diff --git a/README.md b/README.md
index 0d4ce406a..2cd89fde2 100644
--- a/README.md
+++ b/README.md
@@ -6,17 +6,18 @@ APEX Calculus was originally [written in LaTeX](https://github.com/APEXCalculus/
To build the book, use of the [PreTeXt CLI](https://github.com/PreTeXtBook/pretext-cli) is recommended.
+Note that the build time is quite long, and may exceed the time allowed using GitHub codespaces.
+(If you're building on Windows, make sure your computer is not set to sleep after an hour!)
+
Preparation for generating APEX Calculus from source:
Software requirements:
- a recent LaTeX distribution
-- [Sage](https://www.sagemath.org/)
- Python (version 3.8 or later)
- the PreTeXt CLI (do `pip install pretextbook`)
-- [pdf2svg](https://github.com/jalios/pdf2svg-windows)
-- ImageMagick
-Note that Asymptote compilation is done remotely by default, so Asymptote does not need to be installed locally.
+**Note**: ImageMagick may be needed, but can probably be avoided.
+Older versions of the PreTeXt CLI also require `pdf2svg` to build HTML, but soon this will not be required.
Configuration:
@@ -37,13 +38,7 @@ The following variations are also available:
- `pretext build latex-color-print`: in color, but two-sided layout for printing
- `pretext build latex-color-print-novideo`: as above, but without QR codes for videos
-**Do not build PDF directly.** APEX Calculus places figures in the margins in PDF.
-Unfortunately, we do not currently have a mechanism for setting the vertical placement
-of margin figures in the PreTeXt source, so these have to be adjusted by hand.
-Look in the LaTeX source for occurrences of `\listmarginbox`, `\parmarginbox` and `\tcbmarginbox`.
-Each such environment ends with either `{0cm}` or `{-1cm}`.
-
-These values can be adjusted to ensure that items in the margins do not overlap.
-It is unfortunately a rather tedious process at the moment.
+**Do not build PDF directly.** Most of the PDF design is now automated,
+but you may need to adjust the placement of some figures, add page breaks, etc.
At this time, EPUB generation is not fully supported.
diff --git a/ptx/sec_Graphical_Numerical.ptx b/ptx/sec_Graphical_Numerical.ptx
index e1f7084ee..191ee338f 100644
--- a/ptx/sec_Graphical_Numerical.ptx
+++ b/ptx/sec_Graphical_Numerical.ptx
@@ -918,7 +918,7 @@
and then uses the above idea to find approximate values for the solution y at later x-values.
The algorithm is summarized in .
-
+ Euler's Method
Consider the initial value problem
@@ -1864,7 +1864,7 @@
-
+
Graph showing slope field for the given differential equation.
@@ -1917,7 +1917,7 @@
-
+
Graph showing slope field for the given differential equation.
@@ -1972,7 +1972,7 @@
-
+
Graph showing slope field for the given differential equation.
@@ -2020,7 +2020,7 @@
-
+
Graph showing slope field for the given differential equation.
@@ -2183,7 +2183,7 @@
-
+
In the following exercises,
diff --git a/ptx/sec_antider.ptx b/ptx/sec_antider.ptx
index d3b3665ec..ceb58e9fc 100644
--- a/ptx/sec_antider.ptx
+++ b/ptx/sec_antider.ptx
@@ -365,7 +365,7 @@
as we learn.
diff --git a/ptx/sec_cylindrical_spherical.ptx b/ptx/sec_cylindrical_spherical.ptx
index 0081cb9a5..4b5dfd54c 100644
--- a/ptx/sec_cylindrical_spherical.ptx
+++ b/ptx/sec_cylindrical_spherical.ptx
@@ -789,7 +789,7 @@
The following Key Idea gives conversions to/from our three spatial coordinate systems.
-
+ Converting Between Rectangular, Cylindrical and Spherical Coordinates
diff --git a/ptx/sec_hessian.ptx b/ptx/sec_hessian.ptx
index 0bdd2dad3..35e4c9a76 100644
--- a/ptx/sec_hessian.ptx
+++ b/ptx/sec_hessian.ptx
@@ -163,7 +163,7 @@
You've probably already noticed a problem with talking about higher-order polynomials in several variables:
the notation gets really messy, since there are so many more possible terms!
For example, even a relatively simple case like a degree 3 polynomial in 3 variables looks like
-
+ P(x,y,z) \amp = a+bx+cy+dz+ex^2+fxy+gxz+hy^2+kyz+lz^2\amp \quad\quad\quad\quad +mx^3+nx^2y+oxy^2+pxyz+qx^2z+rxz^2+sy^3+ty^2z+uyz^2+vz^3
diff --git a/ptx/sec_hyperbolic.ptx b/ptx/sec_hyperbolic.ptx
index 1c5c92d2d..d4b1d9789 100644
--- a/ptx/sec_hyperbolic.ptx
+++ b/ptx/sec_hyperbolic.ptx
@@ -765,7 +765,7 @@
-
+
Domains and ranges of the hyperbolic and inverse hyperbolic functions
diff --git a/ptx/sec_polar.ptx b/ptx/sec_polar.ptx
index 9127e9ec9..386d588a4 100644
--- a/ptx/sec_polar.ptx
+++ b/ptx/sec_polar.ptx
@@ -1227,7 +1227,7 @@
and to investigate with technology other types of polar functions.
-
+
+ Note that for (1+x)^k, the interval of convergence may contain one or both endpoints,
+ depending on the value of k, and we are using the generalized binomial coefficients
+
+ \binom{k}{n} = \frac{k(k-1)\cdots (k-(n-1))}{n!}
+ .
+ Taylor Seriescommon series
+