From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sun, 12 May 2019 11:59:39 +0000 (+1000)
Subject: [methods] add diagrams to calculus notes
X-Git-Tag: yr12~142
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/ad9657230a5cfc5b07e52463616421526bc73083

[methods] add diagrams to calculus notes
---

diff --git a/methods/calculus-ref.md b/methods/calculus-ref.md
index 2239ed2..0ec4f7b 100644
--- a/methods/calculus-ref.md
+++ b/methods/calculus-ref.md
@@ -1,6 +1,6 @@
 ---
 geometry: margin=2cm
-<!-- columns: 2 -->
+columns: 2
 graphics: yes
 tables: yes
 author: Andrew Lorimer
@@ -16,16 +16,14 @@ 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
 
@@ -51,32 +49,32 @@ Not differentiable at:
 ## 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 ${dy \over dx} = 0$
 
-![](https://cdn.edjin.com/upload/RESOURCE/IMAGE/78444.png){#id .class width=20%}
+![](graphics/stationary-points.png){#id .class width=50%}
 
 **Local maximum at point $A$**  
 - $f^\prime (x) > 0$ left of $A$
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