MATH

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Review Problems for the Final Exam Maxima, minima, convexity, concavity (a) Find all the critical points. (b) Use the second derivative test to determine if they are local maxima or local minima. Find the open intervals on which the function is (c) increasing, (d) convex. 1. f ( x ) = 2 3 x 3 - x 2 - 4 x + 1 2. f ( x ) = 2 x + 18 x 3. f ( x ) = x x 2 +1 4. f ( x ) = - 2 x - 1 / 2 - 3 ln x + 4 x for x > 0 5. Find the critical points of f ( x ) = x ( x - 1) 4 and use the first derivative test to determine if they are maxima or minima. 6. A ball is tossed straight up from an initial height of 15 meters. Suppose that the velocity is given by v ( t ) = 10 - 10 t meters/sec. (a) Determine the height h ( t ). (b) When does the ball reach its maximum height? What is the maximum height? (c) At what time does the ball hit the ground, i.e., h ( t ) = 0. Max-Min word problems. Remember to use the second derivative test. 7. A ship uses 5 x 2 dollars of fuel per hour while traveling at a speed of x miles per hour. The other expenses of operating the ship come to \$2000 per hour. What speed minimizes the cost of a 500 mile trip?

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