From ffab8a8c5db527394322b29a2295918570c05c56 Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sun, 12 Aug 2018 17:41:03 +1000
Subject: [PATCH] general additions

---
 methods/calculus.md |  4 ++++
 physics/waves.md    | 26 ++++++++++++++------------
 spec/calculus.md    | 25 ++++++++++++++++++++++---
 3 files changed, 40 insertions(+), 15 deletions(-)

diff --git a/methods/calculus.md b/methods/calculus.md
index 13888ad..2d029fb 100644
--- a/methods/calculus.md
+++ b/methods/calculus.md
@@ -31,6 +31,10 @@ Instantaneous rate of change is estimated by using two given points on each side
 
 Each point $Q_n<P$ becomes closer to $Q_P$.
 
+## Limits and Continuity
+
+(see spec notes)
+
 ## Position and velocity
 
 Position - location relative to a reference point
diff --git a/physics/waves.md b/physics/waves.md
index d0d795c..a218765 100644
--- a/physics/waves.md
+++ b/physics/waves.md
@@ -6,7 +6,7 @@
 - individual particles have little movement regardless of the distance of wave
 - **transfer of energy without net transfer of matter**
 
-**Nodes** - point of no motion (fixed point on graph)
+**Nodes** - point of no motion (fixed point on graph)  
 **Antinodes** - point of maximum motion (peaks)
 
 **Crests** (peaks) & **troughs** (azimuths)
@@ -25,11 +25,11 @@
 
 ### Measuring mechanical waves
 
-**Amplitude $A$** - max displacement from rest position (0)
-**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)
+**Amplitude $A$** - max displacement from rest position (0)  
+**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)  
 **Frequency $f$** - number of cycles (wavelengths) per second
 
-$T={1 \over f}\quad$(period: time for one cycle)
+$T={1 \over f}\quad$(period: time for one cycle)  
 $v=f \lambda \quad$(speed: displacement per second)
 
 ### Doppler effect
@@ -48,11 +48,12 @@ When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to trave
 ## Interference patterns
 
 When a medium changes character:
+
 - some energy is *reflected*
 - some energy is *absorbed* by new medium
 - some energy is *transmitted*
 
-**Superposition** - stimuli add together at a given point (vector addition)
+**Superposition** - stimuli add together at a given point (vector addition)  
 **Standing wave** - constructive interference at resonant frequency
 
 ### Reflection
@@ -67,9 +68,10 @@ Direction of motion of wave fronts can be shown by arrows, called *rays*, which
 ![](graphics/rays.png)
 
 Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
+
 - Normal: $\perp$ to wall
 - Incident wave front: $\perp$ to incident ray
-- Incident ray: $
+- Incident ray: $\theta_i$
 
 #### Transverse
 - sign of reflected transverse wave is inverted when endpoint is fixed in y-axis (equivalent to $180^\circ=\pi={\lambda \over 2}$ phase change)
@@ -81,11 +83,11 @@ Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
 **Overtone** - a multiple of the fundamental harmonic which produces the same no. of wavelengths at a different frequency (due to constructive interference)
 
 #### Wave has antinodes at both ends:
-$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)  
 $f = {nv \div 2l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
 
 #### Wave has antinode at one end:
-$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)  
 $f = {nv \div 4l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
 
 ## Light
@@ -126,7 +128,7 @@ $\therefore \theta_c = {n_2 \over n_1}$
 ### Double Slit
 
 ![](graphics/double-slit.png)
-**(a) Double slit as theorised by particle model** - "streams" of photons are concentrated in bright spots
+**(a) Double slit as theorised by particle model** - "streams" of photons are concentrated in bright spots  
 **(b) Double slit as theorised by wave model** - waves disperse onto screen (overlapping)
 
 Young's double slit experiment supports wave model:
@@ -141,15 +143,15 @@ Young's double slit experiment supports wave model:
 
 Path difference $pd = |S_1P-S_2P|$ for point $p$ on screen
 
-Constructive interference when $pd = n\lambda$ where $n \in [0, 1, 2, ...]$
+Constructive interference when $pd = n\lambda$ where $n \in [0, 1, 2, ...]$  
 Destructive interference when $pd = (n-{1 \over 2})\lambda$ where $n \in [1, 2, 3, ...]$
 
 Fringe separation:
 $$\Delta x = {{\lambda l }\over d}$$
 
 where
-$\Delta x$ is distance between fringes
-$l$ is distance from slits to screen
+$\Delta x$ is distance between fringes  
+$l$ is distance from slits to screen  
 $d$ is separation between sluts ($=S_1-S_2$)
 
 ## Electromagnetic waves
diff --git a/spec/calculus.md b/spec/calculus.md
index d1a981c..32ea661 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -8,7 +8,7 @@ $L^-$ - limit from below
 
 $L^+$ - limit from above
 
-$\lim_{x \to a} f(x)$ - limit of a point
+$\lim_{x \to a} f(x)$ - limit of a point  
 
 - Limit exists if $L^-=L^+$
 - If limit exists, point does not.
@@ -52,7 +52,7 @@ Can also be used with functions, where $h=\delta x$.
 
 ## First principles derivative
 
-$$\lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx} = f^\prime(x)$$
+$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
 
 $$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
 
@@ -61,11 +61,30 @@ $$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
 $$m_{\operatorname{chord PQ}}=f^\prime(x)$$
 
 first principles derivative:
-$${m_{\operatorname{tangent at P}} =\lim_{h \rigzhtarrow 0}}{{f(x+h)-f(x)}\over h}$$
+$${m_{\operatorname{tangent at P}} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
 
+## Gradient at a point
 
+Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$
 
 
+## Derivatives of $x^n$
+
+For $f: \mathbb{R} \rightarrow \mathbb{R}$ where $f(x)=x^n, x \in \mathbb{N}$
+
+Derivative is $f^\prime(x) = nx^{n-1}$
+
+If $x=$ constant, derivative is $0$
+
+If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$
+
+If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
+
+If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$
+
+$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
+$$=\lim_{h \rightarrow 0}
+
 ## Euler's number as a limit
 
 $$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
-- 
2.49.0