Modeling Accumulations: Introduction to the Issues 11/07/2011

Transcription

1 Modeling Accumulations: Introduction to the Issues /07/0

2 The purpose of calculus is twofold:. to find how something is changing, given what it s doing;. to find what something is doing, given how it s changing. We did () geometrically and algebraically. We did () algebraically. Let s do () geometrically!

3 If you travel at mph for 4 hours, how far have you gone?

4 If you travel at mph for 4 hours, how far have you gone? Answer: 8 miles.

5 If you travel at mph for 4 hours, how far have you gone? Answer: 8 miles. Another way: 3 4 (graph of speed, i.e. graph of derivative)

6 If you travel at mph for 4 hours, how far have you gone? Answer: 8 miles. Another way: Area = (graph of speed, i.e. graph of derivative)

7 If you travel at mph for hours, and mph for hours, how far have you gone? Area = +4= (graph of speed, i.e. graph of derivative)

8 If you travel at.5 mph for hour, mph for hour,.5 mph for hour, mph for hour, how far have you gone? Area = = (graph of speed, i.e. graph of derivative)

9 If you travel at.75 mph for /4 hour,.5 mph for /4 hour,... mph for /4 hour, how far have you gone? Area = = (graph of speed, i.e. graph of derivative)

10 If you travel at t mph for hours, how far have you gone? Area = 4 (it s a triangle) 3 4 (graph of speed, i.e. graph of derivative)

11 Choose another sequence of speeds: 3 4

12 Choose another sequence of speeds: 3 4

13 Choose another sequence of speeds: 3 4

14 Choose another sequence of speeds: 3 4

15 Choose another sequence of speeds: 3 4

16 Choose another sequence of speeds: 3 4

17 Choose another sequence of speeds: y = 8 x,area=??? 3 4

18 Estimate the area under the curve y = 8 x between x =0andx =4: Area =??? 3 4

19 Estimate the area under the curve y = 8 x between x =0andx =4: Estimate : pick the highest point Area 8 3 4

20 Estimate the area under the curve y = 8 x between x =0andx =4: Estimate : pick two points Area +4 = 5 3 4

21 Estimate the area under the curve y = 8 x between x =0andx =4: Estimate 3: pick four points Area =

22 Estimate the area under the curve y = 8 x between x =0andx =4: Estimate 4: pick eight points Area =

23 Estimate the area under the curve y = 8 x between x =0andx =4: Estimate 5: pick sixteen points Area

24 Estimate the area under the curve y = 8 x between x =0andx =4: Estimate 6: pick thirty two points Area

25 Estimating the Area of a Circle with r = - 0 -

26 Estimating the Area of a Circle with r = Divide it up into rectangles: - 0 -

27 Estimating the Area of a Circle with r = Divide it up into rectangles: - 0 -

28 Estimating the Area of a Circle with r = Divide it up into rectangles: Estimate area of the half circle (f (x) = p x ) and mult. by. - 0

29 Estimating the Area of a Circle with r = Divide it up into rectangles: Estimate area of the half circle (f (x) = p x ) and mult. by. height = f(0) = A= - 0 height = f() = 0 base= #rect. Area 4 * = 4* 4*3 4*4 4*5 base=

30 Estimating the Area of a Circle with r = Divide it up into rectangles: Estimate area of the half circle (f (x) = p x ) and mult. by. - 0 #rect. Area 4 * = 4* 4*3 4*4 4*5

31 The Method of Accumulations Big idea: Estimating, and then taking a limit. Let the number of pieces go to i.e. let the base of the rectangle for to 0. This not only gives us a way to calculate, but gives us a proper definition of what we mean by area! Also good for volumes and lengths...

32 A small dam breaks on a river. The average flow out of the stream is given by the following: hours m 3 /s hours m 3 /s hours m 3 /s

33 A small dam breaks on a river. The average flow out of the stream is given by the following:

34 Over each time interval, we estimate the volume of water by Average rate 900 s 500 V = 500m 3 /s*900s

35 Over each time interval, we estimate the volume of water by Average rate 900 s

36 Over each time interval, we estimate the volume of water by Average rate 900 s hours m 3 hours m 3 hours m total=33,39,800

37 A tent is raised and has height given by xy over the grid where 0 < x < and0< y <. What is the volume of the tent?

38 A tent is raised and has height given by xy over the grid where 0 < x < and0< y <. What is the volume of the tent? Estimate via boxes! Volume = base *height. x y height = xy volume * 0 0 0* 0 0 0* * total volume 0

39 A tent is raised and has height given by xy over the grid where 0 < x < and0< y <. What is the volume of the tent? Estimate via boxes! Volume = base *height. x y height = xy volume * * * 4 4* total volume 9 0

40 A tent is raised and has height given by xy over the grid where 0 < x < and0< y <. What is the volume of the tent? Estimate via boxes! Volume = base *height. x y height = xy volume * * * * total volume 4.5 0