AP Calculus AB Take-Home Chapter 5 Name: Sanchez L. Period : 4 Due Date : 3 /14 / 23 Show all work! 10 1. Evaluate the integral ( V9 - x 2 dx a. Calculator answer 706 8 6 10 8 - 6 4 - 2 2 4 6 8 10 b. Now find the answer using the graph of the integrand and area to evaluate the integral. Leave your answer in terms of . 2. A hose is being used to fill a rectangular pool. Water is flowing into the pool at a rate of 5 gallons per minute. The hose is turned on, placed in the pool, and left for 2 hours. Write an integral to represent this situation and evaluate the integral. 3. The table below shows the velocity of a ball on a labyrinth. Time (sec) 5 10 15 20 25 30 Velocity (cm/sec) 0 0.5 1.2 1.8 2.6 2.4 2.2 a ) Find the right Riemann Sum approximation to estimate the distance traveled by the ball using 6 intervals of equal length. Include units! Find the left Riemann Sum approximation to estimate the distance traveled by the ball using 6 intervals of equal length. Include units c) Find the Midpoint Riemann Sum approximation to estimate the distance traveled by the ball using 3 intervals of equal length. Include units! 4. Find - d ( 1 2 + 4 ) dt . dx JILx-1 5. 5 2x2 + x +4 dx= 2 - 1 (A) 139 - 23 (B ) (c) 22 + 7(In 4 - In 2) (D) 24 + 8(In 4 - In 2) 6. Find the average value of f(x) = - from x = 1 to x = e. Write and evaluate a definite integral.7. Match the differential equation ay = 1 -x2 to the correct slope field. dx b ) a) 8. The weight of a population of yeast is given by a differentiable function, W, where W(t) is measured in grams and t is measured in hours. The weight of the yeast population increases with respect to t at a rate that is directly proportional to the weight. At time t = 10 hours, the weight of the yeast is 200 grams and is increasing at the rate of 5 grams per hour. Which of the following is a differential equation that models this situation? (A) W = 5(t - 10) + 200 (B) d w (C) dW - = to W ( D) d = +W 9. Use the differential equation- dy = 2x dx y for the following three questions. a) Construct a slope field for the differential equation at the nine points 2 indicated. b) Let y = f(x) be the particular solution to the differential equation with the initial condition f (1) = -1. Write an equation for the line tangent to the graph of f at (1, -1). c) Find the particular solution y = f(x) to the given differential equation with the -1 O 2 initial condition f (1) =-1. 10. Which slope field has only negative slopes in Quadrant IV? dy b. dy C. dy - xy d. dy e . dy dx V dx dx dx 1 2 dx 2