From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Tue, 28 Aug 2018 01:26:45 +0000 (+1000)
Subject: antidiff applications & photoelectric graphics
X-Git-Tag: yr11~56
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/0101cfe0300704ddcf5cbcac347465d482eae7ff

antidiff applications & photoelectric graphics
---

diff --git a/physics/graphics/energy-levels.png b/physics/graphics/energy-levels.png
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diff --git a/physics/graphics/photoelectric-effect.png b/physics/graphics/photoelectric-effect.png
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diff --git a/physics/graphics/standing-wave-electrons.png b/physics/graphics/standing-wave-electrons.png
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diff --git a/physics/graphics/synchrotron.png b/physics/graphics/synchrotron.png
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diff --git a/physics/light-matter.md b/physics/light-matter.md
index 19c50c3..c14d998 100644
--- a/physics/light-matter.md
+++ b/physics/light-matter.md
@@ -2,6 +2,8 @@
 
 ## Photoelectric effect
 
+![](graphics/photoelectric-effect.png)
+
 ### Planck's equation
 
 $$E=hf,\quad f={c \over \lambda}$$
@@ -90,6 +92,8 @@ $$\lambda = {h \over \rho} = {h \over mv}$$
 - therefore, stable orbits are those where circumference = whole number of e- wavelengths
 - if $2\pi r \ne n{h \over mv}$, interference occurs when pattern is looped and standing wave cannot be established
 
+![](graphics/standing-wave-electrons.png)
+
 ### Photon momentum
 
 $$\rho = {hf \over c} = {h \over \lambda}$$
@@ -104,6 +108,9 @@ $$\rho = {hf \over c} = {h \over \lambda}$$
 - Black lines in spectrum show light not reflected  
 
 ### Emission
+
+![](graphics/energy-levels.png)
+
 - Coloured lines show light being ejected from e- shells  
 - Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$  
 - Bohr's model describes discrete energy levels
@@ -112,6 +119,9 @@ $$\rho = {hf \over c} = {h \over \lambda}$$
 - EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
 
 ## Light sources
+
+![](graphics/synchrotron.png)
+
 - **incandescent:** <10% efficient, broad spectrum
 - **LED:** semiconducting doped-Si diodes
 - - most electrons in *valence band* (energy level)
diff --git a/spec/calculus.md b/spec/calculus.md
index 08cfe8f..f68b829 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -194,7 +194,7 @@ $\int k f(x) dx = k \int f(x) dx$
 | ------------------------------- | ---------------------------- |
 | $k$ (constant)                  | $kx + c$                     |
 | $x^n$         | ${1 \over {n+1}}x^{n+1} + c$ |
-| $a \cdot {1 \over x}$ | $a \cdot \log_e x + c$ |
+| $a x^{-n}$ | $a \cdot \log_e x + c$ |
 | $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
 | $e^k$ | $e^kx + c$ |
 | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
@@ -202,5 +202,15 @@ $\int k f(x) dx = k \int f(x) dx$
 | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 | $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
 | $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+
+## Applications of antidifferentiation
+
+- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
+- the nature of any stationary point of $y=F(x)$ is determined by the way the sign of the graph of $y=f(x)$ changes about its $x$-intercepts
+- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$
+
+To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.