1. A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height h feet is given by the function A, where A(h) is measured in square feet. The function A is continuous and decreases as h increases. Selected values for A(h) are given in the table above.
2. The figure above shows the polar curves r = f(θ) = 1 + sinθcos(2θ) and r = g(θ) = 2cosθ for 0 ≤ θ ≤ π/2. Let R be the region in the first quadrant bounded by the curve r = f(θ) and the x-axis. Let S be the region in the first quadrant bounded by the curve r = f (&theta), the curve r = g(θ), and the x-axis. (a) Find the area of R. (b) The ray θ = k, where 0 < k < &pi/2, divides S into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of k. (c) For each θ, 0 ≤ θ ≤ π/2, let w(θ) be the distance between the points with polar coordinates (f(θ),θ) and (g(θ),θ). Write an expression for w(θ). Find wA, the average value of w(θ) over the interval 0 ≤ θ ≤ π/2. (d) Using the information from part (c), find the value of θ for which w(θ) = wA. Is the function w(θ) increasing or decreasing at that value of θ ? Give a reason for your answer.
3. The function f is differentiable on the closed interval [−6, 5] and satisfies f(−2) = 7. The graph of f', the derivative of f, consists of a semicircle and three line segments, as shown in the figure above.
AP Calculus BC 2017 Free Response Question 5
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