From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sun, 28 Apr 2019 23:39:10 +0000 (+1000)
Subject: [methods] strictly increasing and tangents
X-Git-Tag: yr12~153
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/9d6c5c6f10335aaed3b5df1c4cf1300e726bc59b

[methods] strictly increasing and tangents
---

diff --git a/methods/calculus-ref.md b/methods/calculus-ref.md
index dab682a..2239ed2 100644
--- a/methods/calculus-ref.md
+++ b/methods/calculus-ref.md
@@ -54,6 +54,18 @@ Not differentiable at:
 **Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$)  
 **Secant** $={{f(x+h)-f(x)} \over h}$
 
+$$\tan \Theta = m = f^\prime x$$
+
+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 $x_2 > x_1$
+- 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)