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AP Calculus Problem Set 48 Ch4 I 1112

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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

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                                                                       




                                                           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

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                                                           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




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                                                                 



                                                          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




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                                                          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                       At 
(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
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



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                                                                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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