Calculus Small Changes, Big Ideas

Section 5.6 Valuable theorems

• Emphasize the importance of hypotheses! The Pythagorean theorem is NOT \(a^{2}+b^{2}=c^{2}\text{.}\)

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• Theorem 4. Squeeze Theorem.

  If \(f(x)\le g(x)\le h(x)\) for all \(x\) sufficiently close to \(c\text{,}\) and \(\limit{x\to c}f(x)=\limit{x\to c}h(x)=L\text{,}\) then \(\limit{x\to c}g(x)=L\text{.}\)

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• Theorem 5. Intermediate Value Theorem.

  If \(f\) is continuous on \([a,b]\text{,}\) then for all \(y\) between \(f(a)\) and \(f(b)\text{,}\) there exists \(c\in[a,b]\) such that \(f(c)=y\text{.}\)

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• Theorem 6. Extreme Value Theorem.

  If \(f\) is continuous on \([a,b]\text{,}\) then there exist \(u,v\in[a,b]\) such that \(f(u)\le f(x)\le f(v)\) for all \(x\in[a,b]\text{.}\)

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• Theorem 7. Mean Value Theorem.

  If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{,}\) then there exists \(c\in(a,b)\) such that

  \begin{equation*} f'(c)=\frac{f(b)-f(a)}{b-a}\text{.} \end{equation*}

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