Clicker Question 1

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# Clicker Question 1 - PowerPoint PPT Presentation

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
• 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
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
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.
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
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
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
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.