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now ready for Part B and I've wiped all the salt off of my fingers and I've had a glass of water so now I'm ready for some business find the second derivative of W with respect to T in terms of W use this second derivative to determine whether your answer in Part A is an underestimate or an overestimate for the amount of solid waste that the landfill contains at time T equals 1/4 so let's do the first things first let's find the second derivative in terms of W and we already have the first derivative over here and in Part A I rewrote it just with slightly different notation but I'll just use this just because I need to write down here and I can still refer to this thing so let's just take the derivative of both sides of this of this differential equation with respect to T so you take the derivative of this left-hand side with respect to T you get the second derivative of W the second derivative of W with respect to T as a function of T is equal to now you take the derivative of this with respect to T this is the same thing as 1 over 25 w t minus 12 and so when you take the derivative that constant part drops off and you're just left with you're just left with 1 over 25 times the derivative of W T and we making it let me make that clear this is the same thing as 1 over 25 W 1 over 25 W as a function of T minus 12 125th of 300 is 12 and you take the derivative of this you get 125th times the first derivative of W and then the derivative of this with respect to T is just 0 a constant obviously does not change with respect to T and so we get this right over here now this is the second derivative in terms of the first derivative but the question asks us write the second derivative in terms of W but lucky for us we know how to express this we know how to express this as a function of W they gave that to us in the problem this was given this is just me rewriting it in a different notation so this is going to be the same thing as 1 over 25 1 over 25 times the derivative of W and the derivative of W is this this is the differential equation Liberto so the derivative of you is this over here so one over 25 times 1 over 25 w t or the function W is a function of t minus minus 300 and so we can say the second derivative let me write a little smaller so don't waste space the second derivative of W as a function of T is equal to 1 over 625 times W as a function of T which is a function of t minus 300 so we've done the first part we've found out the second derivative of W in terms of just W now let's try to address the second part of their question so we did the first part now use what we just figured out to determine whether your answer in Part A is an underestimate or an over estimate of the amount of solid waste that the landfill contains at time 1/4 so in Part A we found the slope we found the slope of the tangent line at time equals 0 and we use that slope to extrapolate out to three months or time equals 1/4 1/4 of a year now if the functions double use slope over that fourth was exactly the same as the slope of the tangent line or if that slope did not change then our extrapolation would be exactly right if w's slope is increasing over that time then our then our estimate would be an underestimate of the correct thing and if the slope is decreasing over that time then our estimate would be an over estimate of of the actual amount in the landfill and to figure out whether the slope is increasing or decreasing we just have to look at the value of the second derivative if the second derivative is positive that means our slope is increasing which means that our extrapolation would be an understatement let me just be clear here let me draw this let me make it very clear so let me just draw a random function so let's say that's W so this is a case where w's slope is increasing faster or a W slope is increasing from that starting point so our starting point this was our slope and then W slope keeps increasing from there and in this case our estimate is going to be an under estimate of where W is actually is after a fourth of the year if doubles use slope is exactly the same as our function over the course of the year over the course of the first three months so it would look something like that maybe a diverges later on and in this case our approximation would be a really good it would probably be exact for W and if w's slope for whatever reason goes negative after that point and we they already tell us it's an increasing function so that is that is not likely but or well it could it doesn't have to even go negative if W if W slope decreases it could still stay positive and the way I drew it doesn't really draw it like this just to make it clear so let's say that let's say we find out that the slope looks something like this this is this is the slope of the tangent line at time equals 0 if W slope increases from that point then W might look something like that and then of our our our answer to Part A would be an understatement would be an underestimate of where W actually is after 1/4 of a year let me draw it make it clear to you what I'm doing so this is this is my W axis so this is right at our initial condition of 1400 and that this right over here is our time axis and this is at 1/4 so in the last video we said hey this is sitting right at 1411 if W slope increases from that point then this is an underestimate if WS slope stays the same then this is actually a very good estimate because then we're going to be we're going to hit that point directly with W and if W slope decreases so you can imagine maybe W look something like this its slope it has that height it has that slope of the tangent line when it starts but then the slope decreases and it's still an increasing function but the slope is decreasing and in that case we would have we would have an overestimate and in this this is a situation so this first situation slope is increasing that means that means W prime prime is positive W prime prime is positive the slope is increasing the second derivative is positive this means or this would be a byproduct of or this would cause the second derivative to be negative our slope is decreasing and this would be our second derivative our second derivative is zero if your slope isn't changing your second or if your slope is constant if your first derivative is constant your second derivative is going to be zero so let's just see what our second derivative is at our initial condition and then we'll have a pretty good sense of whether we have an over estimate or an underestimate for Part A so let's just figure out what our second derivative is at time zero is going to be equal to one over six twenty-five times w of zero which we know minus 300 well W zero or the amount of waste we have at time zero they told us in the problem is 1400 1400 tons 1400 minus 1300 is 1100 and then 1100 minus 1100 divided by six 25 is a small number it's one point something but it is a positive number and that's the important thing here so this thing all of this business it is positive it is positive so the second derivative is positive which means the slope is increasing at least right at our starting point our slope is increasing which tells us that our since our slope is increasing from that point and frankly the fact that it's increasing at all and that we know that W is an increasing function tells us that our that our estimate in Part A is an under estimate so this is this is the case that we're dealing with so our estimate in Part A is an underestimate of the actual amount of solid waste in the landfill at time T equals one-fourth

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