Maths1_Quick_Refresher_Formulas

Maths 1: Quick Refresher & Formula Sheet

TIP

How to use: Use this for quick revision before quizzes or exams. It contains all the “Facts” and “Formulas” without the deep explanations.

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Week 1: Sets & Relations

• Cardinality: $|A\cup B|=|A|+|B|-|A\cap B|$.
• Power Set: If $|A|=n$, then $|P(A)|=2^n$.
• Equivalence Relation:
  1. Reflexive: $(a,a)\in R$.
  2. Symmetric: $(a,b)\in R⟹(b,a)\in R$.
  3. Transitive: $(a,b)\in R,(b,c)\in R⟹(a,c)\in R$.
• Function Types:
  • One-to-One: $f(x)=f(y)⟹x=y$.
  • Onto: Range = Codomain.
  • Bijective: One-to-One + Onto.

Week 2: Coordinate Geometry

• Distance: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.
• Section Formula: $P=\frac{mB+nA}{m+n}$.
• Midpoint: $M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$.
• Slope: $m=\frac{y_2-y_1}{x_2-x_1}=\tan \theta$.
• Lines:
  • $y=mx+c$ (Slope-Intercept).
  • $y-y_1=m(x-x_1)$ (Point-Slope).
• Parallel: $m_1=m_2$.
• Perpendicular: $m_1{m}_2=-1$.

Week 3: Quadratics

• Standard: $f(x)=a{x}^2+bx+c$.
• Vertex: $x_v=\frac{-b}{2a}$.
• Roots: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.
• Discriminant ($D$): $b^2-4ac$.
  • $D>0$: 2 Real Roots.
  • $D=0$: 1 Real Root.
  • $D<0$: No Real Roots.

Week 4: Polynomials

• Remainder Theorem: Remainder of $P(x)÷(x-c)$ is $P(c)$.
• Turning Points: Max $n-1$ turns for degree $n$.
• Multiplicity:
  • Odd: Crosses axis.
  • Even: Touches/Bounces.

Week 5: Functions

• Composite: $(f\circ g)(x)=f(g(x))$.
• Inverse: Swap $x$ and $y$, solve for $y$. Graph reflected over $y=x$.
• Even: $f(-x)=f(x)$ (Y-axis symmetry).
• Odd: $f(-x)=-f(x)$ (Origin symmetry).

Week 6: Logarithms

• Definition: ${\log }_{b}x=y⟺b^y=x$.
• Rules:
  • $\log (xy)=\log x+\log y$.
  • $\log (x/y)=\log x-\log y$.
  • $\log (x^k)=k\log x$.
• Change Base: ${\log }_{b}x=\frac{\ln x}{\ln b}$.

Week 7: Limits

• Rational Limit ($n\to \infty$):
  • Top < Bottom $\to$ 0.
  • Top > Bottom $\to$ $\infty$.
  • Equal $\to$ Ratio of coeffs.
• Continuity: LHL = RHL = $f(c)$.

Week 8: Derivatives

• Definition: $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$.
• Rules:
  • Power: $n{x}^{n-1}$.
  • Product: $f^{\prime }g+f{g}^{\prime }$.
  • Quotient: $\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$.
  • Chain: $f^{\prime }(g(x))\cdot g^{\prime }(x)$.
• Tangent Line: $y-f(a)=f^{\prime }(a)(x-a)$.