edits

quarto/integrals/mean_value_theorem.qmd

@@ -144,10 +144,10 @@ $$
 So in particular $K$ is in $[m, M]$. But $m$ and $M$ correspond to values of $f(x)$, so by the intermediate value theorem, $K=f(c)$ for some $c$ that must lie in between $c_m$ and $c_M$, which means as well that it must be in $[a,b]$.
 
 
-##### Proof of second part of Fundamental Theorem of Calculus
+##### Proof of the second part of the Fundamental Theorem of Calculus
 
 
-The mean value theorem is exactly what is needed to prove formally the second part of the Fundamental Theorem of Calculus. Again, suppose $f(x)$ is continuous on $[a,b]$ with $a < b$. For any $a < x < b$, we define $F(x) = \int_a^x f(u) du$. Then the derivative of $F$ exists and is $f$.
+The mean value theorem is exactly what is needed to  formally prove the second part of the Fundamental Theorem of Calculus. Again, suppose $f(x)$ is continuous on $[a,b]$ with $a < b$. For any $a < x < b$, we define $F(x) = \int_a^x f(u) du$. Then the derivative of $F$ exists and is $f$.
 
 
 Let $h>0$. Then consider the forward difference $(F(x+h) - F(x))/h$. Rewriting gives: