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AP Calculus 3.1 – 3.2 Worksheet Name: For #1 – 2, Explain why each of the following functions fails to satisfy the conditions of the Mean Value Theorem. (Hint: you may need to make a graph as part of your explanation.) 1. f(x) = x 1 2 x 3 3, x 1 2. f ( x) 2 x 1, x 1 3. Let f(x) = 1 x 2 , A = (-1, f(-1)), and B = (1, f(1)). Find a tangent to f in the interval (-1, 1) that is parallel to the line AB. In #4 – 6, (a) state whether or not the function satisfies the hypotheses of the Mean Value Theorem on the given interval and (b) if it does, find each value of c on the interval (a, b) that satisfies the equation f ' (c) 4. f(x) = x2 + 2x – 1 on [0, 1] 6. f(x) = x2/3 on [0, 1] 5. f(x) = x1/3 on [-1, 1] f (b) f (a ) ba AP Calculus Name: 3.1 – 3.2 Worksheet For #7 – 8, find any critical numbers of the function. 8. g ( x) 7. h(x) = sin2x + cos x, 0 < x < 2π 4x x 1 2 For #9-12, locate the absolute extrema of the function on the closed interval. 2x , [-2, 2] x 1 9. g(x) = x2 – 2x, [0, 4] 10. f ( x) 11. y = 3 - |t – 3|, [-1, 5] 12. g(x) = sec x, 2 , 6 3 In #13 – 14, graph a function on the interval [-2, 5] having the given characteristics. 13. Absolute maximum at x = -2, absolute 14. Relative minimum at x = -1, critical number (but not an minimum at x = 1, relative maximum at x = 3. absolute extrema) at x = 0, absolute maximum at x = 2, absolute minimum at x = 5 y –5 –4 –3 –2 y 5 5 4 4 3 3 2 2 1 1 –1 –1 1 2 3 4 5 x –5 –4 –3 –2 –1 –1 –2 –2 –3 –3 –4 –4 –5 –5 1 2 3 4 5 x