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--- math104-s21:s:genevievebrooks [2021/05/12 20:45]
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-**Continuity**
+**Continuity**\\
+Definitions:
+(Credit to Kaylene Stocking because I really liked the way she described the definitions)\\
+The limit perspective:  Let $(x_n)$ be a sequence of points in the domain of $f$ that converges to some $x_0 \epsilon S$. The sequence $f(xn)$ must converge to $f(x_0)$
+The bounded rate of change perspective:  Pick any point $x_0 \epsilon S$ and any $\varepsilon > 0$. There must exist some $\delta > 0$ so that moving less than $\delta$ away from $x_0$ results in a change of less than $\varepsilon$ in the value of $f(x)$.\\
+**formal definition**: $f$ is continuous at point $p$ if $\forall \varepsilon > 0 \exists \delta > 0$ such that $$d_y(f(x), f(p)) < \varepsilon$$ $$\forall x \epsilon E$$ for which $$d_x(x, p) < \delta$$ Then if f is continuous at every p in its domain, f is continuous overall.
+The topological perspective:  Let E be an open subset of the range of f. The set of points in the domain of f that f maps into E must also be open.
@@ Line 163: / Line 171: @@
 .2 Theorem: If $f$ is defferentiable at $x$ then $f$ is continuous at $x$.
-**Intermediate Value Theorems**
+**Mean Value Theorems**
 These theorems let us extrapolate data about arbitrary points on a function in a range [a, b] for which f is defined. If a point on f is a local min or local max then its derivative at that point is 0.\\
-Generalized IVT: If $f, g$ are continuous on [a, b] and differentiable on (a, b) then $\exists x \epsilon (a, b)$ such that $$[f(b) - f(a)]g'(x) = [g(b) - g(a)]f'(x)$$
+Generalized MVT: If $f, g$ are continuous on [a, b] and differentiable on (a, b) then $\exists x \epsilon (a, b)$ such that $$[f(b) - f(a)]g'(x) = [g(b) - g(a)]f'(x)$$
+MVT: $\frac{f(b) - f(a)}{b - a} = f'(x)$
+Rolle's Theorem (special case of MVT): If f is continuous and differentiable on (a, b) then $f(a) = f(b)$ then there exists at least one x in (a, b) such that $f'(x) = 0$.
+Continuity of Derivatives Theorem: Suppose $f$ is a real differentiable function on [a, b] and suppose $f'(a) < \lambda < f'(b)$ then there is a point x in (a, b) such that $f'(x) = \lambda$
+-useful corollary: If f is differentiable on (a, b) then f can not have any simple discontinuities on (a, b).
-IVT: \frac{f(b) - f(a)}{b - a} = f'(x)
+**L'Hopital's Theorem**\\
+Let $s$ signify $a$, $a^+$, $a^-$, $\infty$, or $-\infty$ where $a \epsilon \mathbb{R}$ and suppose $f$ and $g$ are differentiable functions for which the following limit exists: $$\lim_{x \rightarrow s} \frac{f'(x)}{g'(x)} = L$$ If $$\lim_{x \rightarrow s} f(x) = \lim_{x \rightarrow s} g(x) = 0$$ or if $$ \lim_{x \rightarrow s} |g(x)| = \infty$$ then $$\lim_{x \rightarrow s} \frac{f(x)}{g(x)} = L$$ In other words, we can take the derivative of the top and bottom when f over g is in some indeterminate form and this will give us the same limit and may be easier to calculate the limit from.
+**Taylor's Theorem**\\
+If $f$ is a smooth function (is infinitely differentiable) then $$\sum_{k = 0}^{\infty} \frac{f^k(c)}{k!}(x - c)^k$$ is the Taylor series of f about c. We can stop k at a certain index to get a partial representation of $f$ with some error. The error formula is: $$R_n(x) = f(x) - \sum_{k = 0}{n - 1} \frac{f^k(c)}{k!}(x - c)^k$$ The remainder depends on f and c. Furthermore we may have that $$f(x) = \sum_{k = 0}^{\infty} \frac{f^k(c)}{k!}(x - c)^k$$ iff $$\lim_{n \rightarrow \infty} R_n(x) = 0$$ But the remainder need not always tend to zero. Hence we can have $f$ not given exactly by its Taylor series.\\
+**The actual Taylor's Theorem** \\
+Let $f$ be defined on (a, b) where a < c < b. Suppose the nth derivative of f(x) exists on (a, b). Then for each $x \neq c$ in (a, b) there is some $y$ between c and x such that $$R_n(x) = \frac{f^n(y)}{n!}(x - c)^n$$

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