1. Calculate the time required to deposit 7g of silver from a solution of Silver Nitrate with a current of 4A.
2. Calculate the current required to deposit 40g of gold from AuCl3 solution in 7 hours.
INORGANIC CHEMISTRY PRESENTED BY GUO CHENG GUANG(aged 11) IN ENGLISH(1-02，S/N: 02).
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
1. A block of material has a density 𝜌 and floats three-fourths submerged in a liquid of unknown density in terms of 𝜌.
2. The density of ice is 917kg/m³, and the approximate density of seawater in which an iceberg floats is 1025kg/m³. What fraction of the iceberg is beneath the water surface.
FLUID DYNAMICS PRESENTED BY GUO CHENG GUANG(aged 11) AND GUO CHENGXI(aged 8) IN ENGLISH(1-05，S/N: 05).
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Finding the area of the region with double integrals. Example ∬(x+y)dxdy, The region bounded by 0⩽y⩽1, y=4x²and y=x².
Calculus--Indefinite Integrals (06-24, S/N: 347) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
What is the difference between gravitational force and electrostatic force?
Gravitational force can only be attractive but electrostatic force can be attractive as well as repulsive.
ELECTROMAGNETIC DYNAMICS PRESENTED BY GUO CHENG GUANG(aged 11) AND GUO CHENGXI(aged 8) IN ENGLISH(1-05，S/N: 05).
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Archimedes' principle: A body floating or submerged in a fluid is buoyed (lifted) upward by a force equal to the weight of the fluid that would be in the volume displaced by the liquid. Archimedes' principle deals with the forces applied to objects by the fluids around the object.
FLUID DYNAMICS PRESENTED BY GUO CHENG GUANG(aged 11) AND GUO CHENGXI(aged 8) IN ENGLISH(1-03，S/N: 03).
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Archimedes' Principle: Physical law of buoyancy fundamental to fluid mechanics, discovered by the ancient Greek mathematician, physicist, engineer, astronomer and inventor Archimedes, the statement that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weighted of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid
Fluid dynamics -- Archimedes's Principle (01-02, S/N: 002) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
In this video, we studied the differential equation that relates speed, distance and time.
Ordinary differential equations--Sep (01-15, S/N: 15) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Double Integral in Cartesian coordinates with respect to y first.
Calculus--Indefinite Integrals (06-21,S/N: 344) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
How to find the area of circle using integrations in calculs? The equation of the circle is x^2+y^2=R^2 and the circle is asymmetric with respect to the x-axis and y-axis, so first, we can solve the area of one quarter of circle and then multiple by 4 in order to get the total area of the circle.
In this video, we used three ways to integrate the area of the circle:
1. By circumference
2. By horizontal bar
3. By vertical bar
Calculus--Indefinite Integrals (06-02,S/N: 325) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
We can use substition to the limit towards one point whenever the function is continuous at that point, that's one of the definition of continuity.If the function is not continuous at the point we can't get the limit by direct substitution.
Find limits of functions at specific points just easily by directly plugging the point into the function.
METHODS OF LIMIT-CALCULATING PRESENTED BY GUO CHENG GUANG AND GUO CHENGXI(1-02，S/N: 71) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Integral can be used to find displacement, areas, volumes, central points and other concepts that arise by combining infinitesimal date.
A definite integral is an integral with upper and lower limits, in other words it has start and end end values or there is an interval.
Calculus--Indefinite Integrals (06-01,S/N: 324) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Locus is if a point moves on a plane satisfying some given geometrical condition then the path trace out by the point in the plane.
The equation to the locus of the moving point is the algebraic form of the given geometrical condition which is satisfy by the coordinate of all points on the locus.
ANALYTIC GEOMETRY PRESENTED BY GUO CHENG GUANG and GUO CHENGXI(01-10,S/N: 10) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using direct substituion with limits: It's easily find limits of functions at points where the functions are continuous, simly plug in the value into the function.
METHODS OF LIMIT-CALCULATING PRESENTED BY GUO CHENG GUANG AND GUO CHENGXI(7-01，S/N: 70) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
5.area of a triangle: a half of base times high【A=(1/2)bh】
ANALYTIC GEOMETRY PRESENTED BY GUO CHENG GUANG and GUO CHENGXI(01-09,S/N: 9) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Calculus--Indefinite Integrals (05-58,S/N: 323) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Prove Heron's Formula or Hero's Formula A=sqrt(s(s-a)(s-b)(s-c)), where a,b and c are side lengths of triangle, s=(a+b+c)/2.
ANALYTIC GEOMETRY PRESENTED BY GUO CHENG GUANG and GUO CHENGXI(01-08,S/N: 8) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using partial fraction and trigonometric substitution integrate ∫((1+x)/sqrt(1-4x^2))dx and ∫((2-x)/sqrt(2x-x^2))dx
Calculus--Indefinite Integrals (05-57,S/N: 322) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
B. Algebra Formulas - Factoring formulas- Difference of squares:
11. a^2-b^2=(a-b)(a+b)
C. Euler's formula
12. e^(ix)=cosx+isinx
13. sinx=(e^(ix)-e^(-ix))/2i
14. cosx=(e^(ix)+e^(-ix))/2
D. Logarithm rules
15. ln(x/y)=lnx-lny
E.The complex inverse trigonometric function
16. arctanz=i/2(ln((1-iz)/(1+iz)))
F. Basic Integration formulae:
17. ∫(1/x)dx=lnx+c
18. ∫(1/(x^2+y^2))dx=1/a(arctan(x/a))+c
Calculus--Indefinite Integrals (05-53-56,S/N: 321) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
In this video, we compared the three groups of concepts about macroscopic and microscopic quantities and units in physical chemistry or theoretical chemistry. First group: Physical and chemical, second group: state and procss, third group: matter, substance, body, object and material.
PHYSICAL CHEMISTRY BY GUO CHENG GUANG (aged 11) and GUO CHENGXI(aged 8) in Engling and Chinese (01-01,S/N: 4). Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Forth method of integration the ∫(1/(sinx+cosx))dx, using the integral by complex variable.
Calculus--Indefinite Integrals (05-56,S/N: 321) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Heron's Formula (Hero's formula): Area of Triangle whose sides have lengths a, b and c is = sqrt(s(s-a)(s-b)(s-c)), s is the semiperimeter of the triangle that is s=(a+b+c)/2.
ANALYTIC GEOMETRY PRESENTED BY GUO CHENG GUANG,LIM ZI JIE,GUO CHENGXI(01-06,S/N: 6) Presented By Guo Cheng Guang (aged 11), LIM ZI JIE(aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Third method of computing the ∫(1/(sinx+cosx))dx, using the integral by substitution.
Calculus--Indefinite Integrals (05-55,S/N: 320) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Second method of solving the ∫(1/(sinx+cosx))dx, using the integral by trigonometric identities.
Calculus--Indefinite Integrals (05-54,S/N: 319) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
The following methods have been used to solve this example:
1.Denominate Rationalization
2. Factorization
3. Partial Fraction
4. Trigonometric Identities
5. Logarithmic Integral
Calculus--Indefinite Integrals (05-53,S/N: 318) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
ANALYTIC GEOMETRY PRESENTED BY GUO CHENG GUANG,LIM ZI JIE,GUO CHENGXI(01-05,S/N: 5) Presented By Guo Cheng Guang (aged 11), LIM ZI JIE(aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
1. Finding the points of intersection of perpendicular line. To find the slope of the slope of the perpendicular line, flip the fraction and change the sign. For any linear equation y=mx+b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal of the original slope.
2. Finding the angle between two line equations.Using α=arctan(m)[Here, α is the angle of inclination, m is the slope. ] and tan(α+β)= [tan(α)+tan(β)]/[1-tan(α)tan(β)]
3. Finding the area of quadrilateral. Using shoelace formula, suppose the quadrilateral has vertices (x1,y1),(x2,y2),(x3,y3),(x4,y4), listed in clockwise order.
ANALYTIC GEOMETRY PRESENTED BY GUO CHENG GUANG,LIM ZI JIE,GUO CHENGXI(01-05,S/N: 5) Presented By Guo Cheng Guang (aged 11), LIM ZI JIE(aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
The integral by substitution technique amounts to applying the Chain Rule in reverse: to compute ∫f(x)g'(x)dx, we let u=g(x), du=g'(x)dx. Then ∫f(x)g'(x)dx= ∫f(u)du=F(u)=F(g(x)) where F is an antidifferenciation of f.
Let's computer ∫(1/sqrt((4+x^2)^3)) dx and ∫(1/sqrt((3+x^2)^5)) dx by making the substitution.
Calculus--Indefinite Integrals (05-50,S/N: 313) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
How does the substitution technique work? Let's solve the problems[1.∫sinxcosxdx; 2. ∫(x^5)(e^2x )dx] step-by-step to make sure our understand the process.
STEP 1: - Determine which expression to be substituted
STEP 2: - Substitute, rewrite the integral in simpler form and integrate
STEP 3: - Substitute back
Calculus--Indefinite Integrals (05-50,S/N: 313) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
We used the Gaussian elimnation method for solving a pair of simultaneous linear equations reduces one equation to one that has only a single variable.
shoelace theorem also known as shoelace algorithm or Gauss's area formula or the surveyor's formula, is an algorithm, a nifty formula for finding the area of a polygon given the coordinates of its vertices, is equal to the absolute value of :Suppose the triangle has vertices (x1,y1),(x2,y2),(x3,y3), listed in clockwise order.
area of triangle =1/2|x1 x2 x3 x1|
|y1 y2 y3 y1|
=1/2(x1*y2+x2*y3+x3*y1-x2*y1-x3*y1-x1*y3)
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Integration by Substitution also called u-substition, is the most easily technique by means of a change of variable. We are able to rewrite the integral into a very siple integral that we can do.
The substitution rule in general: ∫f(x)g'(x)dx= ∫f(u)du, where u=g(x).
In this lesson, we will prove that d(arcsin(x))/dx = 1/sqrt(1-x^2) by definition of differentiation and show that ∫(1/sqrt(1-x^2))dx= arcsin(x)+c by substitution
Calculus--Indefinite Integrals (05-49,S/N: 312) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
We will use the integral by parts or partial integration ∫f(x)g'(x)dx=f(x)g(x)-∫g'(x)f(x) to evaluate ∫[x(tanx)^-1/sqrt((1+x^2)^5)]dx and ∫[ln(x+sqrt(1+x^2))]^2dx.
Calculus--Indefinite Integrals (05-48,S/N: 311) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using the following methods to Exemplify the equation of a straight lines:
1. Slop-intercept Form
2. General Form
3. Two Point Form
4. Two-intercept Form
5. Cross product Form
include finding parallel and perpendicular lines.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Using Integration by parts to solve two indefinite integrals.
Calculus--Indefinite Integrals (05-47,S/N: 310) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Five methods to find the equation of a straight line:
1. Slop-intercept Form
2. General Form
3. Two Point Form
4. Two-intercept Form
5. Cross product Form
LECTURER:
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
LIM ZIJIE (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
Wechat: Go-Singapore
E-mail: forstudentsg@gmail.com
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Calculus--Indefinite Integrals (05-46,S/N: 309) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
To do these three indefinite integrals we all need to use integration by parts.
Calculus--Indefinite Integrals (05-45, S/N: 308) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
In these three indefinite integrals, we all need to use integral by parts. And we used the following methods to solve these integration:
1. Integral by parts
2. Factarization
3. Partial Fraction
4. Binomial Expansion
5. Trigonometric Identities
6. Integration of Trigonometric Integral
7. Logarithmic Integral
Calculus--Indefinite Integrals (05-44,S/N: 307) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
CALCULUS PRESENTED BY GUO CHENG GUANG（05-43,S/N: 306）-- trigonometric integrations. Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Analytic Geometry: In classical mathematics, analytic geometry also called as coordinate geometry or Cartesian geometry, is the study of geometry using the rectangular coordinate system or the Cartesian coordinate system.In modern analytic geometry is included concept of real or complex algeraic variety, is essentially equivalent to real and comple algebraic geometry.
Cartesian corrdinate system: In the plane, two perpendicular lines are chosen and the coordintaes of a point that can be identifired with ordered pairs of real numbers are taken to be signed distances to the lines.
Application of Analytic geometry: Analytic geometry can be widely used in algebraic, differential, discrete, physics, engineering, electrostatic and electrictity theory, designing machine, aviation,space science, sapceflight, computer science and so on.
To do these four indefinite integrals we'll need to use integration by parts. This special rule, integral by parts , is derived from the product rule of derivative, is available for integrating products of two functions.Calculus--Indefinite Integrals (05-42, total: 305) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Introduction to integration by Parts.The fact that this method come from the differential of function.. When working with the method of integral by parts,the differential of a function will be given first, and the function from which it come must be determined.
Calculus--Indefinite Integrals (combine together from 05-41,total:304) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
We used Five Different Integration techniques in this example and we got some different answers, but actually,they have the same answer in the end. We also can verify by differentiation. Why can we get different answers but all of answers are correct? Because each integration we use different constants for each integration and we play fast and loose with these constants of integral.So we need to be careful with the constants of integrtion, and remember we must understand that there are times where the constants of integral are important.
The Five Different Integration Techniques are:
1. Integral by Complex Variable With Factorization
2. Integral by Completing Square
3. Integral by Trigonometric Substitution
4. Integral by Complex Variable With Inverse Function
5. Integral by Gamma Function
The following methods have been used to solve this example:
1. Factarization
2. Partial Fraction
3. Integration of Trigonometric Integral
4. Euler's Formula
5. Expanding
6. Trigonometric Identities
-- Sum-difference Formulas
-- Double Angle Formula
7. Odd Function
8. Substitution
9. Inverse function
10. Logarithm Rules
11. Logarithmic Integral
12. Gamma function
Calculus--Indefinite Integrals (combine together from 05-36 to 05-40) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using gamma function solve integration.Calculus--Indefinite Integrals (05-40, total: 303) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Calculus--Indefinite Integrals (05-39, total: 302) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
We used the following methods to solve the integration:
1. Trigonometric Substitution
2. Completing Square
3. Double Angle Formula
4. Partial Fraction
5. Factarization
6. Logarithmic Integral
7. Integration of Trigonometric Integral
Calculus--Indefinite Integrals (05-38, total: 301) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
In this indefinite Integral, we used the following methods to solve the integration:
1. Trigonometric Substitution
2. Completing Square
3. Factarization
4. Partial Fraction
5. Double Angle Formula
6. Logarithmic Integral
7. Integration of Trigonometric Integral
Calculus--Indefinite Integrals (05-37, total: 300) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Indefinite Integral with complex number. Calculus--Indefinite Integrals (05-36, total: 299) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using three methods(1.Factorization+Partial Fraction 2.Factorization+Partial Fraction+Quadratic formula+Complex numble) 3.Factorization+Partial Fraction+Complex variable) to Solve Indefinite Integration. Calculus--Indefinite Integrals (05-35, total: 298) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using Factorization+Partial Fraction+Complex variable to Solve Indefinite Integration. Calculus--Indefinite Integrals (05-34, total: 297) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using Factorization+Partial Fraction+Complex variable to Integrate. Calculus--Indefinite Integrals (05-32, total: 295) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using Two Methods (1.completing square, 2. quadratic formula with complex variable ) to solving Indefinite Integration in English.CALCULUS PRESENTED BY GUO CHENG GUANG (aged 11) and Guo Chengxi (aged 8) （05-31） Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Using Two Methods of Integration by Partial Fractions.Calculus--Indefinite Integrals (05-30, total: 293) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Integration Using Two Methods of Partial Fractions.Calculus--Indefinite Integrals (05-29, total: 292) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Four methods for Indefinit Integrating.Calculus--Indefinite Integrals (05-28, total: 291) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Four ways to Solve the Indefinite Integral.Calculus--Indefinite Integrals (05-27, total: 290) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Solve the Indefinite Integral with Complex variable.
Calculus--Indefinite Integrals (05-25, total: 288) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Calculus--Indefinite Integrals (05-24, total: 287) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Calculus--Indefinite Integrals (05-23, total: 286) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Calculus--Indefinite Integrals (05-22, total: 285) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw
Calculus--Indefinite Integrals (05-20, total: 283) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Calculus--Indefinite Integrals (05-19, total: 282) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Calculus--Indefinite Integrals (05-17, total: 280) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Calculus--Indefinite Integrals (05-15, total: 277) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Calculus--Indefinite Integrals (05-14, total: 277) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
ELEMENTAL CHEMISTRY PRESENTED BY GUO CHENG GUANG（aged 11) and Guo Chengxi (aged 8)(001-01-01). Periodic Table of Chemical Elements Recited in English and Chinese Periodically and Groupedly and Sung in English.
在这个视频里，郭承光(11岁)和郭承曦(8岁)分别用英文和中文，横背和竖背并用歌唱的方式背诵了完整的化学元素周期表。Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated.
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English, Please subscribe:
Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English, Please subscribe:
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
GUO CHENG GUANG (11 years old, Primary 5 from YUMIN Primary School)
GUO CHENGXI( 8 years old, Primary 2 from YUMIN Primary School)
主讲：郭承光（11岁，育民小学五年级）
助讲：郭承曦（8岁，育民小学二年级）
Content： further differentiation： derivative, inverse differentiation, Partial derivatives, Total differentiation, L'Hopital's Rule, Leibniz Formula, mean value theorem.
