The Rate at Which Rainwater Flows into a Drainpipe

CALCULUS AB FREE-RESPONSE QUESTIONS 1. The rate at which rainwater flows into a drainpipe is modeled by the function R, where R(t) = 20sin((t ^ 2)/35) cubic feet per hour, 1 is measured in hours, and 0 <= t <= 8 The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by D(t) = - 0.04t ^ 3 + 0.4t ^ 2 + 0.96t cubic feet per hour, for 0 <= t <= 8 There are 30 cubic feet of water in the pipe at time = 0. (a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval 0 <= t <= 87 (b) Is the amount of water in the pipe increasing or decreasing at time 13 hours? Give a reason for your answer. (c) At what time 1, 0 <= r <= 8 is the amount of water in the pipe at a minimum? Justify your answer. (d) The pipe can hold 50 cubic feet of water before overflowing. For t > 8 water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time w when the pipe will begin to overflow. 2. Let f and g be the functions defined by f(x) = 1 + x + e ^ (x ^ 2 - 2x) and g(x) = x ^ 4 - 6.5x ^ 2 + 6x + 2 Let R and S be the two regions enclosed by the graphs of f and g shown in the (a) Find the sum of the areas of regions R and S. (b) Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid. (c) Let & be the vertical distance between the graphs of f and g in region S. Find the rate at which a changes with respect to x when x = 1.8 (0,2) 0 #2 (2,4) R S ↑ CALC NON CALC↓↓ #3 1 (minutes) nu(t) 12 200 20 24 40 0 240 -220 150 (meters per minute) 3. Johanna jogs along a straight path. For 0 <= t <= 40 Johanna's velocity is given by a differentiable function v. Selected values of nu(t) where is measured in minutes and v(t) is measured in meters per minute, are given in the table above. (a) Use the data in the table to estimate the value of v' * (16) (b) Using correct units, explain the meaning of the definite integral integrate |nu(t)| dt from 0 to 40 in the context of the problem. Approximate the value of int 0 ^ 40 | nu(t)|c using a right Riemann sum with the four subintervals indicated in the table. (c) Bob is riding his bicycle along the same path. For 0 <= 1 <= 10 Bob's velocity is modeled by B(t) = t ^ 3 - 6t ^ 2 + 300 where t is measured in minutes and B(t) is measured in meters per minute. Find Bob's acceleration at time t = 5 (d) Based on the model B from part (c), find Bob's average velocity during the interval 0 <= t <= 10 4. Consider the differential equation d/dx (y) = 2x - y (a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated. (b) Find (d ^ 2 * y)/(d * x ^ 2) in terms of x and y. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer. (c) Let y = f(x) be the particular solution to the differential equation with the initial condition f(2) = 3 . Does have a relative minimum, a relative maximum, or neither at x =2? Justify your answer. (d) Find the values of the constants m and b for which y = nux + b is a solution to the differential equation. #47 2+ 1+ #5 -1 2 - 12 3 /4 Graph of f' 5. The figure above shows the graph of f' the derivative of a twice-differentiable function f_{1} on the interval [-3, 4]. The graph of f' has horizontal tangents at x = - 1 x = 1 and x = 3 The areas of the regions bounded by the x-axis and the graph of f' on the intervals (-2, 1] and [1, 4] are 9 and 12, respectively. (a) Find all x-coordinates at which ƒ has a relative maximum. Give a reason for your answer. (b) On what open intervals contained in - 3 < x < 4 is the graph of f both concave down and decreasing? Give a reason for your answer. (c) Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer. (d) Given that f(t) = 3 write an expression for f(x) that involves an integral. Find (4) and f(- 2) 6. Consider the curve given by the equation y ^ 3 - xy = 2 It can be shown that d/dx (y) = y/(3y ^ 2 - x) (a) Write an equation for the line tangent to the curve at the point (-1, 1). (b) Find the coordinates of all points on the curve at which the line tangent to the curve at that point is vertical. (c) Evaluate (d ^ 2 * y)/(d * x ^ 2) at the point on the curve wherex x = - 1 and y = 1