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author: Andrew Lorimer
$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
-Average 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 $\overleftrightarrow{PQ}$
-
-On CAS: (Action|Interactive) -> Calculation -> Diff -> $f(x)$ or $y=\dots$
+On CAS: (Action|Interactive) $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ $f(x)$ or $y=\dots$
## Instantaneous rate of change
-Secant - line passing through two points on a curve
-Chord - line segment joining two points on a curve
+**Secant** - line passing through two points on a curve
+**Chord** - line segment joining two points on a curve
-Estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change.
+Estimated by using two given points on each side of the concerned point.
## Limits & continuity
## Tangents & gradients
**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$
-**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$)
+**Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)
**Secant** $={{f(x+h)-f(x)} \over h}$
-$$\tan \Theta = m = f^\prime x$$
+$$\tan \theta = m = f^\prime (x)$$
-where $\Theta$ is the angle that tangent line makes with +ve direction of $x$-axis
+where $\theta$ is the angle that tangent line makes with +ve direction of $x$-axis
## Strictly increasing
-- Function $f$ is **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
-- Function $f$ is **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
+- $f$ is **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
+- $f$ is **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
- If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing**
- If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing**
- Endpoints are included, even where gradient $=0$
### Solving on CAS
-**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line)
-**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
+**In main**: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)
+**In graph**: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
## Stationary points
Stationary where $m=0$.
Find derivative, solve for ${dy \over dx} = 0$
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**Local maximum at point $A$**
- $f^\prime (x) > 0$ left of $A$
