AP Calculus AB - Important Formulas/Theorems

$\lim_{h\rightarrow0}\ \frac{f\left(x+h\right)-f\left(x\right)}{h}$

$\lim_{h\rightarrow0}\ \frac{f\left(h\right)-f\left(x\right)}{h}$

$\lim_{h\rightarrow0}\ \frac{f\left(x+h\right)+f\left(x\right)}{h}$

$\lim_{h\rightarrow0}\ \frac{f\left(h\right)+f\left(x\right)}{h}$

y-values

first derivative values

second derivative values

x-values

$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$f\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$f\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$

$f'\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$

A closed interval

A differentiable function

A continuous function

A twice-differentiable function

$y-f\left(x_1\right)=f'\left(x_1\right)\left(x-x_1\right)$

$y=f'\left(x_1\right)\left(x-x_1\right)$

$y-f\left(y_1\right)=f'\left(x_1\right)\left(x-x_1\right)$

$y=f\left(x_1\right)\left(x-x_1\right)$

$\int_a^bf\left(x\right)=F\left(b\right)-F\left(a\right)\$ where F is the antiderivative

$\int_a^bf\left(x\right)=f'\left(b\right)-f'\left(a\right)\$

$\int_a^bF\left(x\right)=f\left(b\right)-f\left(a\right)\$ where F is the antiderivative

$\int_a^bF\left(x\right)=f'\left(b\right)-f'\left(a\right)\$ where F is the antiderivative

$\frac{d}{dx}\int_0^xf\left(t\right)dt=f\left(x\right)$

$\frac{d}{dx}\int_0^xf\left(t\right)dt=F\left(x\right)$ where F is the antiderivative

$\frac{d}{dx}\int_0^xF\left(t\right)dt=f\left(x\right)$ where F is the antiderivative

$\frac{d}{dx}\int_0^xf\left(t\right)dt=f'\left(x\right)$

 $\pi\int_{x=a}^{x=b}\left(top\ -bottom\right)^2dx\ \$  

 $\pi\int_{x=a}^{x=b}\left(top\ -bottom\right)dx\ \$  

 $\int_{x=a}^{x=b}\left(top\ -bottom\right)^2dx\ \$  

 $\int_{x=a}^{x=b}\left(top\ -bottom\right)dx\ \$  

 $\pi\int_{y=a}^{y=b}\left(right\ -left\right)^2dy\ \$  

 $\pi\int_{y=a}^{y=b}\left(right\ -left\right)dy\ \$  

 $\int_{y=a}^{y=b}\left(right\ -left\right)^2dy\ \$  

 $\int_{y=a}^{y=b}\left(right\ -left\right)dy\ \$  

 $\pi\int_{x=a}^{x=b}R^2-r^2\ dx\ \$  

 $\pi\int_{x=a}^{x=b}\left(R-r\right)^{2\ }dx\ \$  

 $\int_{x=a}^{x=b}\left(R-r\right)^2dx\ \$  

 $\int_{x=a}^{x=b}R^{2\ }-r^{2\ }dx\ \$