From ad9657230a5cfc5b07e52463616421526bc73083 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 12 May 2019 21:59:39 +1000 Subject: [PATCH] [methods] add diagrams to calculus notes --- methods/calculus-ref.md | 28 ++++++++++++------------- methods/exact-values-1.png | Bin 5475 -> 0 bytes methods/exact-values-2.png | Bin 6260 -> 0 bytes methods/graphics/exact-values-1.png | Bin 8111 -> 5475 bytes methods/graphics/exact-values-2.png | Bin 6445 -> 6260 bytes methods/graphics/stationary-points.png | Bin 0 -> 20998 bytes 6 files changed, 13 insertions(+), 15 deletions(-) delete mode 100644 methods/exact-values-1.png delete mode 100644 methods/exact-values-2.png create mode 100644 methods/graphics/stationary-points.png 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 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 $m=0$. 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