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Summary of common formula notes for solving problems in Higher Mathematics

2022-07-28 22:50:00 【Da Zha Xie Gemini】

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List of articles

One 、 The equivalent infinitesimal

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Two 、 A sequence of equal differences

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3、 ... and 、 Geometric series

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Four 、 Indefinite integral

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Replace

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5、 ... and 、 Basic function image

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6、 ... and 、 Trigonometric functions

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One full sine, two sine, three tangent and four cosine

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Inverse trigonometric function formula （ Commonly used ）

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The relationship between trigonometric functions

Multiply diagonally to 1
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Special angle

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Angle and π Transformation

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The formula of the sum of two corners

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Double angle formula

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Triple angle formula （ Is not important ）

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Half angle formula

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Universal formula

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And differential product

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7、 ... and 、 definite integral

Mean value theorem of integral

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Even times odd zero

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Interval representation

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Wallis formula ( Ignition formula )

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Periodic function

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Induction formula

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Anomalous integral ( Generalized integral )

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Add

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8、 ... and 、 Differential equations

First order linear differential equation

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Second order homogeneous linear differential equations with constant terms

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n Homogeneous linear differential equations with constant terms of order

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Second order nonhomogeneous linear differential equations with constant terms

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Nine 、 Taylor formula

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Common McLaughlin unfolds

In fact, these expansions are used most , The former is often used in the summation function of series 5 individual
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Ten 、 Asymptotes

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Horizontal asymptote

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Vertical asymptote

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Oblique asymptote

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11、 ... and 、 Vector algebra and space analytic geometry

Modules of vectors

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Direction angle and direction cosine

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Unit vector

The mold length is 1 Vector

Basis,

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Projection

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The product of quantity （ Point multiplication ）

What is involved in the operation is the vector , The result is a number
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Vector product （ Cross riding ）

What is involved in the operation is the vector , The result is still a vector
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Twelve 、 Directional derivatives and gradients

Directional derivative ( It's a number )

cos Is the direction cosine
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gradient

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The gradient direction is the direction in which the function grows fastest or the direction in which the directional derivative takes the maximum

The relationship between directional derivative and gradient ( Modules of gradients )

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13、 ... and 、 Standard geometry surface area / Volume formula

First statement ：
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Prism and pyramid

prism

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Pyramid

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Cylinder 、 Cone 、 The ball

Cylinder

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Cone

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The ball

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fourteen 、 Theorem

Intermediate value theorem

The English expression is ：The value between m and M.
Observe , Only these two theorems belong to closed interval
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Zero point theorem

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Rolle mean value theorem

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Lagrange mean value theorem

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Cauchy mean value theorem

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15、 ... and 、 Green's formula

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positive ： Anti-clockwise
reverse ： Clockwise （ Clock direction ）[ You need to add a minus sign ]
Because the content is relatively small, it is not too difficult , Add it here directly .
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sixteen 、 Unconditional extremum

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seventeen 、 Volume of revolution formula

Insert picture description here The volume formula of rotating body can be derived from the element method , details ： Application of element method

eighteen 、 Define derivation

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nineteen Triple integral

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Add

1 Polar coordinates

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2 One variable quadratic equation

a x 2 + b x + c = 0 ax^2+bx+c=0 ax2+bx+c=0
△ = b 2 − 4 a c △=b^2-4ac △=b2−4ac
△ > 0 x 1 , 2 = − b ± b 2 − 4 a c 2 a △>0\quad x_{1,2}=\frac{-b±\sqrt{b^2-4ac}}{2a} △>0x1,2​=2a−b±b2−4ac​​
△ = 0 x 1 = x 2 △=0\quad x_1=x_2 △=0x1​=x2​ Heavy root
△ < 0 △<0\quad △<0 Two negative roots   for example ： i 2 = − 1 − 9 = ± 3 i i^2=-1\quad \sqrt{-9}=±3i i2=−1−9​=±3i
Relationship between root and coefficient ： x 1 + x 2 = − b a , x 1 × x 2 = c a x_1+x_2=-\frac{b}{a},x_1×x_2=\frac{c}{a} x1​+x2​=−ab​,x1​×x2​=ac​

example ： solve 4 y ′ ′ + 4 y ′ + 5 y = 0 4y''+4y'+5y=0 4y′′+4y′+5y=0
Explain ：
4 λ 2 + 4 λ + 5 = 0 4λ^2+4λ+5=0 4λ2+4λ+5=0
△ = b 2 − 4 a c = 16 − 4 × 5 = 16 − 80 < 0 △=b^2-4ac=16-4×5=16-80<0 △=b2−4ac=16−4×5=16−80<0
λ 1 , 2 = − 4 ± − 64 2 × 4 = − 4 ± i 8 8 = − 1 2 ± i = α ± β i λ_{1,2}=\frac{-4±\sqrt{-64}}{2×4}=\frac{-4±i8}{8}=-\frac{1}{2}±i=α±βi λ1,2​=2×4−4±−64​​=8−4±i8​=−21​±i=α±βi
namely α = − 1 2 , β = 1 α=-\frac{1}{2},β=1 α=−21​,β=1
Sum up , The general explanation is y = e α x ( C 1 c o s β x + C 2 s i n β x ) y=e^{αx}(C_1cosβx+C_2sinβx) y=eαx(C1​cosβx+C2​sinβx)
        = e − 1 2 x ( C 1 c o s x + C 2 s i n x ) =e^{-\frac{1}{2}x}(C_1cosx+C_2sinx) =e−21​x(C1​cosx+C2​sinx)

3 Anti trigonometric function

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4 power function

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5 Exponential function

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6 Logarithmic function

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7 Multiplication and factorization

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8 Summation of sequences

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9 Analytic geometry

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10 Elementary Geometry

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11 inequality

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本文为[Da Zha Xie Gemini]所创，转载请带上原文链接，感谢
https://yzsam.com/2022/209/202207282057479087.html

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