(k∈Z)

Representation of angle in angle system;

sin( 180 -α)=sinα(k∈Z)

cos( 180 -α)=-cosα(k∈Z)

tan( 180 -α)=-tanα(k∈Z)

cot( 180 -α)=-cotα(k∈Z)

sec( 180 -α)=-secα(k∈Z)

csc( 180 -α)=cscα(k∈Z)

Formula 5:

Representation of angles in arc system;

sin(2π-α)=-sinα(k∈Z)

cos(2π-α)=cosα(k∈Z)

tan(2π-α)=-tanα(k∈Z)

cot(2π-α)=-cotα(k∈Z)

sec(2π-α)=secα(k∈Z)

csc(2π-α)=-cscα(k∈Z)

Representation of angle in angle system;

sin(360 -α)=-sinα(k∈Z)

cos(360 -α)=cosα(k∈Z)

tan(360 -α)=-tanα(k∈Z)

cot(360 -α)=-cotα(k∈Z)

Seconds (360-α)= Seconds α(k∈Z)

csc(360 -α)=-cscα(k∈Z)

Equation 6:

Representation of angles in arc system;

sin(π/2+α)=cosα(k∈Z)

cos(π/2+α)=—sinα(k∈Z)

tan(π/2+α)=-cotα(k∈Z)

cot(π/2+α)=-tanα(k∈Z)

sec(π/2+α)=-cscα(k∈Z)

csc(π/2+α)=secα(k∈Z)

Representation of angle in angle system;

sin(90 +α)=cosα(k∈Z)

cos(90 +α)=-sinα(k∈Z)

tan(90 +α)=-cotα(k∈Z)

cot(90 +α)=-tanα(k∈Z)

sec(90 +α)=-cscα(k∈Z)

csc(90 +α)=secα(k∈Z)

⒉

Representation of angles in arc system;

sin(π/2-α)=cosα(k∈Z)

cos(π/2-α)=sinα(k∈Z)

tan(π/2-α)=cotα(k∈Z)

cot(π/2-α)=tanα(k∈Z)

sec(π/2-α)=cscα(k∈Z)

csc(π/2-α)=secα(k∈Z)

Representation of angle in angle system;

sin (90 -α)=cosα(k∈Z)

cos (90 -α)=sinα(k∈Z)

tan (90 -α)=cotα(k∈Z)

cot (90 -α)=tanα(k∈Z)

sec (90 -α)=cscα(k∈Z)

csc (90 -α)=secα(k∈Z)

three

Representation of angles in arc system;

sin(3π/2+α)=-cosα(k∈Z)

cos(3π/2+α)=sinα(k∈Z)

tan(3π/2+α)=-cotα(k∈Z)

cot(3π/2+α)=-tanα(k∈Z)

sec(3π/2+α)=cscα(k∈Z)

csc(3π/2+α)=-secα(k∈Z)

Representation of angle in angle system;

sin(270 +α)=-cosα(k∈Z)

cos(270 +α)=sinα(k∈Z)

tan(270 +α)=-cotα(k∈Z)

cot(270 +α)=-tanα(k∈Z)

sec(270 +α)=cscα(k∈Z)

csc(270 +α)=-secα(k∈Z)

four

Representation of angles in arc system;

sin(3π/2-α)=-cosα(k∈Z)

cos(3π/2-α)=-sinα(k∈Z)

tan(3π/2-α)=cotα(k∈Z)

cot(3π/2-α)=tanα(k∈Z)

sec(3π/2-α)=-secα(k∈Z)

csc(3π/2-α)=-secα(k∈Z)

Representation of angle in angle system;

sin(270 -α)=-cosα(k∈Z)

cos(270 -α)=-sinα(k∈Z)

tan(270 -α)=cotα(k∈Z)

cot(270 -α)=tanα(k∈Z)

sec(270 -α)=-cscα(k∈Z)

csc(270 -α)=-secα(k∈Z)

(4) Inverse trigonometric function

Arcsine (-x)=- Arcsine

arccos(-x)=π-arccosx

Arctangent (-x)=- arctangent

arccot(-x)=π-arccotx

Arc sin x+ arc cos x=π/2

Arc tan x+ arc cot x=π/2

(5) sequence

Arithmetic progression's general formula: an-a 1-(n- 1) d

Sum of the top n items in arithmetic progression: sn = [n (a1+an)]/2 = na1+[n (n-1) d]/2.

Geometric series formula: an = a1* q (n-1);

Sum of the first n terms of geometric series: Sn = a1(1-q n)/(1-q) = (a1-a1-q n)/(1-q)

=a 1/( 1-q)-a 1/( 1-q)*q^n(n≠ 1)

The sum of the first n terms of some series:

1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2

1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)=n^2

2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1)

1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+…+n^2=n(n+ 1)(2n+ 1)/6

1^3+2^3+3^3+4^3+5^3+6^3+…n^3=(n(n+ 1)/2)^2

1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3

(6) multiplication and factorization

factoring

a^2-b^2=(a+b)(a-b)

a^2 2ab+b^2=(a b)^2

a^3+b^3=(a+b)(a^2-ab+b^2)

a^3-b^3=(a-b)(a^2+ab+b^2)

a^3 3a^2b+3ab^2 b^3=(a b)^3

Multiplication formula

Reversing the left and right sides of the factorization formula above is the multiplication formula.

(7) Trigonometric inequality

-|a|≤a≤|a|

| a |≤b & lt； = & gt-b≤a≤b

| a |≤b & lt； = & gt-b≤a≤b

| a |-| b |≤| a+b |≤| a |+| b | | a |≤b & lt； = & gt-b≤a≤b

|a|-|b|≤|a-b|≤|a|+|b|

| z1|-| z2 |-| Zn |≤| z1+z2+...+Zn |≤| z1|+| z2 |+| zinc |

| z1|-| z2 |-| Zn |≤| z1-z2-...-Zn |≤| z1|+| z2 |+| zinc |

| z1|-| z2 |-| Zn |≤| z1z2 ... Zn |≤| z1| +| z2 |+| Zn |

(8) One-variable quadratic equation

The solution of the unary quadratic equation wx1=-b+√ (B2-4ac)/2ax2 =-b-√ (B2-4ac)/2a.

The relationship between root and coefficient (Vieta theorem) x1+x2 =-b/a; x 1*x2=c/a

Discriminant △ = b 2-4ac = 0, then the equation has two equal real roots.

△& gt； 0, the equation has two unequal real roots.

△& lt； 0, then the equation has two * * * yoke roots d (no real roots).

Basic attribute

If a>0, and a≠ 1, m >；; 0, N>0, then:

1.a^log(a)(b)=b

2.log(a)(a)= 1

3 . log(a)(MN)= log(a)(M)+log(a)(N)；

4 . log(a)(M÷N)= log(a)(M)-log(a)(N)；

5.log(a)(M^n)=nlog(a)(M)

6. log (a) [m (1/n)] = log (a) (m)/n Helen's formula: if the three sides a, b, c and the semi-circumference p of a triangle are known, then S = √ [P (p-a) (p-b)].

(Helen Qin Jiushao formula) (p= (a+b+c)/2) factorial of permutation and combination: n! = 1× 2× 3× …× n, (n is an integer not less than 0) specifies 0! = 1。 All permutation numbers of M elements with N different elements, A(n, m)= n! /(n - m)！ (m is a superscript, n is a subscript, and both are integers not less than 0, m ≤ n). Taking m elements from n different elements at a time, no matter what order they are combined into a group, is called combination. Species number of all different combinations C(n, m)= A(n, m)/m! =n！ /[m！ (n-m)！ ]

(m is a superscript, n is a subscript, and they are all integers not less than 0, m≤n)◆ Properties of combination number: c (n, k) = c (n- 1, k)+c (n- 1, k-1); For the combination number C(n, k), n and k are converted into binary respectively. If n corresponding to a binary bit is 0 and k is 1, then c (n, k) is an even number; Otherwise it is odd integer binomial theorem (binomial

Theorem) (a+b) n = c (n, 0) × a n× b 0+c (n, 1) × a (n- 1 )× b+c (n, 2 )× a (n-2). 0)× 1^n+c(n, 1)× 1^(n- 1)× 1+c(n,2)× 1^(n-2)× 1^2+...+C(n,n)× 1^n

= (1+ 1) n = 2 n Definition of calculus limit: Let the function f(x) have a centripetal neighborhood defined at point X. If there is a constant a, there is always a positive number δ for any given positive number ε (no matter how small it is), so that when x satisfies inequality 0.

The corresponding function values f(x) all satisfy the inequality: | f (x)-a |

-∫ u' (x) v (x) dx。 Taylor mean value theorem of Taylor formula of unary function: If f(x) has the first derivative of n+ 1 in the open interval (a, b), when the function is in this interval, it can be expanded into a function about (x-). ? (x-x0)^2,+f'''(x0)/3！ ? N derivative of (x-x0) 3+ ... +f? (x0)/n！ ? (x-x0) n+rn where rn = f (n+1) (ξ)/(n+1)! ? (x-x0) (n+ 1) is the remainder of Lagrange type, where ξ is between x and x0. The form of definite integral is ∫f(x) dx.

(The upper limit A is written above ∫, and the lower limit B is written below ∫). It is called definite integral because the value obtained after its integration is definite, a number, not a function. Newton-Leibniz formula: If F'(x)=f(x), then ∫f(x) dx (upper limit A and 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 in the original function and the lower limit in the original function. Differential equation Any equation that represents the relationship between the derivative and the independent variable of an unknown function is called a differential equation. If an unknown function in a differential equation contains only one independent variable, this equation is called eigenvalue method of ordinary differential equation, which is a general method to solve homogeneous linear differential equation with constant coefficients. For example, the general solution of the second-order homogeneous linear differential equation y''+py'+qy=0: Let the characteristic equation r*r+p*r+q=0 be r 1, r2. 1 If the real root r 1 is not equal to R2y = c1* e (r1x)+C2 * e (r2x) .2 If the real root r = r1= R2y = (c/kloc-)