AP Calc AB Theorems

Squeeze Theorem

A theorem in calculus that establishes the limit of a function that is 'squeezed' between two other functions. If the two bounding functions converge to the same limit at a point, then the squeezed function must also converge to that limit.

“h(x) and g(x) have continuity for all x because we know that h(c) = p = g(c) and the limit as x→c of h(x) = t = the limit as x→c of g(x). By the squeeze theorem, the limit as x→c of f(x) = t and f(c) = p. Therefore, f(x) has continuity for all x.”

Intermediate Value Theorem

A fundamental theorem in calculus that states if a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b) at least once. This means for any value k between f(a) and f(b), there exists a c in (a, b) such that f(c) = k.

“f(x) has continuity, so IVF applies. f(a) = k and f(b) = m, where k<L<m. Because L does exist in between [k, m], there is a value, C, such that f(c) = L.”

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