From 9d6c5c6f10335aaed3b5df1c4cf1300e726bc59b Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 29 Apr 2019 09:39:10 +1000 Subject: [PATCH] [methods] strictly increasing and tangents --- methods/calculus-ref.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) 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) -- 2.43.2