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Worksheet: Riemann Sum

Q1:

Suppose that 𝐹 β€² ( π‘₯ ) = 3    and 𝐹 ( 0 ) = 7 . Find two estimates of 𝐹 ( 1 ) to three decimal places using the left and then the right endpoint methods with 8 rectangles.

  • A7.769, 7.843
  • B7.843, 7.686
  • C0.7689, 0.686
  • D7.769, 7.686
  • E7.686, 7.769

Q2:

Calculate the midpoint rule estimate of ο„Έ π‘₯ + 2 π‘₯ 4 0 2 d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A16, an underestimate
  • B48, an overestimate
  • C28, an overestimate
  • D28, an underestimate
  • E16, an overestimate

Q3:

Calculate the right endpoint estimate of ο„Έ π‘₯ + 2 π‘₯ 4 0 2 d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A48, an underestimate
  • B16, an underestimate
  • C16, an overestimate
  • D48, an overestimate
  • E13, an underestimate

Q4:

The table gives sampled values of an increasing function 𝑓 . Use the data to give a lower and upper bound for ο„Έ 𝑓 ( π‘₯ ) π‘₯ 2 5 1 0 d .

π‘₯ 10 13 16 19 22 25
𝑓 ( π‘₯ ) βˆ’ 1 1 βˆ’ 3 1 4 8 10
  • Alower bound: βˆ’ 1 2 , upper bound: 66
  • Blower bound: βˆ’ 6 0 , upper bound: 3
  • Clower bound: βˆ’ 1 , upper bound: 20
  • Dlower bound: βˆ’ 3 , upper bound: 60
  • Elower bound: βˆ’ 2 0 , upper bound: 1

Q5:

The table gives sampled values of an increasing function 𝑓 . Use the data to give a lower and upper bound for ο„Έ 𝑓 ( π‘₯ ) π‘₯ 2 2 1 2 d .

π‘₯ 12 14 16 18 20 22
𝑓 ( π‘₯ ) βˆ’ 1 4 βˆ’ 1 1 βˆ’ 5 1 6 15
  • Alower bound: βˆ’ 5 2 , upper bound: 20
  • Blower bound: βˆ’ 1 2 , upper bound: 46
  • Clower bound: βˆ’ 2 3 , upper bound: 6
  • Dlower bound: βˆ’ 4 6 , upper bound: 12
  • Elower bound: βˆ’ 6 , upper bound: 23

Q6:

Calculate the left endpoint estimate of ο„Έ π‘₯ + 2 π‘₯ 4 0 2 d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A16, an overestimate
  • B18, an underestimate
  • C48, an underestimate
  • D16, an underestimate
  • E48, an overestimate