Highschool Math

This section covers topics beyond the undergraduate level, introducing formulas and theorems in practical college entrance exam skills not commonly found in standard textbooks or workbooks. It is recommended for examinees to go through them thoroughly. Note that the curriculum referenced here is based on the “2007 Revised Curriculum.”

Practical College Entrance Exam Skills

• Translation and Derivative Relationships of Trigonometric Functions
• Frequently Used Definite Integrals of Quadratic Functions $\int_{\alpha}^{\beta}{(x-\alpha)(x-\beta)dx}=-\frac{{(\beta-\alpha)}^3}{6}$
• Using Harmonic Mean to Calculate Average Speed $v = \frac{2ab}{a+b}$
• Formula to Calculate the Distance Between Two Parallel Lines $d=\frac{|2k|}{\sqrt{m^2+1}}$
• If Two Events are Independent, Their Complements are Also Independent
• If Two Events are Mutually Exclusive, They are Dependent
• Quickly Finding the Extremes of a Quadratic Function
• Integration of Natural Logarithms Raised to a Power $\int {(\ln x)^n} dx=x{(\ln x)^n}-\int n{(\ln x)^{n-1}}dx$
• Area of a Triangle Enclosed by a Line and the Axes
• Inverse of a Fraction Function and the Shape of the Inverse of a 2x2 Square Matrix
• Repeated Squaring Formula for Rotation Matrices
• Simplified Formula for the Sum of the Elements of the Product of Two Square Matrices
• Easily Calculating the Equivalent Resistance in a Parallel Circuit
• Quickly Rationalizing Fractions Containing Roots
• Easily Squaring Two-Digit Numbers Ending in 5
• Expansion Formula for Double Radicals $\sqrt{a + b \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}$

Mathematics Curriculum

• Number of Subsets of a Finite Set with n Elements
• Deriving the Quadratic Formula by Rote
• Proof that √2 is Irrational
• Pythagoras’ Theorem
• Gabi’s Theorem
• If an Infinite Series Converges, the Infinite Sequence Converges to 0
• Calculating the Sum of an Arithmetic Series
  • Partial Sums of an Arithmetic Series are also Arithmetic
• Calculating the Sum of a Geometric Series
  • Partial Sums of a Geometric Series are also Geometric
• Calculating the Sum of Square Numbers
• Binomial Theorem
• General Term of the Fibonacci Sequence
• Base Change Formula for Logarithms
• Various Integration Methods for Trigonometric Functions
• Relationship Between Area Calculated by Riemann Sum and Definite Integral
• Calculating Probabilities for Poker Hands
• Calculating the Area of an Ellipse Using Integration
• Derivation of the Equation of a Tangent to a Parabola
• Property of a Line Passing Through the Focus of a Parabola
• The Product of the Slopes of Two Perpendicular Lines is Always -1
• Internal and External Division Points on a Straight Line
• Multiplication Formula Table
• Equation of a Tangent to a Circle with Slope $m$
• Equation of Tangents from a Point on a Circle
• Proof of $\sin^2 + \cos^2 = 1$
• For Small Central Angles, the Length of the Arc and the Chord are Approximately Equal
• Derivation of the Equation of an Ellipse
• Relationship Between the Roots and Coefficients of Quadratic/Cubic/n-th Degree Equations

Physics Curriculum

• Formula for the Velocity of a Falling Object $v = \sqrt{2gh}$
• Uniformly Accelerated Motion and Its Graphs
• Uniform Motion and Its Graphs
• Simple Proof of the Conservation of Momentum (High School Level)
• Collision of Two Bodies and Coefficient of Restitution
• Perfectly Elastic Collisions and Conservation of Kinetic Energy