From: Andrew Lorimer Date: Sat, 18 Aug 2018 06:55:14 +0000 (+1000) Subject: logarithmic derivatives X-Git-Tag: yr11~65 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/e8e9140bc7e4ad3cfdcbc9e0d0877ee9095b226e?hp=0a209ebb258b05cf52b57e5d8bcf62af0c05e1b5 logarithmic derivatives --- diff --git a/methods/calculus.md b/methods/calculus.md index fb19fdb..589bf2f 100644 --- a/methods/calculus.md +++ b/methods/calculus.md @@ -48,3 +48,23 @@ Instantaneous velocity - calculated the same way as averge $\Delta$ $$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$ **Tangent line** of function $f$ at point $M(a, f(a))$ is the line through $M$ with gradient $f^\prime(a)$. + +## Tangents and gradients + + +### Tangent of a point + +For a point $P(q,r)$ on function $f$, the gradient of the tangent is the derivative $dy \over dx$ of $f(q)$. Therefore the tangent line is defined by $y=mx+c$ where $m={dy \over dx}$. Substitute $x=q, \hspace{0.5em} y=q$ to solve for $c$. + +### Normal + +Normal $\perp$ tangent. + +$$m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$$ + +Normal line for point $P(q,r)$ on function $f$ is $y=mx+c$ where $m={-1 \over m_{\tan}}$. To find $c$, substitute $(x, y)=(q,r)$ and solve. + +### Solving on CAS + +**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line) +**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line. diff --git a/physics/light-matter.md b/physics/light-matter.md new file mode 100644 index 0000000..04a90a4 --- /dev/null +++ b/physics/light-matter.md @@ -0,0 +1,28 @@ +# Light and matter + +## Photoelectric effect + +### Planck's equation + +$$E=hf$$ + +where +$E$ is energy of a quantum of light (J) +$f$ is frequency of EM radiation +$h$ is Planck's constant ($6.63 \times 10^{-34}\operatorname{J s}$) + + +### Electron diffraction patterns + +$$W=qV$$ + +(work for accelerating electon of charge $q$ with voltage $V$) + +$$\lambda = {h \over mv}$$ + +(de Broglie's equation) + +Solving wavelength of electrons from gun: +1. + +774 abc melbourne \ No newline at end of file diff --git a/spec/calculus.md b/spec/calculus.md index 1f3d404..2b68bf5 100644 --- a/spec/calculus.md +++ b/spec/calculus.md @@ -93,7 +93,7 @@ where $u$ and $v$ are functions of $x$ $$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$ -## Chain rule +## Chain rule for $(f\circ g)$ $$(f \circ g)^\prime = (f^\prime \circ g) \cdot g^\prime$$ @@ -151,4 +151,16 @@ $$\ln x = \log_e x$$ ### Differentiating logarithms $${d \over dx} \log_b x = {1 \over x \ln b}$$ +## Solving $e^x$ + +| $f(x)$ | $f^\prime(x)$ | +| ------ | ------------- | +| $\sin x$ | $\cos x$ | +| $\sin ax$ | $a\cos ax$ | +| $\cos x$ | $-\sin x$ | +| $\cos ax$ | $-a \sin ax$ | +| $e^x$ | $e^x$ | +| $e^{ax}$ | $ae^{ax}$ | +| $\log_e x$ | $1 \over x$ | +| $\log_e {ax}$ | $1 \over x$ |