1 files changed, 52 insertions, 0 deletions
diff --git a/notes/Sep-15.org b/notes/Sep-15.org
new file mode 100644
index 0000000..d5bf371
--- /dev/null
+++ b/notes/Sep-15.org
@@ -0,0 +1,52 @@
+* Taylor Series Approx.
+Suppose f has $\infty$ many derivatives near a point a. Then the taylor series is given by
+
+$f(x) = \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
+
+For increment notation we can write
+
+$f(a + h) = f(a) + f'(a)(a+h - a) + \dots$
+
+$= \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{h!} (h^n)$
+
+Consider the approximation
+
+$e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |f'(a) - \frac{1}{h}(f(a + h) - f(a))|$
+
+Substituting...
+
+$= |f'(a) - \frac{1}{h}((f(a) + f'(a) h + \frac{f''(a)}{2} h^2 + \cdots) - f(a))|$
+
+$f(a) - f(a) = 0$... and $distribute the h$
+
+$= |-1/2 f''(a) h + \frac{1}{6}f'''(a)h^2 \cdots|$
+
+** With Remainder
+We can determine for some u $f(a + h) = f(a) + f'(a)h + \frac{1}{2}f''(u)h^2$
+
+and so the error is $e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |\frac{h}{2}f''(u)|$
+
+- [https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series]
+ + > Taylor's Theorem w/ Remainder
+
+
+** Of Deriviatives
+
+Again, $f'(a) \approx \frac{f(a+h) - f(a)}{h}$,
+
+$e = |\frac{1}{2} f''(a) + \frac{1}{3!}h^2 f'''(a) + \cdots$
+
+$R_2 = \frac{h}{2} f''(u)$
+
+$|\frac{h}{2} f''(u)| \leq M h^1$
+
+$M = \frac{1}{2}|f'(u)|$
+
+*** Another approximation
+
+$\text{err} = |f'(a) - \frac{f(a) - f(a - h)}{h}|$
+
+$= f'(a) - \frac{1}{h}(f(a) - (f(a) + f'(a)(a - (a - h)) + \frac{1}{2}f''(a)(a-(a-h))^2 + \cdots))$
+
+$= |f'(a) - \frac{1}{h}(f'(a) + \frac{1}{2}f''(a)h)|$
+