Elem Calc w/trig II Spring 08 Class time: TR 9:30 am to 10:45 am Classroom: Whit 257 Helmi Temimi: PhD Teaching Assistant, 463 C

Transcription

1 Elem Calc w/trig II Spring 08 Class time: TR 9:30 am to 10:45 am Classroom: Whit 257 Helmi Temimi: PhD Teaching Assistant, 463 C temimi@vt.edu Office Hours 11:00 am 12:00 pm Course contract, Syllabus

2 Riemann Sums and Accumulated Change

3 Math 1016 Given data, find the rate of change. Math 2015 Given a rate of change, find the total change. Example: Velocity and Distance Traveled

4 Recovering Distance From Velocity If we travel 2 feet per second for 4 seconds, Then we traveled a total of 8 feet. Why?

5 Recovering Distance From Velocity Model Train Data: t (sec) v (ft/sec) How far does it travel between t =4 and t =8?

6 Recovering Distance From Velocity Three ways to answer: t (sec) v (ft/sec)

7 Recovering Distance From Velocity Three ways to answer: t (sec) v (ft/sec) Use the first velocity:

8 Recovering Distance From Velocity Three ways to answer: t (sec) v (ft/sec) Use the first velocity: (2 feet/sec)(4 sec) = 8 feet

9 Recovering Distance From Velocity Three ways to answer: t (sec) v (ft/sec) Use the first velocity: (2 feet/sec)(4 sec) = 8 feet Use the middle velocity:

10 Recovering Distance From Velocity Three ways to answer: t (sec) v (ft/sec) Use the first velocity: (2 feet/sec)(4 sec) = 8 feet Use the middle velocity: (3 feet/sec)(4 sec) = 12 feet

11 Recovering Distance From Velocity Three ways to answer: t (sec) v (ft/sec) Use the first velocity: (2 feet/sec)(4 sec) = 8 feet Use the middle velocity: (3 feet/sec)(4 sec) = 12 feet Use the final velocity:

12 Recovering Distance From Velocity Three ways to answer: t (sec) v (ft/sec) Use the first velocity: (2 feet/sec)(4 sec) = 8 feet Use the middle velocity: (3 feet/sec)(4 sec) = 12 feet Use the final velocity: (3.5 feet/sec)(4 sec) = 14 feet

13 A Useful Aside: Suppose we know the train is speeding up. We Claim: It must have gone at least 8 feet, and less than 14 feet. Is it right? Why or why not?

14 A Useful Aside: Suppose we know the train is speeding up. We Claim: It must have gone at least 8 feet, and less than 14 feet. Is it right? Why or why not? Using first velocity: 8 feet Using middle velocity: 12 feet Using final velocity: 14 feet

15 It s Right! If it s speeding up, the first speed is the lowest, and the last is the highest.

16 It s Right! If it s speeding up, the first speed is the lowest, and the last is the highest. If it s always faster than 2 ft/sec, the train travels more than (2 feet/sec)(4 sec) = 8 feet

17 It s Right! If it s speeding up, the first speed is the lowest, and the last is the highest. If it s always faster than 2 ft/sec, the train travels more than (2 feet/sec)(4 sec) = 8 feet If it s always going slower than 3.5 ft/sec, the train travels less than (3.5 feet/sec)(4 sec) = 14 feet

18 But what if it s not speeding up? Then anything might happen! The total distance might be less than 8 feet

19 But what if it s not speeding up? Then anything might happen! The total distance might be less than 8 feet Or more than 14 feet!

20 Subintervals How far just from t =4 to t =6? t (sec) v (ft/sec)

21 Subintervals How far just from t =4 to t =6? t (sec) v (ft/sec) Using the starting v:

22 Subintervals How far just from t =4 to t =6? t (sec) v (ft/sec) Using the starting v: Using the ending v:

23 Subintervals What about from t =6 to t =8? t (sec) v (ft/sec)

24 Subintervals What about from t =6 to t =8? t (sec) v (ft/sec) Using the starting v:

25 Subintervals What about from t =6 to t =8? t (sec) v (ft/sec) Using the starting v: Using the ending v:

26 Subintervals The total distance traveled is approximately t (sec) v (ft/sec)

27 Subintervals The total distance traveled is approximately t (sec) v (ft/sec) If we use the first velocity in each subinterval:

28 Subintervals The total distance traveled is approximately t (sec) v (ft/sec) If we use the first velocity in each subinterval: (2 x 2) + (3 x 2) = 4ft + 6 ft = 10 ft. Called a Left-hand sum.

29 Subintervals The total distance traveled is approximately t (sec) v (ft/sec)

30 Subintervals The total distance traveled is approximately t (sec) v (ft/sec) If we use the last velocity in each subinterval:

31 Subintervals The total distance traveled is approximately t (sec) v (ft/sec) If we use the last velocity in each subinterval: (3 x 2) + ( 3.5 x 2) = = 13 feet Called a Right-hand sum.

32 Left and Right hand sums are both examples of Riemann sums, after this guy. Georg Friedrich Bernhard Riemann September 17, 1826 July 20, 1866

33 Question: 1) If the train is speeding up, is either sum an over or underestimate of the distance?

34 Question: 1) If the train is speeding up, is either sum an over or underestimate of the distance? The left-hand sum is an underestimate; The right-hand sum is an overestimate.

35 Question: 2) What if the train were slowing down instead?

36 Question: 2) What if the train were slowing down instead? If the train were slowing down, The left-hand sum would be an overestimate; The right-hand sum would be an underestimate.

37 Improving our Estimates We can improve our estimates by:

38 Improving our Estimates We can improve our estimates by: averaging left and right hand sums, or

39 Improving our Estimates We can improve our estimates by: averaging left and right hand sums, or using more subintervals.

40 More Data Suppose we collect additional data about our train s velocity: t v

41 Left-Hand Sum Table t v Interval t t v Distance #1 [4, 5.5]

42 Left-Hand Sum Table Which subinterval t v Interval t t v Distance #1 [4, 5.5]

43 Left-Hand Sum Table t v Interval t t v Distance #1 [4, 5.5] What t values

44 Left-Hand Sum Table t v Subinterval width Interval t t v Distance #1 [4, 5.5]

45 Left-Hand Sum Table t v Interval t t v Distance #1 [4, 5.5] Velocity for this subinterval

46 Left-Hand Sum Table t v Interval t t v Distance #1 [4, 5.5] Traveled in this subinterval

47 Left-Hand Sum t v Interval t t v Distance #1 [4, 5.5] #2 [5.5, 6] 0.5? 1.35 #3 [6, 6.2] #4 [6.2, 7]? #5????

48 Left-Hand Sum t v Interval t t v Distance #1 [4, 5.5] #2 [5.5, 6] #3 [6, 6.2] #4 [6.2, 7] #5 [7, 8]

49 Left-Hand Sum Interval t t v Distance #1 [4, 5.5] #2 [5.5, 6] #3 [6, 6.2] #4 [6.2, 7] #5 [7, 8] Total: 10.73

50 Right-Hand Sum t v Interval t t v Distance #1 [4, 5.5] 1.5? 4.05 #2 [5.5, 6] #3 [6, 6.2] #4 [6.2, 7] #5 [7, 8]

51 Right-Hand Sum t v Interval t t v Distance #1 [4, 5.5] #2 [5.5, 6] #3 [6, 6.2] #4 [6.2, 7] #5 [7, 8]

52 Right-Hand Sum Interval t t v Distance #1 [4, 5.5] #2 [5.5, 6] #3 [6, 6.2] #4 [6.2, 7] #5 [7, 8] Total: Average of LHS and RHS: feet

53 Pictures On the velocity graph: Velocity is height t is width So v(t) t is the area of a rectangle! Note: Height depends on which velocity we use!

54 Pictures Riemann sum rectangles, t = 4 and n =1:

55 Pictures Riemann sum rectangles, t = 2 and n =2:

56 Pictures Riemann sum rectangles, n =5 ( t varies)

57 Accumulated Change Velocity is the rate of change of position. We can apply Riemann sums to any other rate of change. A Riemann sum applied to a rate of change estimates the total change over the interval.

58 Example: A Reservoir Filling at variable rate, starting with 1,000 gallons at noon. Total volume at 1:00? Time (minutes past noon) Rate (gal/min)

59 Example: A Reservoir First, estimate the change: Left-hand sum: Right-hand sum: Best Estimate: Time (minutes past noon) Rate (gal/min)

60 Example: A Reservoir Total volume at 1:00 is the change added to the original volume:

61 Example: A Reservoir Total volume at 1:00 is the change added to the original volume: 1, ,310 = 2,310 gallons

62 Summary Approximate total change from rate of change using left/right hand sums. For increasing functions, left hand sums underestimate and right hand overestimate. Left hand sum: multiply function at left of each subinterval by width of subinterval. Riemann sums look like areas of rectangles.

63 Assignment Go to the Math 2015 Homepage: 1. Read the Announcements - after every class 2. Read and complete the Assignment

64 In-Class Assignment A car starts moving at time t=0 and goes faster and faster. Its velocity is shown in the following table. Estimate how far the car travels during the 12 seconds? t (sec) Velocity (ft/sec)