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Name_______________________________Period_______ Category Preparation Completeness Organiz./Neat Timeliness Total Points earned AP Calculus Problem Set 48 Upon completion, circle one of the following to assess your current understanding: Completely understand Mostly understand Sort of understand Don’t understand 1. The graph of the function f is shown below. Sketch the graph of each approximation method. f x dx using a right Riemann f x dx using a left Riemann 6 6 a) Approximate b) Approximate 0 0 sum with 3 subintervals of equal length. sum with 3 subintervals of equal length. y y x x f x dx using a midpoint Riemann d) Approximate f x dx using a trapezoidal 6 6 c) Approximate 0 0 sum with 3 subintervals of equal length. sum with 3 subintervals of equal length. y y x x 2. The graph of the function f is shown below. Sketch the graph of each approximation method. f x dx using a right Riemann f x dx using a left Riemann 8 8 a) Approximate b) Approximate 0 0 sum with 4 subintervals of equal length. sum with 4 subintervals of equal length. y y x x f x dx using a midpoint Riemann d) Approximate f x dx using a trapezoidal 8 8 c) Approximate 0 0 sum with 4 subintervals of equal length. sum with 4 subintervals of equal length. y y x x 3. A post office delivered letters between 7 a.m. (t = 0) and 5 p.m. (t = 10). The number of letters delivered t hours after noon is modeled by a differentiable function L for 0 t 10 . L t 0 for all t. Values of L t , in thousands of letters, at various time t are shown in the table below. t 0 3 7 8 10 (hours) L t 0 1 4 5 9 (thousands of letters) L t dt 10 a) Use a trapezoidal sum with the four subintervals given by the table to approximate 0 L t dt ? 10 Does this approximation overestimate or underestimate the actual value of 0 Give a reason for your answer. 1 10 L t dt 10 0 b) Use a left Riemann sum with the four subintervals given by the table to approximate 1 10 L t dt , in terms of the number of letters delivered. 10 0 Using correct units, explain the meaning of 4. During the time interval 0 t 24 hours, snow accumulates on a driveway at the rate N t cubic feet per hour. The table below gives values of N t , a strictly monotonic differentiable function, for selected values of t. 1 24 a) Using correct units, explain the meaning of N t dt and N t dt in terms 24 t N t 0 24 0 0 0 of the snow in the driveway. 4 12 b) Use a midpoint Riemann sum with three subintervals of equal length to approximate N t dt . Show the computations that lead to your answer. 24 8 20 0 12 25 c) Use a right Riemann sum with three subintervals of equal length to approximate 16 30 1 12 N t dt . Show the computations that lead to your answer. Does this 12 0 20 50 24 55 approximation overapproximate or underapproximate the actual value of 1 12 N t dt ? Justify your answer. 12 0 5. The velocity of a particle moving along the x-axis is modeled by a differentiable function, v, in meters per second. Selected values of v t are given in the table below. t 0 8 10 15 27 47 (seconds) v t 2 4 -8 -6 -1 3 (meters per second) v t dt 47 a) Using correct units explain the meaning of in the context of this problem. 10 v t dt 47 b) Use a trapezoidal sum with the three subintervals indicated by the table to approximate 10 v t dt in the context of this problem. Use a right 47 c) Using correct units explain the meaning of 10 v t dt 47 Riemann sum with the three subintervals indicated by the table to approximate 10 6. Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable strictly monotonic function C for 0 t 9 . Values of C t are various times t are shown in the table below. t (hours) 0 1 3 4 7 8 9 C t (people) 190 180 140 120 100 80 0 1 9 C t dt in the context of this problem. 8 1 a) Using correct units, explain the meaning of 1 9 b) Use a left Riemann sum with five subintervals to estimate C t dt . Does this approximation 8 1 1 overestimate or underestimate the actual value of C t dt . Give a reason for your answer. 9 8 1 7. A spherical hot air balloon expands as air inside the balloon is heated. The table below gives selected values of the rate of change of the radius of the balloon over the time interval 0 t 18 . t (minutes) 0 3 6 9 12 15 18 r t (feet per minute) 7 5 4 3 2 2 1 r t dt 18 a) Using correct units, explain the meaning of in the context of this problem. 0 r t dt . 18 b) Use a right Riemann sum with six subintervals of equal width to approximate 0 1 18 r t dt in the context of this problem. 18 0 c) Using correct units, explain the meaning of 1 18 r t dt . 18 0 d) Use a midpoint Riemann sum with three subintervals of equal width to approximate 8. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function A of time t. A t 0 for all t. A table of selected values of A t , for the time interval 0 t 100 , is shown below. t At (minutes) (gallons per minute) 0 10 20 50 30 60 70 80 90 85 100 87 A t dt in the context of the problem. 100 a) Using correct units, explain the meaning of 0 A t dt using a trapezoidal sum with the five subintervals indicated 100 b) Approximate the value of 0 A t dt ? Justify your answer 100 by the data in the table. Is this approximation less than the value of 0 c) Use a right Riemann sum with the five subintervals indicated by the data to approximate A t dt Is this approximation greater than the value of A t dt ? Justify your answer. 100 100 0 0 f x dx , and a right Riemann sum underapproximates 5 9. If a trapezoidal sum overapproximates 0 f x dx which of the following could be the graph of y f x ? 5 0 (A) (B) (C) y y y x x x (D) (E) y y x x 10. The graph of the function f is shown below. Of the following, which has the least value? y f x dx 5 (A) 1 f x dx 5 (B) Left Riemann sum approximation of 1 (C) Right Riemann sum approximation of f x dx 5 1 (D) Midpoint Riemann sum approximation of f x dx 5 1 (E) Trapezoidal sum approximation of f x dx 5 1 x

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posted: | 7/28/2012 |

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