methods / summary.mdon commit [spec] minor clarifications to calculus section of sac notes (065aae3)
   1# Methods - Semester 1
   2
   3## Simulatenous equations
   4Methods of solving:
   5- substitution (state one variable in terms of the other)
   6- subtraction (subtract one equation from the other, substitute resulting equation into the other)
   7
   8## Linear inequatlities
   9
  10- Flip operator when multiplying / dividing by <0
  11
  12## Coordinate geometry
  13
  14Regarding points $(x_1,y_1)(x_2y_2)$:
  15
  16**Midpoint:** $m=({{\Sigma x}\over 2}, {{\Sigma y}\over 2})$
  17
  18**Distance:** $d=\sqrt{(\Delta x)^2+(\Delta y)^2}$
  19
  20**Gradient:** $m={\operatorname{rise}\over\operatorname{run}}={\Delta y \over \Delta x}$
  21
  22**Line through points:** $y={\Delta y \over \Delta x}(x+x_2)-y_2$
  23
  24- parallel lines: $m_1=m_2$
  25- perpendicular lines: $m_1m_2=-1$
  26
  27## Polynomials
  28
  29**Binomial expansion:** $(a+b)(c+d) = a(c+d) + b(c+d)$
  30**Cubic expansion:** $(a+b)^3=a^3+3a^2b+3ab^2+b^3$
  31**Quartic expansion:** $(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$
  32
  33**Perfect square expansion:** $(a+b)^2=a^2+2ab+b^2$
  34**Difference of perfect squares:** $a^2-b^2=(a+b)(a-b)$
  35
  36**Factorising quadratics:** $x^2+bx+c = (x+e)(x-f)$
  37where $e \times f = c, \quad e+f = b$
  38
  39**Remainder theorem:** if ${P({-\alpha \over \beta})}=0,$ then ${\beta x+ \alpha}$ is a factor of $P(x)$
  40
  41**Factor theorem:** if $P(\alpha)=0,$ then $x-\alpha$ is a factor of $P(x)$
  42
  43**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$
  44**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$
  45
  46**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$
  47**Discriminant:** $\Delta=b^2-4ac$
  48
  49**Solving** $y_1=ax^2+bx+c_1, \quad y_2=mx+c_2$:
  50- $ax^2+(b-m)x+(c_1-c_2)=0$
  51- Solve for $x$, substitute into $y_2$
  52
  53**Axis of symmetry:** $x={-b\over2a}$ (when written as $y=ax^2+bx+c$)
  54
  55**Determining quadratic rules:**
  56$(1)\quad y=a(x-e)(x-f)$ &nbsp;two $x$-intercepts, one point
  57$(2)\quad y=a(x-h)^2+k$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;turning point, one point
  58$(3)\quad y=ax^2+bx+c$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;three points
  59
  60## Graphs
  61
  62In general:
  63- turning point / centre point at $(h,k)$
  64- asymptotes at $x=h$ and $y=k$
  65- $a+$ dilates graph away from centre point
  66- $a<0$ reflects graph across $x=0$
  67- $h$ is the horizontal ($x$) shift, $k$ is the vertical ($y$) shift
  68
  69**Rectangular hyperbola:** $\quad y={a\over x-h}+k$
  70- as $x\rightarrow \pm\infty, \quad y \rightarrow 0^\pm$
  71- as $x \rightarrow 0^{\pm}, \quad y \rightarrow \pm \infty$
  72
  73**Truncus:**$\quad y={a\over (x-h)^2}+k$
  74- as $x \rightarrow \pm \infty, \quad y \rightarrow 0^+$
  75- as $x \rightarrow 0^{\pm}, \quad y \rightarrow \infty$
  76
  77**Square root:**$\quad y=a\sqrt{x-h}+k$
  78- parabola rotated 90 degrees
  79
  80**Square root negative:**$\quad y=a\sqrt{-(x-h)}+k$
  81- reflection of $y=\sqrt{x}$ across $y$-axis
  82
  83**Circle:**$\quad (x-h)^2+(y-x)^2=r^2$
  84- factorised: $x^2+y^2-2hx-2ky+c=0$
  85- or: $y=\pm\sqrt{r^2-x^2}$
  86
  87**Semicircles** - take +ve or -ve square root
  88- $y=\pm\sqrt{r^2-x^2}\quad$(top or bottom)
  89- $y=\pm\sqrt{r^2-y^2}\quad$(left or right)
  90
  91**Cubic:** $\quad f(x)=a(x-h)^3+k$
  92- all cubics have >1 root
  93
  94**Inverse cubic:** $\quad f^{-1}(x)=x^{1\over 3}$
  95- inverse of $\quad f(x)=a(x-h)^3+k$
  96
  97**Quartic:** $\quad f(x)=a(x-h)^4+k$
  98
  99
 100## Set notation
 101
 102- set difference: $\quad A \setminus B=\{x:x\in A, x\notin B\}$
 103- interval notation: $(a,b)=\{x : a \lt x \lt b\},\quad$[a,b]=\{x : a \lte x \lte b\}$
 104
 105## Functions
 106$$f:\operatorname{dom}(f) \rightarrow \mathbb{R},\quad f(x)=\dots$$
 107- function - one $y$ (image) value per $x$ (preimage)
 108- 1:1 function - unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is)
 109
 110**Domain $\operatorname{dom}(f)$:** set of all $x$ values in function
 111- maximal (implied) domain - largest domain for which the rule is defined
 112- restricted domain: $f(x)=\dots,\quad\operatorname{dom}(f)$
 113
 114**Range $\operatorname{ran}(f)$:** set of all $y$ values in function
 115
 116**Piecewise functions:** each domain has a corresponding equation
 117
 118**Inverse functions:** $f^-1(x)=y$ if $f(y)=x,\quad$ for $x\in \operatorname{ran}(f), y\in \operatorname{dom}(f)$
 119
 120**Methods of factorising cubics:**
 121- extract common factor
 122- factor theorem
 123- polynomial division or equating coefficients
 124- sum or difference of two cubes
 125- quadratic formula
 126
 127**Polynomials:** $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$
 128- degree ($n$) of $P(x)$ is the highest power of $x$
 129
 130## Bisection method
 131If $f(x)=0$ has a solution $\alpha$ between $[a_1,b_1]$:
 132- $f(a_1)\lt 0,\quad f(b_1)>0$
 133- calculate $f(c_1)=f({{a_1+b_1}\over 2})$
 134- if $f(c_1)\lt 0$, a root lies between $c_1$ and $b_1$
 135- if $f(c_1)\gt 0$, a root lies between $a_1$ and $c_1$
 136
 137## Matrices
 138- addition is only defined when dimensions are equal
 139- multiplication is only defined when columns in first = rows in second
 140- identity - equal to one
 141- inverse: $A^{-1}={1\over{ad-bc}}\times A\prime$
 142- determinant: $\det(A)=ad-bc$
 143
 144## Probability
 145- mutually exclusive: $\Pr(A\cap B)=0$
 146- independent: $\Pr(A|B)=\Pr(A)$
 147- addition rule: $\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)$
 148- multiplication rule: $\Pr(A\cap B)=\Pr(A|B)\times \Pr(B)$
 149- law of total probability: $\Pr(A)=\Pr(A|B)\Pr(B)+\Pr(A|B\prime)\Pr(B\prime)$
 150- conditional probability: $\Pr(A|B)={{\Pr(A\cap B)\over\Pr(B)}}$
 151
 152## Combinatorics
 153
 154- Arrangements of $n$ in $r$ is given by $n!\over{(n-r)!}$
 155- Combinations of $n$ in $r$ is given by $n! \over{r!(n-r)!}$