geometric proofs with vectors

diff --git a/spec/calculus-ref.md b/spec/calculus-ref.md
index 88ad3717fc09e5cc5f2e7e3d2dd1329502e2abc6..eba250f95774c4f2687d9794e7f722b466ff6521 100644 (file)
--- a/spec/calculus-ref.md
+++ b/spec/calculus-ref.md
@@ -4,8 +4,11 @@ geometry: margin=2cm
 graphics: yes
 tables: yes
 author: Andrew Lorimer
+classoption: twocolumn
+
 ---
 
+<!-- \renewcommand{\arraystretch}{2} -->
 # Spec - Calculus
 
 ## Gradients
@@ -33,11 +36,13 @@ $$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
 
 ## Derivatives
 
+
+
 | $f(x)$ | $f^\prime(x)$ |
-| ------ | ------------- |
+| --- | --- |
 | $kx^n$ | $knx^{n-1}$ |
 | $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
-| $c$    | $0$ |
+| $c$ | $0$ |
 | ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
 | $uv$ | $u{dv \over dx} + v{du \over dx}$ |
 | $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
@@ -50,6 +55,8 @@ $$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
 
 
 
+
+
 ## Product rule for $y=uv$
 
 $${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
@@ -65,7 +72,7 @@ $$\int f(x) dx = F(x) + c$$
 - area enclosed by curves
 
 | $f(x)$ | $\int f(x) \cdot dx$ |
-| --------------- | ------------------ |
+| ----|--- |
 | $k$ (constant) | $kx + c$ |
 | $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
 | $a x^{-n}$ | $a \cdot \log_e x + c$ |
@@ -92,9 +99,12 @@ $$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$
 **acceleration $a$** - change in velocity  
 **speed** - magnitude of velocity  
 
+
+
 | | no |
 | - | -- |
 | $v=u+at$ | $s$ |
 | $s=ut + {1 \over 2} at^2$ | $v$ |
 | $v^2 = u^2 + 2as$ | $t$ |
 | $s= {1 \over 2}(u+v)t$ | $a$ |
+