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anonymous
 5 years ago
Need help with the stepbystep of using the Midpoint Rule to find the distance traveled in the following scenario: v(t) = 3t5, 0<x<3
anonymous
 5 years ago
Need help with the stepbystep of using the Midpoint Rule to find the distance traveled in the following scenario: v(t) = 3t5, 0<x<3

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yes :D, therefore, the displacement is just the antiderivative of that function, right? My question is, how do I utilize the Midpoint Rule to find the total distance traveled, as opposed to the displacement?

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.0correct you could find distance traveled as \[\int\limits_{0}^{3}v(t) dt\] do you mean the mean value theorem

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0well, for the displacement, I used the integral and the evaluation theorem.. but I'm stumped for the distance. do I used the mean value theorem? I was given a clue, the "midpoint rule," but I'm not sure how to approach it..

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I must say, I actually don't think the midpoint rule has anything to do with this, haha.. do you guys agree? I am totally lost then

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.0Since the displacement does equal total distance then there must be a change in direction. find where v(t) = 0 as the turnaround point

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.0If t* is the time of direction change then you are looking for x(t*) + (x(t*)x(3)) where x(t) is antiderivative of v

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0To find th distance traveled you would normally use the arc length equation: \[\int\limits_{0}^{3}\sqrt{1+[v'(t)]^{2}}dt\] which yields 3sqrt(10) However this time you will be doing an approximation using the midpoint rule. Imagine the line y=3t5 and draw three rectangles under, spanning from 0 to 1, then 1 to 2, then 2 to 3. Then take the midpoint of each of these intervals. 1/2, 3/2 etc. If you name the midpoints t1, t2, t3 you can drive the following formula for the length: You know the distance between the sucessive t values is always 1 since this is what i described in the drawing above. So by pythagoren theorem L=sqrt(1+[v(t)v(t1)]^2). Evaluate this for t=3/2 and t=5/2 normally but half the values when you evaluate for n=1/2 and t=7/2 because the interval they measure is too big (for example with the t=1/2 you are calculating the length of the interval 1/2 to 1/2 when you actually want the lower bound to be 0) Sum all these values and you get 3sqrt(10)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Xavier, that's what I originally got! But the answer is actually "41/6 meters." I have an idea if the displacement is the total area under the graph between 0 and 3 (i.e. the positive area  the negative area), would the distance traveled be just the positive area??

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I need to go to bed.. I'm going to try this in the morning. Thanks so much for the help guys. I'm a fan. :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I thought of the distance travelled as the length of the line not the area under. I don't know why its 41/6. I mean if you take advantage of the fact that its linear and use the distance formula you get sqrt[(30)^2(4(5))^2]=3sqrt(10).

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.0use the method i used i got 41/6

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ah, dumbcow I didn't even see that post! I'll try that! thank you!

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Wow...v(t) is the velocity. I feel dumb for missing that for this long....thought it was just any old function. Got 41/6 too. Sorry

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.0"f the displacement is the total area under the graph between 0 and 3 (i.e. the positive area  the negative area), would the distance traveled be just the positive area??" that is correct

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0If i think about that one logically i'd say no. If i go forward then walk backwards the same distance, the displacement is 0. So is the area under the graph. But the distance i walked is the sum of the positive and abs(negative area)

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.0sorry i didn't read that carefully enough you are right xavier its pos area + neg area

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Well the positive value of the neg area. I checked using triangles: .5*5*(5/3)+.5(3(5/3))*4=41/6. I guess since you know the area is going to be negative on that on that certain interval you can do: \[\int\limits_{0}^{5/3}(3x5 )dx + \int\limits_{5/3}^{3} (3x5) dx = 41/6\] except with all the xs replaces with ts >__< Also you need to use the midpoint rule instead of definite integrals: http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0thx guys, got it!!! you can give yourselves huge pats on the back!!
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