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Good day students will come two-part three on our rectangle approximations of now let's take a look at the instructions for the question we are to estimate the distance and object with a given time velocity chart covers in the first ten seconds of new movements using the left rectangular approximation okay so we are to use this time velocity chart provided right here to estimate the distance covered in the first ten seconds. So in what way are we going to use LRAM in this calculations so to do that lets take a look at some formulas first and then we can see the justification of our application of LRAM and the situation right so when connecting position velocity acceleration and jerk we talked about the savants continuum SB AJ representing position velocity expiration jerk away go down in this direction a differentiate in the anti-differentiate when going in the other direction okay so let's see what it is go from position to velocity how do you do that the have position of if you differentiate position with respect to time that gives you the velocity function right now what of if I wanted to get my position isolated how can I get position is a velocity as indicated in this the charts right here so let's start off by multiplying both site in this equation by DT okay and that yields DS equals the dt. Right so to get rid of that is the components were going to anti-differentiate so we're going to take the anti-derivative of both sides of the definite integral of both sides in the we going to have a cancellation action happening here is interval oriented derivative counsel that will the derivative on the left side will have the position S equals the integral of velocity the DT okay so the integrates the jerk function to get the expiration in the integrated solution function you get the velocity and if you integrate the velocity function you get skewed position function as indicated here right sell it way to find a position from 0 to 10 as indicated in this problem we going to calculate the definite integral from 0 to 10 of the velocity function right since we do not have a velocity function is going to be making an approximation as indicated in the problem is going to find the Valley of this definite integral using the left rectangular approximation okay from 0 to 10 right so what is LRAM let's go ahead and review that formula real quick in the search and its applicability to this situation, under consideration here okay so LRAM left rectangular approximation let's see we have X of an interval's it's going to be given by the with times F of the pick all the points to the left of the right endpoint okay so is can a look something like this the with times F of X one plus F of X to all the way to the X value before the right endpoint which we can in the case with X sub and minus one okay now you have to be really careful to note that this formula is for and equal sub interval's right and equal sub interval's now let's let's represent discharges in our interval notation interval chart and then see if we can use is from our here okay right so we can make an x-axis, interval this would help meet a certain my my my X my inputs into my LRAM formula okay so it does say this one is for over to the left a little bit right rights I have a number line, going to partition it from were going from zero all the way to eleven so let's say this is zero zero companies two boxes here case zero one two three four five six seven eight nine ten eleven rights is that's eleven so let me take a tour these numbers here pick up the one to take a look at so let's see where going from zero all the way to two I just want to put the numbers that show up on our chart here so don't get confused so the from zero two two and then from two two three and there from 3 to 6 four five six from 6 to 9 seven eight nine from 9 to 10 and there from 10 to 11 know what you notice about this interval that we have right here this is one interval this is another this is another interval is another interval
Tagged under: area,curve,estimating,finite,sum,average,dummy,variable,error bounds,Riemann,simpson' rule,sigma,upper,limit,bound,partition,norm,trapezoidal,region,definite,integral,integration,antiderivative,approximating,hand,left,LrAM,Method,numerical integration
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