From 7090965ce342a888e3f5d975836917ce5090a97f Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Thu, 23 Aug 2018 21:23:35 +1000 Subject: [PATCH] calculus (specialist & methods) --- methods/calculus.md | 15 +++++++++++++++ spec/calculus.md | 16 ++++++++++++---- 2 files changed, 27 insertions(+), 4 deletions(-) diff --git a/methods/calculus.md b/methods/calculus.md index fb3761f..c28a498 100644 --- a/methods/calculus.md +++ b/methods/calculus.md @@ -77,3 +77,18 @@ Normal line for point $P(q,r)$ on function $f$ is $y=mx+c$ where $m={-1 \over m_ Stationary where $m=0$. Find derivative, solve for ${dy \over dx} = 0$ + +### Type of stationary points + +![](https://cdn.edjin.com/upload/RESOURCE/IMAGE/78444.png) + +**Local maximum at point $A$** +- $f^\prime (x) > 0$ left of $A$ +- $f^\prime (x) < 0$ right of $A$ + +**Local minimum at point $B$** +- $f^\prime (x) < 0$ left of $B$ +- $f^\prime (x) > 0$ right of $B$ + +**Stationary** point of inflection at $C$ + diff --git a/spec/calculus.md b/spec/calculus.md index 8476412..f98aee7 100644 --- a/spec/calculus.md +++ b/spec/calculus.md @@ -189,8 +189,16 @@ $$\int xn = {x^{n+1} \over n+1} + c$$ $\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$ $\int k f(x) dx = k \int f(x) dx$ -| $f(x)$ | $\int f(x) \cdot dx$ | -| ------ | -------------------- | -| $k$ (constant) | $kc + c$ | -| $x^n (n \in J\\\{-1\})$ | ${1 \over {n+1}}x^{n+1} + c$ | +| $f(x)$ | $\int f(x) \cdot dx$ | +| ------------------------------- | ---------------------------- | +| $k$ (constant) | $kc + c$ | +| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ | +| $1 \over x$ | $\log_e x + c$ | +| $e^kx$ | ${1 \over k} e^{kx} + c$ | +| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ | +| $\cos kx$ | ${1 \over k} \sin (kx) + c$ | +| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ | +| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)| +| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ | + -- 2.43.2