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.