methods / calculus.mdon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (efa4da6)
   1# Calculus
   2
   3## Planner
   4
   51. 16A Recognising relationships and 16B Constant rate of change
   62. 16C Average rate of change and 16D Instantaneous rate of change
   73. 17F Limits and continuity
   84. 17A First principles
   95. 17B Rules for differentiation and 17C Negative integers
  106. 17D Graphs of derivatives
  117. 18A Tangents and normals
  128. 18B Rates of change
  139. 18C and 18D Stationary point
  1410. 18E Applications of Max and Min
  1511. Revision
  1612. Test
  17
  18
  19## Average rate of change
  20
  21$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
  22
  23Average rate of change between $x=[a,b]$ given two points $P(a, f(a))$ and $Q(b, f(b))$ is the gradient $m$ of line $\overleftrightarrow{PQ}$
  24
  25On CAS: (Action|Interactive) -> Calculation -> Diff -> $f(x)$ or $y=\dots$
  26
  27## Instantaneous rate of change
  28Tangent to a curve at a point - has same slope as graph at this point.
  29Values for $\Delta$ are always approximations.
  30
  31Secant - line passing through two points on a curve  
  32Chord - line segment joining two points on a curve
  33
  34Instantaneous rate of change is estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change.
  35
  36Each point $Q_n<P$ becomes closer to $Q_P$.
  37
  38## Limits and Continuity
  39
  40(see spec notes)
  41
  42## Position and velocity
  43
  44Position - location relative to a reference point  
  45Average velocity - average rate of change in position over time  
  46Instantaneous velocity - calculated the same way as averge $\Delta$
  47
  48## Derivatives
  49
  50**Derivative** denoted by $f^\prime(x)$:
  51
  52$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  53
  54**Tangent line** of function $f$ at point $M(a, f(a))$ is the line through $M$ with gradient $f^\prime(a)$.
  55
  56For $f(x)=x^n, \hspace{0.5em} f^\prime (x) = nx^{n-1}$
  57
  58## Tangents and gradients
  59
  60
  61### Tangent of a point
  62
  63For a point $P(q,r)$ on function $f$, the gradient of the tangent is the derivative $dy \over dx$ of $f(q)$. Therefore the tangent line is defined by $y=mx+c$ where $m={dy \over dx}$. Substitute $x=q, \hspace{0.5em} y=q$ to solve for $c$.
  64
  65### Normal
  66
  67Normal $\perp$ tangent.
  68
  69$$m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$$
  70
  71Normal line for point $P(q,r)$ on function $f$ is $y=mx+c$ where $m={-1 \over m_{\tan}}$. To find $c$, substitute $(x, y)=(q,r)$ and solve.
  72
  73### Solving on CAS
  74
  75**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line)  
  76**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
  77
  78## Stationary points
  79
  80Stationary where $m=0$.  
  81Find derivative, solve for ${dy \over dx} = 0$
  82
  83### Type of stationary points
  84
  85![](https://cdn.edjin.com/upload/RESOURCE/IMAGE/78444.png)
  86
  87**Local maximum at point $A$**  
  88- $f^\prime (x) > 0$ left of $A$
  89- $f^\prime (x) < 0$ right of $A$
  90
  91**Local minimum at point $B$**  
  92- $f^\prime (x) < 0$ left of $B$
  93- $f^\prime (x) > 0$ right of $B$
  94
  95**Stationary** point of inflection at $C$
  96