From 1b07812d144dd0fb7c3e54423a7200eab0135a54 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 11 Sep 2018 16:57:41 +1000 Subject: [PATCH] general additions --- chem/acids-bases.md | 9 ++++ english/persuasive.md | 0 methods/stuff.md | 12 +++-- spec/calculus-ref.md | 100 ++++++++++++++++++++++++++++++++++++++++++ spec/calculus-ref.pdf | Bin 0 -> 173477 bytes 5 files changed, 118 insertions(+), 3 deletions(-) create mode 100644 english/persuasive.md create mode 100644 spec/calculus-ref.md create mode 100644 spec/calculus-ref.pdf diff --git a/chem/acids-bases.md b/chem/acids-bases.md index 32a0d25..50c0919 100644 --- a/chem/acids-bases.md +++ b/chem/acids-bases.md @@ -81,3 +81,12 @@ $$[H_3O^+] \times [OH^-] = 10^{-14} \quad \text{(hydrolysis constant)}$$ Ionic bases dissolving in $H_2O$ - ionic compounds dissociate into constituent ions. Not ionised. e.g. $NaOH_{(s)}\stackrel{\mathrm{H_2O}}{\longrightarrow}Na^+_{(aq)}+OH^−_{(aq)}$ + +## pH values + +Acid/base/neutral not equivalent to pH (logarithmic scale) + +In water at 25C: $[H_3O^+] \times [OH^-]=10^{-7} \therefore \text{pH}=7$ + +$$\text{pH} = -\log[H_3O^+]=-\log[H^+]$$ +$$[H_3O^+]=10^{-\text{pH}}$$ diff --git a/english/persuasive.md b/english/persuasive.md new file mode 100644 index 0000000..e69de29 diff --git a/methods/stuff.md b/methods/stuff.md index 9335409..1eb560c 100644 --- a/methods/stuff.md +++ b/methods/stuff.md @@ -1,4 +1,6 @@ -# Index laws +# random methods shit + +## Index laws $a^m \times a^n = a^{m+n}$ $a^m \div a^n = a^{m-n}4$ @@ -6,6 +8,10 @@ $(a^m)^n = a^{_mn}$ $(ab)^m = a^m b^m$ ${({a \over b})}^m = {a^m \over b^m}$ -# Fractional indices +## Fractional indices + +$^n\sqrt{x}=x^{1/n}$ + +## Logarithms -$^n\sqrt{x}=x^{1/n}$ \ No newline at end of file +$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$ diff --git a/spec/calculus-ref.md b/spec/calculus-ref.md new file mode 100644 index 0000000..88ad371 --- /dev/null +++ b/spec/calculus-ref.md @@ -0,0 +1,100 @@ +--- +geometry: margin=2cm + +graphics: yes +tables: yes +author: Andrew Lorimer +--- + +# Spec - Calculus + +## Gradients + +$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$ + +## Limit theorems + +1. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$ +2. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$ +3. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$ +4. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$ + + +## First principles derivative + +$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$ + + +## Tangents & gradients + +**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$ +**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$) +**Secant** $={{f(x+h)-f(x)} \over h}$ + +## Derivatives + +| $f(x)$ | $f^\prime(x)$ | +| ------ | ------------- | +| $kx^n$ | $knx^{n-1}$ | +| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ | +| $c$ | $0$ | +| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ | +| $uv$ | $u{dv \over dx} + v{du \over dx}$ | +| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ | +| $\sin ax$ | $a\cos ax$ | +| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ | +| $\cos ax$ | $-a \sin ax$ | +| $e^{ax}$ | $ae^{ax}$ | +| $\log_e {ax}$ | $1 \over x$ | +| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ | + + + +## Product rule for $y=uv$ + +$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$ + +## Logarithms + +$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$ + +## Integration + +$$\int f(x) dx = F(x) + c$$ + +- area enclosed by curves + +| $f(x)$ | $\int f(x) \cdot dx$ | +| --------------- | ------------------ | +| $k$ (constant) | $kx + c$ | +| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ | +| $a x^{-n}$ | $a \cdot \log_e x + c$ | +| $e^{kx}$ | ${1 \over k} e^{kx} + c$ | +| $e^k$ | $e^kx + c$ | +| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ | +| $\cos kx$ | ${1 \over k} \sin (kx) + c$ | +| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ | +| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)| +| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ | +| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ | +| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ | + + +## Definite integrals + +$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$ + +## Kinematics + +**position $x$** - distance from origin or fixed point +**displacement $s$** - change in position from starting point (vector) +**velocity $v$** - change in position with respect to time +**acceleration $a$** - change in velocity +**speed** - magnitude of velocity + +| | no | +| - | -- | +| $v=u+at$ | $s$ | +| $s=ut + {1 \over 2} at^2$ | $v$ | +| $v^2 = u^2 + 2as$ | $t$ | +| $s= {1 \over 2}(u+v)t$ | $a$ | diff --git a/spec/calculus-ref.pdf b/spec/calculus-ref.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d2ede0431dbcba2933be77687bee634f16351467 GIT binary patch literal 173477 zcma&LL$D|e%wTzK+qP}nwr$(CZQHhO+qUtp)n8A|KbxLKXO&b|StNDpgj7L9jFyp( z6^e9WWpo3Ig^_@Pz~0CTiiZb^UdGhU+{J=`iJ6s+;Qt9IdNE5I7gHw!dNCVA7gG^a zV|x=*C_X+YXBQ_^Lt7}1tywK?d+l*Vzx8?!Uq}fQ^X@ljC}M#vEDHq`$=Y%*zXC}u 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