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Transcript
3.2 Rolle’s Theorem •Understand and use Rolle’s Theorem. •Understand and use the Mean Value Theorem. 1. Sketch a rectangular coordinate plane. Label the points (1,3) and (5,3). Draw the graph of a differentiable function f that starts at (1,3) and ends at (5,3). a) Is there at least one point on the graph for which f’ =0? b) Can you draw a graph where every point’s derivative is zero? c) Would it be possible to draw a graph where there is no point for which f’ = 0? Rolle’s Theorem Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a) = f(b) then there is at least one number c in (a,b) such that f’(c) = 0. Determine whether Rolle’s Theorem can be applied to the function. If it can find all values of c where f’(c) = 0. 1. f(x) = x² - 5x + 4 [1,4] 2. f(x) = (x² -1)/ x [-1,1] 3. f(x) = sec(x) [ -π/4, π/4 ] Mean Value Theorem If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that f ’(c) = Two interpretations 1) Guarantees there is a tangent line that is parallel to the secant line that goes through a and b. 2) Implies that there must be a point at which the instantaneous rate of change is equal to the average rate of change. Remember that polynomial, rational and trigonometric functions are all differentiable at all points in their domains.