Calculus Theorems

Intermediate Value Theorem

If 𝑓 is a continuous function whose domain contains the interval [a, b], then it takes on any given value between 𝑓(𝑎) and 𝑓(𝑏) at some point within the interval.

Extreme Value Theorem

If a real-valued function 𝑓 is continuous on the closed and bounded interval [𝑎,𝑏], then 𝑓 must attain a maximum and a minimum, each at least once. That is, there exist numbers 𝑐 and 𝑑 in [𝑎,𝑏] such that:𝑓(𝑐)≥𝑓(𝑥)≥𝑓(𝑑)∀𝑥∈[𝑎,𝑏].

Mean Value Theorem

For a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

Rolle’s Theorem

Any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero.