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Clicker Question 1

Clicker Question 1. What is ? A. x tan( x 2 ) + C B. ½ (sec( x 2 ) + tan( x 2 )) + C C. ln |sec( x 2 ) + tan( x 2 )| + C D. ½ tan( x 2 ) + C E. ½ ln |sec( x 2 ) + tan( x 2 )| + C. Clicker Question 2. What is ? A. x 2 tan(x) + C

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Clicker Question 1

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  1. Clicker Question 1 • What is ? • A. x tan(x2) + C • B. ½ (sec(x2) + tan(x2)) + C • C. ln |sec(x2) + tan(x2)| + C • D. ½ tan(x2) + C • E. ½ ln |sec(x2) + tan(x2)| + C

  2. Clicker Question 2 • What is ? • A. x2 tan(x) + C • B. x tan(x) + sec2(x) + C • C. x tan(x) – ln |sec(x)| + C • D. ½ x2 tan(x) + C • E. x tan(x) + ln |sec(x)| + C

  3. Numerical (or Approximate) Integration (2/14/14) • To evaluate a definite integral, we always hope to apply the Fundamental Theorem by finding an antiderivative and then evaluating it at the endpoints. But this isn’t always possible or practical. • Another option always available is to do numerical integration to approximate the answer. Here we use Riemann sums.

  4. Getting Good Approximations • In general, the more data you use (i.e., the more rectangles you measure), the better the estimate. • Methods (in increasing order of accuracy for a fixed number of rectangles) • Left or right-hand endpoints • Trapezoid Rule (average of the above) • Midpoint Rule • Simpson’s Rule

  5. Simpson’s Rule • It turns out that the Midpoint Rule is about twice as accurate as the Trapezoid Rule and the errors are in opposite directions. • Hence we form a “weighted average”:Simpson’s Rule = (2 Midpoint + Trapezoid) / 3 • Simpson’s Rule is in general extremely accurate even for small n (i.e., few rectangles). • Can get Simpson directly from the data using(1 + 4 + 2 + 4 + 2 +…+ 2 + 4 + 1) / 6

  6. Clicker Question 3 • Suppose a definite integral has the following estimates for 10 subdivisions of the interval:Left-hand sum: 12.5Right-hand sum: 16.5Midpoint Rule sum: 13.0What is the Simpson’s Rule estimate? • A. 13.5 B. 13.0 C. 13.75 D. 14.0 E. 13.25

  7. Assignment for Monday • Finish up Hand-in #2 (due Monday in class) • Section 7.7 goes into much more detail on approximate integration than we need to, and defines Simpson’s Rule slightly differently, so just work from our notes please. • Do Exercise 3 on page 516 (note: no known way to use FTC on this one), but also compute the Simpson’s Rule estimate with n = 4 (i.e., use 9 data points) both using the weighted average of Trap and Mid and the (1-4-2-4-2-4-2-4-1) / 6 method.

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