## Theorem List

### Basic

Rule Name Theorem
$$f$$ is continuous at $$c$$
$$f$$ is continuous on $$[a,b]$$
There is at least one number $$c$$ such that $$f(c)=k$$
$$f$$ has both a maximum and minimum on the interval
There exists a number $$c$$ in $$(a,b)$$ such that $$f'(c)=\dfrac {f(b)-f(a)}{b-a}$$
There exists a number $$c$$ in $$(a,b)$$ such that $$f'(c)=0$$
$$\lim \limits_{x \to c} \dfrac {f(x)}{g(x)}=\lim \limits_{x \to c} \dfrac {f'(x)}{g'(x)}$$

### Derivative

Rule Name Theorem
$$f'(x)= \lim \limits_{h \to 0} \dfrac {f(x+h)-f(x)}{h}$$
$$f'(a)= \lim \limits_{x \to a} \dfrac {f(x)-f(a)}{x-a}$$

### Integral

Rule Name Theorem
$$\lim \limits_{n \to \infty}\sum_{i=1}^{n}f(c_i)\Delta x_i = \int_a^b f(x)dx$$
$$\int_a^b f(x)dx = F(b) - F(a)$$
There exists a number $$c$$ in the closed interval $$[a,b]$$ such that $$\int_a^b f(x)dx = f(c)(b-a)$$
The average value of $$f$$ on the interval is $$\dfrac {1}{b-a}\int_a^b f(x)dx$$
$$\int_a^b F'(x)dx = F(b) - F(a) =$$ net change in $$F$$ from a to b
$$\dfrac {d}{dx} \int_a^x f(t)dt=f(x)$$