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Worksheet: The Trapezoidal Rule

Q1:

The following table shows how the midpoint and trapezoidal rule perform on estimating ๏„ธ 1 1 + ๐‘ฅ ๐‘ฅ ๏Šง ๏Šฆ ๏Šจ d . The error is the difference from the actual value of the integral ๐œ‹ 4 .

Intervals Mid E r r ร— 1 0 ๏Šฎ Trap E r r ร— 1 0 ๏Šฎ
2 0.790588 โˆ’ 5 1 8 9 8 3 . 6 6 0 3 0.775 1 0 3 9 8 1 6 . 3 4
8 0.7857236 โˆ’ 3 2 5 4 3 . 6 6 0 2 6 0.7847471 6 5 1 0 6 . 3 3 9 7 4
32 0.785418 โˆ’ 1 9 8 3 . 6 6 0 2 5 5 0.785357 4 1 1 6 . 3 3 9 7 4 5
128 0.785399 โˆ’ 8 3 . 6 6 0 2 5 5 1 7 0.785396 216.3397448
512 0.785398 16.33974482 0.785398 16.33974482

What appears to be true of the ratio of successive errors E r r E r r ( ๐‘› ) ( 4 ๐‘› ) for the midpoint rule?

  • A The error is 16 times as much for 4 ๐‘› intervals as for 4 ๐‘› intervals.
  • B The error is 16 times as much for 4 ๐‘› intervals as for ๐‘› intervals.
  • C The error is 4 times as much for ๐‘› intervals as for 4 ๐‘› intervals.
  • D The error is 16 times as much for ๐‘› intervals as for 4 ๐‘› intervals.
  • E The error is 4 times as much for 4 ๐‘› intervals as for ๐‘› intervals.

What appears to be true of the ratio of successive errors E r r E r r ( ๐‘› ) ( 4 ๐‘› ) for the trapezoidal rule?

  • AThe error is 16 times as much for ๐‘› intervals as for 4 ๐‘› intervals.
  • BThe error is 4 times as much for ๐‘› intervals as for 4 ๐‘› intervals.
  • CThe error is 16 times as much for 4 ๐‘› intervals as for 4 ๐‘› intervals.
  • DThe error is 16 times as much for 4 ๐‘› intervals as for ๐‘› intervals.
  • EThe error is 4 times as much for 4 ๐‘› intervals as for ๐‘› intervals.

The midpoint rule underestimates the integral and the trapezoidal rule overestimates it. What geometric property of the graph of ๐‘“ ( ๐‘ฅ ) = 1 1 + ๐‘ฅ ๏Šช explains this?

  • AThe graph is concave down.
  • B The graph is concave up.

For a fixed number of intervals ๐‘› , what appears to be the relation between the midpoint rule and the trapezoidal rule errors?

  • AThe trapezoidal error is โˆ’ 2 times the midpoint error.
  • BThe trapezoidal error is โˆ’ 4 times the midpoint error.
  • CThe trapezoidal error is 1 2 times the midpoint error.
  • DThe trapezoidal error is โˆ’ 1 2 times the midpoint error.
  • EThe trapezoidal error is 2 times the midpoint error.

Simpsonโ€™s rule can be expressed as the weighted average 2 + 3 M i d T r a p . Using the table above with ๐‘› = 8 , we get a Simpsonโ€™s error of 0 . 0 3 ร— 1 0 ๏Šฑ ๏Šฎ . Using technology, find the actual error to 3 decimal places.

  • A 0 . 5 9 ร— 1 0 ๏Šฑ ๏Šฎ
  • B ร— 1 0 ๏Šฑ ๏Šฎ
  • C0.059
  • D 0 . 0 5 9 ร— 1 0 ๏Šฑ ๏Šฎ
  • E0.59

Q2:

For a fixed function on a given interval, let T r a p ( ๐‘› ) be the estimated integral using the trapezoidal rule with ๐‘› subintervals. Use the following diagram to relate the concavity of ๐‘“ with how well T r a p ( ๐‘› ) estimates ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž d .

  • A T r a p d ( ๐‘› ) = ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž if ๐‘“ is concave down on [ ๐‘Ž , ๐‘ ] .
  • B T r a p d ( ๐‘› ) โ‰ฅ ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž if ๐‘“ is concave down on [ ๐‘Ž , ๐‘ ] .
  • CThere is no conclusion about the comparison between the estimate and actual values.
  • D T r a p d ( ๐‘› ) โ‰ค ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž if ๐‘“ is concave down on [ ๐‘Ž , ๐‘ ] .

Q3:

Consider a function whose graph ๐‘ฆ = ๐‘“ ( ๐‘ฅ ) is concave up on the interval ๐น ๐ธ .

The line ๐ด ๐ต that determines points ๐ด and ๐ต is the tangent line to ๐‘ฆ = ๐‘“ ( ๐‘ฅ ) over the midpoint ๐‘€ of segment ๐น ๐ธ .

Which quadrilateral has the area given by T r a p ( 1 ) , the trapezoidal rule estimate of the integral ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๏Œค ๏Œฅ d ?

  • A ๐ด ๐ต ๐ธ ๐น
  • B ๐ด ๐ต ๐ธ ๐น โˆ— โˆ—
  • C ๐ถ ๐น ๐ธ ๐ต
  • D ๐ถ ๐ท ๐ธ ๐น
  • E ๐ถ ๐น ๐ธ ๐ต โˆ—

Which quadrilateral has the area given by M i d ( 1 ) , the midpoint rule estimate of the integral ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๏Œค ๏Œฅ d ?

  • A ๐ด ๐ต ๐ธ ๐น โˆ— โˆ—
  • B ๐ถ ๐น ๐ธ ๐ต
  • C ๐ด ๐ต ๐ธ ๐น
  • D ๐ถ ๐ท ๐ธ ๐น
  • E ๐ถ ๐น ๐ธ ๐ต โˆ—

Why is Area ( ๐ด ๐ต ๐ธ ๐น ) = โˆ— โˆ— Area ( ๐ด ๐ต ๐ธ ๐น ) ?

  • Abecause the triangles ๐‘‹ ๐ด ๐ด โˆ— and ๐‘‹ ๐ต ๐ต โˆ— are congruent
  • Bbecause ๐‘‹ ๐ด ๐ด โˆ— and ๐‘‹ ๐ท ๐ต โˆ— are equal
  • Cbecause the quadrilaterals have the common base ๐ธ ๐น
  • Dbecause ๐ด ๐ต = ๐ด ๐ต โˆ— โˆ—

What relationship can you deduce about the numbers M i d ( 1 ) , T r a p ( 1 ) , and ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๏Œค ๏Œฅ d in the case where the graph is concave up?

  • A M i d d T r a p ( 1 ) < ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ < ( 1 ) ๏Œค ๏Œฅ
  • B M i d d T r a p ( 1 ) > ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ < ( 1 ) ๏Œค ๏Œฅ
  • C M i d d T r a p ( 1 ) > ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ > ( 1 ) ๏Œค ๏Œฅ
  • D M i d d T r a p ( 1 ) < ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ > ( 1 ) ๏Œค ๏Œฅ

Q4:

Calculate the trapezoidal rule estimate of ๏„ธ ๐‘ฅ + 2 ๐‘ฅ 4 0 2 d with ๐‘› = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A48, an overestimate
  • B28, an underestimate
  • C16, an underestimate
  • D32, an overestimate
  • E28, an overestimate

Q5:

For a fixed function on a given interval, let T r a p ( ๐‘› ) be the estimated integral using the trapezoidal rule with ๐‘› subintervals. Use the following diagram to relate the concavity of ๐‘“ with how well T r a p ( ๐‘› ) estimates ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž d .

  • A T r a p d ( ๐‘› ) = ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž if ๐‘“ is concave up on [ ๐‘Ž , ๐‘ ] .
  • B T r a p d ( ๐‘› ) โ‰ค ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž if ๐‘“ is concave up on [ ๐‘Ž , ๐‘ ] .
  • CThere is no conclusion about the comparison between the estimate and actual values.
  • D T r a p d ( ๐‘› ) โ‰ฅ ๏„ธ ๐‘“ ( ๐‘ฅ ) ๐‘ฅ ๐‘ ๐‘Ž if ๐‘“ is concave up on [ ๐‘Ž , ๐‘ ] .