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Review 6

Review 6. Riemann Sums and Trapezoidal Rule Definite integrals (limits). Trapezoidal Rule when the intervals are the same. You calculate h, then:. Trapezoidal Rule when the intervals are different. You calculate individual trapezoids and add them all up. LRAM

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Review 6

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  1. Review 6 • Riemann Sums and Trapezoidal Rule • Definite integrals (limits)

  2. Trapezoidal Rule when the intervals are the same. • You calculate h, then:

  3. Trapezoidal Rule when the intervals are different. • You calculate individual trapezoids and add them all up.

  4. LRAM Calculate h, then add up all of the heights, starting on the far left and ending just prior to the endpoint.

  5. RRAM Calculate h, then add up all of the heights, starting on the far right and ending just prior to the endpoint.

  6. MRAM Calculate h, then add up all of the heights by finding the midpoint of each individual interval.

  7. A test plane flies in a straight line with positive velocity v(t) in miles per minute. Selected values are given. • All are counting by 5 minutes, so h = 5 mintues – even intervals.

  8. A. Give an estimate for the total miles traveled using a left sum. • Always write out the integral. • Total miles traveled in 40 minutes.

  9. B. Give an estimate for the total miles traveled using a right sum. • Total miles traveled in 40 minutes.

  10. C. Give a total estimate using 4 sub intervals of equal length using a midpoint sum. • Total miles traveled in 40 minutes.

  11. D. Use a trapezoidal sum to calculate the total distance traveled. • Total miles traveled in 40 minutes.

  12. 2008 #2 revisted. • Concert tickets are sold starting at noon (t = 0) and are sold out in 9 hours. The number of people waiting in line can be modeled by the following table.

  13. A. Find the total number of people waiting in line using a left sum. • people

  14. B. Find the total number of people waiting in line using a right sum. • people

  15. C. Find the total number of people waiting in line using a midpoint sum with three intervals. • Midpoint – 3 intervals so the midpoints are inbetween 0-3, 3-7, 7-9. The first midpoint will be 3 wide, then 4 wide, then 2 wide. people

  16. D. Use a trapezoidal sum to estimate the number of people waiting in line for the first 4 hours. people

  17. Average? 1/(b-a) times the answer • Find the average number of people waiting in line for the first 4 hours using a trapezoidal sum. • people

  18. Average Value of f(x) on [a, b]

  19. Fundamental Theorem • of Calculus Part 2

  20. Solving for area under the curve • Find the anti-derivative • Evaluate with the limits • Units? The time goes away if it’s velocity, or the power decreases on the time if it’s acceleration. • Units – unit’s squared – measurement squared. • Might be pulling information from a graph! It’s the area under the curve! • If a starting point is given, then add up all of the areas as is! + and – as given.

  21. Fundamental Theorem of Calculus, Part 1 • means

  22. Example a • The graph of f is given. Let g be the function given by • This means that f(t)=g’(x). So whenever you need a value for g’(x) you just read the graph. g(x) will be the area under the curve, starting at 2.

  23. Example a (continued) • Find g(3), g’(3) and g’’(3). • Write the tangent line for x = 3. • g(3) is the integral, g’(3) is on the graph, g’’(3) is the slope of the curve that goes through 3.

  24. Example a continued • So g(3) • g’(3) = 2 (from the graph) • g’’(3) is • The point is (3, 3) with a tangent slope of 2 so the tangent line is y = 2(x – 3) +3

  25. Example b • The graph of f consists of six line segments. Let g be the function given by • Again, f(t) = g’(x) • The function is starting at 0, so all area calculations must start at 0.

  26. Example b (continued) • Find g(4), g’(4), g’’(4) • g(4) is the integral which will be made up of a triangle that has negative area and a trapezoid. • g’(4) is from the graph and is 0 • g’’(4) is the slope of the curve going through 4

  27. Example C • The graph of f consists of three line segments. Let g be the function given by • And let h be the function given by • Same pattern as previous 2 examples.

  28. Example c (continued) • Find g(1) and g’(1) • g(1) is the integral • g’ (1) is from the graph which is 2.

  29. Find all intervals where h is decreasing. • Since the integral has x in the lower limit, the function must be read right to left, or you make the integral negative and switch places with the limits. Thus, anytime the curve is above the x-axis, the actual function is decreasing. You must mention that h’ = -f whenever f >0,. So h decreases from (0, 2)

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