methods / calculus.mdon commit tangents & normals, light-matter interactions (08a48a5)
   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
  21Average 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 $\overleftrightarrow{PQ}$
  22
  23## Instantaneous rate of change
  24Tangent to a curve at a point - has same slope as graph at this point.
  25Values for $\Delta$ are always approximations.
  26
  27Secant - line passing through two points on a curve  
  28Chord - line segment joining two points on a curve
  29
  30Instantaneous rate of change is estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change.
  31
  32Each point $Q_n<P$ becomes closer to $Q_P$.
  33
  34## Limits and Continuity
  35
  36(see spec notes)
  37
  38## Position and velocity
  39
  40Position - location relative to a reference point  
  41Average velocity - average rate of change in position over time  
  42Instantaneous velocity - calculated the same way as averge $\Delta$
  43
  44## Derivatives
  45
  46**Derivative** denoted by $f^\prime(x)$:
  47
  48$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  49
  50**Tangent line** of function $f$ at point $M(a, f(a))$ is the line through $M$ with gradient $f^\prime(a)$.
  51
  52## Tangents and gradients
  53
  54
  55### Tangent of a point
  56
  57For 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 $c$.
  58
  59### Normal
  60
  61Normal $\perp$ tangent.
  62
  63$$m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$$
  64
  65Normal 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.
  66
  67### Solving on CAS
  68
  69**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line)  
  70**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.