updates

quarto/derivatives/taylor_series_polynomials.qmd

@@ -76,7 +76,7 @@ ImageFile(imgfile, caption)
 ## The secant line and the tangent line
 
 
-We approach this general problem **much** more indirectly than is needed. We introducing notations that are attributed to Newton and proceed from there. By leveraging `SymPy` we avoid tedious computations and *hopefully* gain some insight.
+We approach this general problem **much** more indirectly than is needed. We introduce notations that are attributed to Newton and proceed from there. By leveraging `SymPy` we avoid tedious computations and *hopefully* gain some insight.
 
 
 Suppose $f(x)$ is a function which is defined in a neighborhood of $c$ and has as many continuous derivatives as we care to take at $c$.
@@ -102,7 +102,7 @@ $$
 tl(x) = f(c) + f'(c) \cdot(x - c).
 $$
 
-The key is the term multiplying $(x-c)$ for the secant line is an approximation to the related term for the tangent line. That is, the secant line approximates the tangent line, which is the linear function that best approximates the function at the point $(c, f(c))$. This is quantified by the *mean value theorem* which states under our assumptions on $f(x)$ that there exists some $\xi$ between $x$ and $c$ for which:
+The key is the term multiplying $(x-c)$ for the secant line this is an approximation to the related term for the tangent line. That is, the secant line approximates the tangent line, which is the linear function that best approximates the function at the point $(c, f(c))$. This is quantified by the *mean value theorem* which states under our assumptions on $f(x)$ that there exists some $\xi$ between $x$ and $c$ for which:
 
 
 $$
@@ -153,7 +153,7 @@ As in the linear case, there is flexibility in the exact points chosen for the i
 ---
 
 
-Now, we take a small detour to define some notation. Instead of writing our two points as $c$ and $c+h,$ we use $x_0$ and $x_1$. For any set of points $x_0, x_1, \dots, x_n$, define the **divided differences** of $f$ inductively, as follows:
+Now, we take a small detour to define some notation. Instead of writing our two points as $c$ and $c+h,$ we use $x_0$ and $x_1$. For any set of points $x_0, x_1, \dots, x_n$, recursively define the Newton **divided differences** of $f$ inductively, as follows:
 
 $$
 \begin{align*}