[spec] formatting for calculus rules

diff --git a/spec/calculus-rules.tex b/spec/calculus-rules.tex
index ec3372640c264410ddebc10a532a561902233b36..2b6b712b0e5b2c49d47f40949525c35558e3dfa8 100644 (file)
--- a/spec/calculus-rules.tex
+++ b/spec/calculus-rules.tex
@@ -30,14 +30,17 @@
 \end{tabularx}
 
 \vfill
 \end{tabularx}
 
 \vfill

-
-\subsubsection*{Index identities}
-
-\(b^{m+n}=b^m \cdot b^n\)\\
-\((b^m)^n=b^{m \cdot n}\)\\
-\((b \cdot c)^n = b^n \cdot c^n\)\\
-\({a^m \div a^n} = {a^{m-n}}\)
-
+\vtop to 5cm {
+  \flushbottom
+  \subsubsection*{Index identities}
+  \begin{align*}
+    b^{m+n} &= b^m \cdot b^n \\
+    (b^m)^n &= b^{m \cdot n} \\
+    (b \cdot c)^n  &=  b^n \cdot c^n \\
+    {a^m \div a^n}  &=  {a^{m-n}}
+  \end{align*}
+}
+  

 
 \subsection*{Antiderivatives}
 
 
 \subsection*{Antiderivatives}
 


@@ -73,9 +76,13 @@
 Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\)
 
 \vfill
 Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\)
 
 \vfill

-
-\subsubsection*{Logarithmic identities}
-
-\(\log_b (xy)=\log_b x + \log_b y\)\\
-\(\log_b x^n = n \log_b x\)\\
-\(\log_b y^{x^n} = x^n \log_b y\)
+\vtop to 5cm {
+  \flushbottom
+  \subsubsection*{Logarithmic identities}
+  \begin{align*}
+    \log_b (xy) &= \log_b x + \log_b y \\
+    \log_b\left(\frac{x}{y}\right) &= \log_b(x) - \log_b(y) \\
+    \log_b x^n &= n \log_b x \\
+    \log_b y^{x^n} &= x^n \log_b y
+  \end{align*}
+}

diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index d66ece975983a2cf40ff524d6d65e934a4a16a08..225cc8c01920230b2558af383d7d7304c9612027 100644 (file)
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -23,6 +23,7 @@
 \usepackage{pgfplots}
 \usepackage{pst-plot}
 \usepackage{rotating}
 \usepackage{pgfplots}
 \usepackage{pst-plot}
 \usepackage{rotating}

+%\usepackage{showframe} % debugging only

 \usepackage{subfiles}
 \usepackage{tabularx}
 \usepackage{tcolorbox}
 \usepackage{subfiles}
 \usepackage{tabularx}
 \usepackage{tcolorbox}