diff --git a/docs/introduction.tex b/docs/introduction.tex
index 7151b12..0296838 100644
--- a/docs/introduction.tex
+++ b/docs/introduction.tex
@@ -20,7 +20,7 @@
 
 Fredrik Schaffalitzky when he was at Google started the development of Ceres, and even though much has changed since then, many of the ideas from his original design are still present in the current code.
 
-Amongst Ceres' users at Google two deserve special mention: William Rucklidge and James Roseborough. William was the first user of Ceres. He bravely took on the task of porting production code to an as-yet unproven optimization library, reporting bugs and helping fix them along the way. James is perhaps the most sophisticated used of Ceres at Google. He has reported and fixed bugs and helped evolve the API for the better.
+Amongst Ceres' users at Google two deserve special mention: William Rucklidge and James Roseborough. William was the first user of Ceres. He bravely took on the task of porting production code to an as-yet unproven optimization library, reporting bugs and helping fix them along the way. James is perhaps the most sophisticated user of Ceres at Google. He has reported and fixed bugs and helped evolve the API for the better.
 
 Nathan Wiegand contributed the MacOS port.
 \clearpage
diff --git a/docs/tutorial.tex b/docs/tutorial.tex
index da9c29e..fad4ea7 100644
--- a/docs/tutorial.tex
+++ b/docs/tutorial.tex
@@ -17,11 +17,11 @@
 
 Perhaps the simplest example of such a problem is the problem of Ordinary Linear Regression, where given observations $(x_1,y_1),\hdots, (x_k,y_k)$, we wish to find the line $y = mx + c$, that best explains $y$ as a function of $x$. One way to solve this problem is to find the solution to the following optimization problem
 \begin{equation}
-		\arg\min_{m,c} \sum_{i=1}^k (y_i - m x_i - c)^2
+		\arg\min_{m,c} \sum_{i=1}^k (y_i - m x_i - c)^2.
 \end{equation}
 With a little bit of calculus, this problem can be solved easily by hand. But what if, instead of a line we were interested in a more complicated relationship between $x$ and $y$, say for example $y = e^{mx + c}$. Then the optimization problem becomes
 \begin{equation}
-		\arg\min_{m,c} \sum_{i=1}^k \left(y_i - e^{m x_i + c}\right)^2
+		\arg\min_{m,c} \sum_{i=1}^k \left(y_i - e^{m x_i + c}\right)^2.
 \end{equation}
 This is a  non-linear regression problem and solving it by hand is much more tedious.  Ceres is designed to help you model and solve problems like this easily and efficiently.
 
@@ -236,7 +236,7 @@
 \label{sec:tutorial:datafitting}
 The examples we have seen until now are simple optimization problems with no data. The original purpose of least squares and non-linear least squares analysis was fitting curves to data. It is only appropriate that we now consider an example of such a problem. Let us fit some data to the curve
 \begin{equation}
-	y = e^{mx + c}
+	y = e^{mx + c}.
 \end{equation}
 
 The full code and data for this example can be found in