Rolle's TheoremTheorem 1. Let $f$ be a function that satisfies the following conditions:(1) $f$ is continuous on $[a, b]$.(2) $f$ is differentiable on $(a, b)$(3) $f(a) = f(b)$Then there is a number $c \in (a, b)$ such that $f'(c) = 0$.Proof. We may think of three cases. (1) If $f(x) = k$ for any constant $k$, then $c$ can be taken to be any number in $(a, b)$.(2) If $f(x) > f(a), \forall x \in ..

Fermat's TheoremTheorem 1. If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.Proof. Without loss of generality, suppose that $f$ has a local maximum at $c$. This means that $f(c) \geq f(c+h)$ for $h$ which is sufficiently close to $0$. If $h > 0$, we have $$\frac{f(c+h) - f(c)}{h} \leq 0 \\ \Longrightarrow \lim_{h \rightarrow 0^+} \frac{f(c+h)-f(c)}{h} = f'(c)..

Absolute Maximum and MinimumDefinition 1. Let $c$ be a number in the domain $D$ of a function $f$. Then $f(c)$ is the(1) absolute maximum value of $f$ on $D$ if $f(c) \geq f(x), \forall x \in D$.(2) absolute minimum value of $f$ on $D$ if $f(c) \leq f(x), \forall x \in D$.$f$의 maximum과 minimum은 extreme value of $f$, 즉 $f$의 극값이라고 부르기도 한다.  Local Maximum and MinimumDefinition 2. The number $f(c)$ ..

Definition Definition. Let $x_1, x_2 \in$ an interval $I$ such that $x_1 f(x_2). $$ Theorem Theorem. Let $f: I \longrightarrow \mathbb{R}$ be a differentiable function on $I$. Then for $\forall x \in I$, $$f'(x) > 0 \Longrightarrow f \text{ is incre..

Definitions 1. $$\text{sinh} x = \frac{e^x - e^{-x}}{2} \qquad \text{csch} x = \frac{1}{\text{sinh} x} \\ \text{cosh} x = \frac{e^x + e^{-x}}{2} \qquad \text{sech} x = \frac{1}{\text{cosh} x} \\ \text{tanh} x = \frac{\text{sinh} x}{\text{cosh} x} \qquad \text{coth} x = \frac{\text{cosh} x}{\text{sinh} x}$$ 중심이 원점이고 반지름이 1인 원, 즉 단위원이 좌표평면 상에 있을 때 직각삼각형을 만들어서 각도에 따라 값이 변하는 cos, sin 값을 이용해 원 위의 점을 ..