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
\(\dfrac {d}{dx} \int_a^x f(t)dt=f(x)\)