Derivative #
(Landau notation)
Suppose $(a _ n) _ {n = 1}^\infty, (c _ n) _ {n = 1}^\infty \in \C$ and $(b _ n) _ {n = 1}^\infty \in (0, \infty)$.
- $a _ n = O(b _ n) \iff \exists M > 0, \forall n \ge 1: |a _ n| \le M b _ n$.
- $a _ n = o(b _ n) \iff \lim _ {n \to \infty} a _ n / b _ n = 0$.
- $a _ n = c _ n + O(b _ n) \iff a _ n - c _ n = O(b _ n)$.
- $a _ n = c _ n + o(b _ n) \iff a _ n - c _ n = o(b _ n)$.
Suppose $S$ is some topological space. Suppose $a(x): S \to \C$ and $b(x): S \to (0, \infty)$.
- $a(x) = o(b(x))$ as $x \to a$ iff $\lim _ {x \to a} a(x) / b(x) = 0$.
Derivative #
Consider $f: [a, b] \to \R$. Define $f^ * : [a, b] \to \R$ for $c \in [a, b]$ as $$ f^ * (x) := \frac{f(x) - f(c)}{x - c} $$
- $f^ * $ has a removable discontinuity at $c$.
When $f'(c): = \lim _ {x \to c} f^ * (x)$ exists in $\R$. $f$ is called differentiable at $c$. And $f'(c)$ is the derivative of $f$ at $c$.
- $f$ is continuous at $c$ if $f$ is differentiable at $c$.
- $f': F \to \R$ is called the derivative of $f$ where $f'(x)$ exists.
- The $n$-th derivative of $f$, denoted by $f^{(n)}$, is the derivative of $f^{(n-1)}$.
We extend the definition of $f'(c)$ when $f$ is continuous at $c$ and $\lim _ {x \to c} f^ * (x)$ exists in $\eR$.
For $c = a$, $f'(c)$ is called the right-derivative, and also denoted as $f' _ {+}(c)$. $f(x)$ is called right-differentiable if $f' _ +(c)$ is defined.
Derivative chain rule #
Consider the following situation:
- Suppose $f: S \to \R$ is differentiable at $c \in S^\circ$.
- Suppose $g: T \to \R$ is differentiable at $f(c) \in T^\circ$.
- Suppose $h = g \circ f: S \to \R$.
Then $h$ is differentiable at $c$ and $h'(c) = g'(f(c))f'(c)$.
- $f(x) = f(c) + f'(c)(x - c) + R _ 0(x-c)$ for $x \in B(c, \delta _ 0)$.
- $g(y) = g(f(c)) + g'(f(c)) (y - f(c)) + R _ 1(y - f(c))$ for $y \in B(f(c), \delta _ 1)$.
- w.l.o.g. take $\delta _ 0$ small enough where $f[B(c, \delta _ 0)] \subseteq B(f(c), \delta _ 1)$.
- $h(x) = g(f(c)) + g'(f(c))f'(c) (x - c) + g'(f(c)) R _ 0(x - c) + R _ 2(x - c)$.
- Where $R _ 2(x - c) := R _ 1(f'(c) (x - c) + R _ 0(x - c))$.
- For $x \in B(c, \delta _ 2)$ with $\delta _ 2$ small enough, we have $\lim _ {x \to c}\frac{R _ 2(x - c)}{x - c} = 0$.
- Just notice that for $\delta _ 2$ small enough $f'(c) (x - c) + R _ 0(x - c)$ is bounded by $K\abs{x - c}$.
Derivative Arithmetic #
For $f, g: S \to \R$ differentiable at $c \in S^\circ$, rules for $(f + g)', (fg)', (f/g)'$ in Calculus are true.
See any Calculus book for the derivations of these rules.
Sets of Differentiable Functions #
For $f:S \subseteq \R \to \R$.
- $f \in C^k[S]$ means $f^{(k)}$ exists and is continuous on $S$.
- $f \in C^\infty[S]$ means $f$ is infinitely differentiable on $S$.
- $f \in D[S]$ means $f$ is differentiable on $S$.
- $f \in D'[S]$ means $f'(x) \in \eR$ is defined on $S$.
Extrema of real functions #
For $f:S \subseteq \R \to \R$.
- Minima and maxima of $f$ are extrema.
- $f$ have a local maximum at local maximum point $a$ if for all points in $B _ S(a, \delta)$, $f(x) \le f(a)$.
- $f$ have a global maximum at global maximum point $a$ if $\forall x \in S: f(x) \le f(a)$.
Mean Value Theorems #
Rolle #
Suppose $f: [a, b] \to \R$. Suppose $f \in C[a, b]$ and $f \in D'(a, b)$, and $f(a) = f(b)$.
There is some $c \in (a, b)$ such that $f'(c) = 0$.
- w.l.o.g. suppose $f(a) = f(b) = 0$, and $\sup f(x) > 0$.
- Suppose for $x \in (a, b)$, $f(x) = \max f[a, b]$.
- Since $f[a, b]$ is compact.
- $f _ +'(x) \le 0$ and $f' _ -(x) \ge 0$. So $f'(x) = 0$.
Mean Value Theorem #
Suppose $f: [a, b] \to \R$. Suppose $f \in C[a, b]$ and $f \in D'(a, b)$.
Then there is some $c \in (a, b)$ such that $f(b) - f(a) = f'(c) (a - b)$.
- Take $g(x) = f(x) - (x-a)(f(b) - f(a))/(b -a)$ then apply (Rolle).
Immediately results from MVT I:
- If $\forall x \in (a, b): f'(x) \gt 0$, $f$ is strictly increasing on $[a, b]$.
- If $\forall x \in (a, b): f'(x) = 0$, $f$ is constant on $[a, b]$.
- Suppose $f, g \in D'(a, b)$ and $f, g \in C[a, b]$. If $f' - g' = 0$ on $(a, b)$ then $f - g = 0$ on $[a, b]$.
Cauchy Mean Value Theorem #
Suppose $f, g: [a, b] \to \R$.
Suppose $f, g \in C[a, b]$ and $f, g \in D'(a, b)$, and $f'(x)$ and $g'(x)$ are not simultaneously infinite.
For some $c \in (a, b)$, $f^{\prime}(c)[g(b)-g(a)]=g^{\prime}(c)[f(b)-f(a)]$.
- Define $h(x) = f(x)[g(b) - g(a)] - g(x)[f(b) - f(a)]$.
- Notice that $h(a) = h(b) = f(a) g(b) - f(b) g(a)$.
- $h \in C[a, b]$ and $h \in D'(a, b)$.
- The result follows from Rolle.
Intermediate Value Theorem of Derivative #
Intermediate Value Theorem of Derivative #
Consider $f: (a, b) \to \R$. Suppose $f \in C(a, b)$ and $f \in D'(a, b)$.
$f'(a, b)$ obtains all intermediate values.
- Define the triangle region $S := \c{a < x < y < b: x, y \in \R}$ where $S \subseteq \R^2$.
- Clearly $S$ is open and connected.
- Consider function $g(x, y): S \to \R$. Defined as
$$
g(x, y) := \frac{f(y) - f(x)}{y - x}
$$
- $g$ is continuous on $S$.
- Since $S$ is connected, $g[S]$ obtains all intermediate values.
- According to intermediate value theorem, $g[S] \subseteq f'(a, b)$.
- Notice that $f'(a, b) = \overline{g[S]}$.
- Even though $g[S] \neq f'(a, b)$ in general. $g[S]$ is clearly dense in $f'(a, b)$.
- So $f'(a, b)$ obtains all intermediate values.
Discontinuities of derivative #
Consider $f: (a, b) \to \R$. Suppose $f \in D'(a, b)$.
$f'$ does not have jump or removable discontinuities.
Monotonicity and derivative #
For $f: (a, b) \to \R$. Suppose $f \in D'(a, b)$.
Suppose $0 \notin f'(a, b)$ then $f$ is strictly monotonic on $[a, b]$.
- Either $f'(a, b) \subseteq (0, \infty]$ or $f'(a, b) \subseteq [-\infty, 0)$.
Suppose $f'$ is monotonic on $(a, b)$, $f' \in C(a, b)$.
- Since $f'$ is monotonic, there can only be jump and removable discontinuities.
Convexity #
Convex real functions #
Suppose $g: I \to \R$ where $I$ is an open interval in $\R$. $g$ is called convex if $$ \forall x, y \in I, \forall a \in [0, 1]: g(ax + (1 - a)y) \le ag(x) + (1 - a)g(y) $$
- Intuitively, the function is always below any line segment.
Suppose $g(x)$ is convex the following are rather apparent:
- $g _ +'$ exists on $I$.
- $\forall 0 < a < b: [g(x+a)-g(x)]/a \leq [g(x+b)-g(x)]/b$.
- $g' _ -$ exists on $I$.
- $\forall 0 < a < b: [g(x) - g(x-b)]/b \geq [g(x) - g(x-a)]/a$.
- $g(x)$ is continuous on $I$. Since $g' _ +$ and $g' _ -$ exists on $I$.
- $\forall x \in I: g' _ -(x) \le g' _ +(x)$.
- For $a > 0$ and $b > 0$, $[g(x) - g(x - a)]/a \le [g(x + b) - g(x)]/b$.
- $g' _ +, g' _ -$ are increasing on $I$.
- $g _ +'(x _ 1) \le [g(x _ 2) - g(x _ 1)]/(x _ 2 - x _ 1) \le g _ -'(x _ 2) \le g _ +'(x _ 2)$.
- For any $c \in I$ there exists a linear $L _ c(x): I \to \R$ where $\forall x \in I: L _ c(x) \le g(x)$.
- There exists some $g _ -'(x) \le k \le g _ +'(x)$.
- Clearly $\forall x\in I: g(x) \ge g(c) + k(x - c)$.
- Such a function $L _ c(x)$ is called a line of support for $g$ at $c$.
- $g(x) = \sup _ {c \in I \cap \Q}L _ c(x)$.
Taylor's Theorem #
Taylor polynomial #
Consider $f: [a, b] \to \R$. Suppose $f \in D^{n}[a, b]$. Where $n \in \N$.
Define n-th order Taylor polynomial $P _ n(x): [a, b] \to \R$: $$ P _ n(x) := \sum _ {k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k = f(a) + f'(a)(x - a) + \frac{f''(a)}{2} (x - a)^2 + \cdots $$ The n-th order Taylor remainder is $R _ n(x) := f(x) - P _ n(x)$.
Lagrange remainder #
Consider $f: [a, b] \to \R$. Suppose $f \in D^{n}[a, b]$, and $f \in D^{n+1}(a, b)$. $$ \forall x \in (a, b],\exist \xi _ x \in (a, x): R _ n(x) = \frac{f^{(n+1)}(\xi _ x)}{(n+1)!}(x - a)^{n+1} $$
- Define $F(x) := R _ n(x) = f(x) - P _ n(x)$. And $G(x):= (x - a)^{n + 1}$.
- $F(a) = F'(a) = \cdots = F^{(n)}(a) = 0$.
- $G(a) = G'(a) = \cdots = G^{(n)} = 0$.
- $F^{(n+1)}(x) = f^{(n+1)}(x)$ for $x \in (a, b)$.
- $G^{(n+1)}(x) = (n+1)!$
- By Cauchy's MVT, we have (division should be expanded to products): $$ \frac{F(x)}{G(x)} = \frac{F'(c _ 1)}{G'(c _ 1)} = \cdots = \frac{F^{(n+1)}(c _ {n+1})}{G^{(n+1)}(c _ {n+1})}; \quad a < c _ {n+1} < \cdots < c _ 1 < x $$