The Theorems of Differential Calculus
Below are statements of theorems from differential calculus with accompanying visualizations.
The Squeeze Theorem
If:[br][list][*][math]f[/math], [math]g[/math], and [math]h[/math] are all continuous functions over [math][a,b][/math],[/*][*][math]c\in\left[a,b\right][/math],[/*][*][math]g[/math] is between [math]f[/math] and [math]h[/math] on [math][a,b][/math], and [br][/*][*][math]\lim_{x\to c}f\left(x\right)=\lim_{x\to c}h\left(x\right)=L[/math],[/*][/list]then [math]\lim_{x\to c}g\left(x\right)=L[/math].
The Intermediate Value Theorem
If:[br][list][*][math]f[/math] is continuous over [math][a,b][/math] and[/*][*][math]y[/math] is a number between [math]f(a)[/math] and [math]f(b)[/math][/*][/list]then there is a number [math]x[/math] in [math]\left(a,b\right)[/math] such that [math]f\left(x\right)=y[/math].
The Extreme Value Theorem
If:[br][list][*][math]f[/math] is continuous over [math][a,b][/math],[/*][/list]then there are numbers [math]c[/math] and [math]d[/math] in [math]\left[a,b\right][/math] such that [math]f\left(c\right)\le f\left(x\right)[/math] for every [math]x[/math] in [math][a,b][/math] and [math]f\left(d\right)\ge f\left(x\right)[/math] for every [math]x[/math] in [math][a,b][/math].
The Mean Value Theorem
If:[br][list][*][math]f[/math] is continuous over [math][a,b][/math] and[/*][*][math]f[/math] is differentiable over [math](a,b)[/math], [/*][/list]then there is a number [math]x\in\left(a,b\right)[/math] such that [math]f'(x)=\frac{f\left(b\right)-f\left(a\right)}{b-a}[/math].
L'Hopital's Rule
If:[br][list][*][math]f[/math] and [math]g[/math] are both differentiable near [math]c[/math],[/*][*][math]\lim_{x\to c}f\left(x\right)=\lim_{x\to c}g\left(x\right)=0[/math] or [math]\pm\infty[/math], and [/*][*][math]\lim_{x\to c}\frac{f'\left(x\right)}{g'\left(x\right)}[/math] exists,[/*][/list]then [math]\lim_{x\to c}\frac{f\left(x\right)}{g\left(x\right)}=\lim_{x\to c}\frac{f'\left(x\right)}{g'\left(x\right)}[/math].