From: Andrew Lorimer Date: Sun, 2 Jun 2019 11:23:27 +0000 (+1000) Subject: [spec] minor clarifications to calculus section of sac notes X-Git-Tag: yr12~114 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/065aae322d2a498a0add41d139d053d89cf0b709 [spec] minor clarifications to calculus section of sac notes --- diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex index 39ab74e..f28326a 100644 --- a/spec/spec-collated.tex +++ b/spec/spec-collated.tex @@ -873,7 +873,7 @@ \rowcolor{shade2} & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\ \hline - \(\frac{dy}{dx}>0\) & + \(\dfrac{dy}{dx}>0\) & \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}& \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0.1, xmax=4, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))}; \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}& \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1.5, xmax=1.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\ @@ -1090,7 +1090,7 @@ \subsubsection*{Gradient at a point on parametric curve} - \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\] + \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\] \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]