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4.2 Practice Name___________________________________ Determine whether the function satisfies the hypotheses of the Mean Value Theorem on the given interval. 1) f(x) = x1/3 on [-2, 5] 2) g(x) = x3/4 on [0, 3] Give an appropriate answer. 3) Find the value or values of c that satisfy 48 f(b) - f(a) on the interval [3, 16]. = f′(c) for the function f(x) = x + x b- a Answer the question. 4) A marathoner ran the 26.2 mile New York City Marathon in 2.7 hrs. Did the runner ever exceed a speed of 9 miles per hour? Use analytic methods to find the local extrema. x-1 5) h(x) = 2 x + 5x + 10 Use analytic methods to find those values of x for which the given function is increasing and those values of x for which it is decreasing. 6) f(x) = 7x2 - 5x Find all possible functions with the given derivative. 7) f'(x) = 5e5x Find the function with the given derivative whose graph passes through the point P 8) f'(x) = x2 + 5, P(3, 51) Sketch a graph of a function y = f(x) that has the given properties. 9) a) Differentiable everywhere except x = 0 b) Continuous for all real numbers c) f'(x) < 0 on (-∞, 0) d) f'(x) > 0 on (0, ∞) e) f(-2) = f(2) = 5 f) y-intercept and x-intercept at (0, 0) 1 Answer Key Testname: APCALCULUS4_2PRACTICE 1) 2) 3) 4) No Yes 4 3 Yes, the Mean Value Theorem implies that the runner attained a speed of 9.7 mph, which was her average speed throughout the marathon. 5) Local minimum at x = -3; local maximum at x = 5 5 5 6) Increasing on , ∞ , decreasing on -∞, 14 14 7) e5x + C 8) f(x) = x3 + 5x + 27 3 9) Possible Answer: 2