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BC 1-2 Spring 2012 Problem Set #7 Name Due: Thursday, 5/17. (at beginning of class) Please show appropriate work – no big calculator leaps – except as indicated. Work should be shown clearly, using correct mathematical notation. Please show enough work on all problems (unless specified otherwise) so that others could follow your work and do a similar problem without help. Collaboration is encouraged, but in the end, the work should be your own. 1. Let f ( x) x x for x 0. Find all values of x for which f ( x) f ( x) . 3 2. Write the integral x 2 e x dx as a limit of Riemann sums. Do not evaluate the limit. 1 x d 3. Evaluate: t sin t dt dx x2 3 BC 1-2 Spring 2012 Problem Set #7 Name Due: Thursday, 5/17. (at beginning of class) 4a. What is the maximum area of a rectangle that can be drawn inside a 3-4-5 triangle with one of the rectangle’s sides along one of the legs of the triangle? 4b. What is the maximum area of a rectangle that can be drawn inside a 3-4-5 triangle with one of the rectangle’s sides along the hypotenuse of the triangle? BC 1-2 Spring 2012 Problem Set #7 Name Due: Thursday, 5/17. (at beginning of class) 5. (AP 2011) A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially empty, and rain enters the can during a 60-day period. The height of water in the can is modeled by the function S, where S(t ) is measured in millimeters and t is measured in days for 0 ≤ t ≤ 60. The rate at which the height of the water is rising in the can is given by S′(t ) = 2sin(0.03t ) + 1.5. [You may use a calculator on this problem]. (a) According to the model, what is the height of the water in the can at the end of the day period? 60- (b) According to the model, what is the average rate of change in the height of water in the can over the 60-day period? Show the computations that lead to your answer. Indicate units of measure. (c) Assuming no evaporation occurs, at what rate is the volume of water in the can changing at time t = 7 ? Indicate units of measure. (d) During the same 60-day period, rain on Monsoon Mountain accumulates in a can identical to the one in Stormville. The height of the water in the can on Monsoon 1 (3t 3 30t 2 330t ) . The Mountain is modeled by the function M, where M (t ) 400 height M( t) is measured in millimeters, and t is measured in days for 0 ≤ t ≤ 60. Let D(t ) = M′(t ) − S′(t ). Apply the Intermediate Value Theorem to the function D on the interval 0 ≤ t ≤ 60 to justify that there exists a time t, 0 < t < 60, at which the heights of water in the two cans are changing at the same rate. BC 1-2 Spring 2012 Problem Set #7 Name Due: Thursday, 5/17. (at beginning of class) 2008 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B) 6. Let g be a continuous function with g (2) 5 . The graph of the piecewise-linear function g , the derivative of g, is shown above for 3 x 7 . (a) Find the x-coordinate of all points of inflection of the graph of y g ( x) for 3 x 7 . Justify your answer. (b) Find the absolute maximum value of g on the interval 3 x 7 . Justify your answer. (c) Find the average rate of change of g ( x ) on the interval 3 x 7 . (d) Find the average rate of change of g ( x) on the interval 3 x 7 . Does the Mean Value Theorem applied on the interval 3 x 7 guarantee a value of c, for 3 c 7 , such that g (c ) is equal to this average rate of change? Why or why not?