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 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