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ET 4.4

ET 4.4. Calculate the indefinite Integral. Calculate the definite Integral. What have we done so far?. Calculated. Indefinite integral. Calculated. Definite Integral. = 12 - 9 + 9. = 12. How is what we learned in the first half of the class connected to the second half?.

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ET 4.4

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  1. ET 4.4 Calculate the indefinite Integral Calculate the definite Integral

  2. What have we done so far? Calculated Indefinite integral

  3. Calculated Definite Integral

  4. = 12 - 9 + 9 = 12

  5. How is what we learned in the first half of the class connected to the second half? Slope of the Tangent Differential Calculus& Integral Calculus Area of some rectangles under a curve. Inverses

  6. Fundamental Theorem of Calculus Where is the c? + c + c = 12

  7. TI-89 or 2nd 7 F3 2: fnInt ( function, variable of integration, lower limit, upper limit)

  8. Graphical Approach Lower Limit? Upper Limit? X = 0 X = 2

  9. Assignments 4.4 • Day 1: 1-41 (eoo) • Day 2: 45-51 odd, 55, 57, 58, 59ab, 61, 65 • Day 3: 67-73, 75, 79, 85, 87, 89, 93 • Day 4: 97, 99-106

  10. READ: Let’s refresh your memory… Amounts to calculating the slope using the endpoints. Casually Stated: There is a point within an interval that gives the average rate of change for the entire interval.

  11. Inscribed rectangle: Area is less than the area under the curve. There exists a point c between a and b such that the area of the rectangle is equal to the area under the curve. a b Circumscribed rectangle: Area is greater than the area under the curve. c a b a b

  12. Mean Value Theorem for Integrals If f is continuous on [a, b], Then there exists a “c” in [a, b] such that Solve for f(c) ( c , f(c) ) f(c) Called the average value of f an [a, b] c a b b - a

  13. Find the average value of f(x) = 3x2 -2x on [ 1, 4 ] There is some c f(c)=16. ( 8/3, 16) f(x) = 3x2 -2x 16 = 3x2 -2x 3x2 -2x -16 = 0 (3x - 8)(x + 2)= 0 x = 8/3 x = - 2 = 16 What does 16 mean?

  14. The average value is not necessarily unique. c c a b

  15. Speed of Sound • 4 x + 34 • 295 • .75x + 278.5 • 1.5 x + 254.5 • 1.5 x + 404.5 • 0 < x < 11.5 • 11.5 < x < 22 • 22 < x < 32 • 32 < x < 50 • 50 < x < 80 (0, 341) s 350 340 330 320 310 300 290 280 S(x) = (50, 329.5) (32, 302.5) Speed of sound (m/sec) (11.5, 295) (22, 295) Interesting Note At first when you encounter higher altitudes the speed of sound decreases, then it levels off, then it speeds up, but even higher slows down again. (80, 284.5) 10 20 30 40 50 60 70 80 x Altitude (km)

  16. = 1/80 (3657 + 3097.5 + … + 9210) = 308 meters per second If you look up speed of sound you’ll find 330 m/sec 740 mi/hr This is at sea level Chuck Yeager was credited with being the first man to break the sound barrier in level flight on 14 October 1947, flying at an altitude of 45,000 ft. If he was flying any lower he wouldn’t have broke the sound barrier that day.

  17. Assignments 4.4 • Day 1: 1-41 (eoo) • Day 2: 45-51 odd, 55, 57, 58, 59ab, 61, 65 • Day 3: 67-73, 75, 79, 85, 87, 89, 93 • Day 4: 97, 99-106

  18. An alternative to finding these separately… Definite Integral as a function of x

  19. The Definite Integral as a number The Definite Integral as a function of x

  20. Now to find these integrals just substitute x=5, x=8, etc. Etc. Check work… Take derivative and get original function. Humm: derivative gave us f(x) not f(t). Although not an anticipated answer … the result leads us to the second fundamental theorem of calculus.

  21. The Second Fundamental Theorem of Calculus What do we call this integral? It is a function of x What will F’(x) equal? = f(x)

  22. NOPE What do you think F’(x) will equal? = 1/(x2)2 = 1/x4 Upper limit is x2 so we might anticipate f(x2) d/dx (x2) = 2x Expected F’(x) d/dx(Upper Limit) 1/x4 2x

  23. GivenFind F’(x). 3 3x Like the Chain Rule F’(x) = (f(upper limit) - f(lower limit)) (upper limit)’ Remind you of anything?

  24. Assignments 4.4 • Day 1: 1-41 (eoo) • Day 2: 45-51 odd, 55, 57, 58, 59ab, 61, 65 • Day 3: 67-73, 75, 79, 85, 87, 89, 93 • Day 4: 97, 99-106

  25. Area under the curve is ½ the rectangle w/area 1 93. Area under the curve 0 (0, 0) ½ (1, ½ ) 1 (2, 1) ½ (3, ½ ) 0 (4, 0)

  26. ET 4.4d What is the difference between total distance an object has traveled and displacement of an object?

  27. The velocity function, in feet per second, is given for a particle moving along a straight line. Find the displacement. This is a distance to the left.

  28. The velocity function, in feet per second, is given for a particle moving along a straight line. Find the total distance the particle travels over the given interval. |-47/3| + 32/3

  29. Hints #103) Given position function x(t) = t3- 6t2 + 9t – 2 such that 0 < t < 5. Asked to find the total distance the particle travels in 5 units of time. Note: Total distance is the area under the velocity function. How do you get from the position function to the velocity function? Derivative. x’(t)

  30. Hints #103) Givenrate of liters/min. function = 500 - 5t Asked to find the total water that flows out in the first 18 minutes. Note: Rate means you were given a velocity function. Rather than total time, the question is asking for total water, which is the area under the velocity function.

  31. Assignments 4.4 • Day 1: 1-41 (eoo) • Day 2: 45-51 odd, 55, 57, 58, 59ab, 61, 65 • Day 3: 67-73, 75, 79, 85, 87, 89, 93 • Day 4: 97, 99-106

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