[spec] minor clarifications to calculus section of sac notes

diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index 39ab74ecb586908cb18c2851ac08382cc756a136..f28326aa90558aaa021f64b22b3849ec46640ca7 100644 (file)
--- 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
                       \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}\\
                       \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}
 
 
                   \subsubsection*{Gradient at a point on parametric curve}
 

-                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\]
+                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]

 
                   \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
 
 
                   \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]