4.2 - The Mean Value Theorem

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## 4.2 - The Mean Value Theorem

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**4.2 - The Mean Value Theorem**TheoremsIf the conditions (hypotheses) of a theorem are satisfied, the conclusion is known to be true. If the hypotheses of a theorem are not satisfied, the conclusion may still be true, but not guaranteed.**Rolle’s Theorem**• Let f be a function that satisfies the following three hypotheses: • f is continuous on the closed interval [a, b]. • f is differentiable on the open interval (a, b). • f (a) = f (b) • Then there is a number c in (a, b) such that • f′(c) = 0.**Examples: Rolle’s Theorem**• Explain why the conclusion to Rolle’s Theorem is not guaranteed for the function • f (x) = x / (x – 3) on the interval [1, 6]. 2. Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.**The Mean Value Theorem**• Let f be a function that satisfies the following two hypotheses: • f is continuous on the closed interval [a, b]. • f is differentiable on the open interval (a, b). • Then there is a number c in (a, b) such that**Example: Mean Value Theorem**3. Verify that the function satisfies the two hypotheses of Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Mean Value Theorem.**Theorem**If f ′(x) = 0 for all x in an interval (a, b), then f is constant on (a, b). Corollary If f ′(x) = g′(x) for all x in an interval (a, b), then f – g is constant on (a, b); that is, f(x) = g(x) + c where c is a constant. Example: If f (x) = x2 + 3 and g(x) = x2 + 7, find f ′(x), g ′(x), and (f – g)(x).**Proof By Contradiction**• Assume that something is true. • Show that under your assumptions, the conditions of a known theorem are satisfied. This guarantees the conclusion of that theorem. • Show that the conclusion of the theorem is, in fact, not true under the assumptions. • Since the conclusion of the theorem must be true if the assumptions were satisfied, the only conclusion left is that the assumptions must be incorrect.**Example**Show that the equation 2x – 1 – sin x = 0 has exactly one real root.