# 4_2 - P where the tangent line is parallel to the secant...

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Section 4.2: The Mean Value Theorem Instructor: Ms. Hoa Nguyen Rolle’s Theorem Let f be a function that satisﬁes the following three hypotheses: 1. f is continuous on the closed interval [ a, b ]. 2. f is diﬀerentiable on the open interval ( a, b ). 3. f ( a ) = f ( b ) Then there is a number c in ( a, b ) such that f 0 ( c ) = 0. Example 2 of Section 4.2 (textbook): The Mean Value Theorem Let f be a function that satisﬁes the following two hypotheses: 1. f is continuous on the closed interval [ a, b ]. 2. f is diﬀerentiable on the open interval ( a, b ). Then there is a number c in ( a, b ) such that: f 0 ( c ) = f ( b ) - f ( a ) b - a or, equivalently, f ( b ) - f ( a ) = f 0 ( c ) · ( b - a ). Geometric meaning : There is at least one point P ( c, f ( c )) on the graph where the slope of the tangent line is the same as the slope of the secant line AB . In other words, there is a point
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Unformatted text preview: P where the tangent line is parallel to the secant line AB . Another way to interpret the Mean Value Theorem is that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval. Example 3 of Section 4.2 (textbook): Example 5 of Section 4.2 (textbook): 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. The above corollary says that, if two functions have identical derivatives on an interval, they must diﬀer by a constant. Example 6 of Section 4.2 (textbook): 1...
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