00:01
In the given question we are told that the total worldwide box office receipts for a long -running movie is approximated by a function which is given as t of x.
00:13
So t of x is given to be equal to 120 times x squared divided by x square plus 4.
00:24
And now what we are told to find is how fast are the total receipts changing for one year, four year and seven years after the release.
00:35
So what we could do over here is we have to find essentially what we have to find us.
00:43
In the question we are told that we should find how fast the total receipts which is given by t of x.
00:56
Total receives which is t of x is changing is changing so what we essentially have to find is a rate right and the rate that we have to find is the rate at which t of x is changing so what we have to do is to just differentiate t of x with respect to x and the function that we get as a result over here can be used to find how fast the total receipts are changing.
01:28
So to integrate, i mean to differentiate 120x squared divided by x square plus 4, what we could do first is we could take this fraction over here and write it as 120x square times 1 by x square plus 4 and now we are going to use the product rule in order to differentiate this so what is the product rule let's write it over here according to the product role when we are differentiating two functions two functions that are products we could differentiate it in a way that we would have let's say you and v or let's take p and q we are differentiating p and q with respect to x p is a function and q is a function and first what we could do is we could differentiate q with respect to x keeping p there and p not just keeping p without doing anything and in the next step we can differentiate p without differentiating uh so this is a method that we follow to differentiate a product of two functions.
03:03
So we are going to do the same over here.
03:05
Let's take this as p and this as q.
03:08
And then what we would have is we would have d by d x of 120x squared times 1 by x squared plus 4 would be equal to 120x squared.
03:24
Would be differentiated as 200 and 40 times x times we have we are differentiating p first so if we differentiate p first so let's go following the formula let's differentiate 1 by x squared plus first and if we are doing that what we would have is we would differentiate 1 by x square plus 4 so we keep p as such and we will have 1 by x squared would be minus 1 by x square plus 4 whole squared times when we differentiate x square plus 4 it is 2x plus 0 so this is what we would have in total.
04:22
So this is just p times dq by dx okay and in the next case we are going to take q over here and keep it just there and differentiate the function p which is 120 x square and this would be 240 times x so now on simplifying we will have minus 240 x cubed divided by x square plus 4 whole square plus 240 x divided by x square plus 4.
05:00
So this is what we would have as the differential of t of x.
05:07
Right.
05:08
So now we could simplify this function, but we don't actually have to do this.
05:13
We just have to take this function that is the differential of t of x and take the numbers 1, 4 and 7 that has been given in the question and substitute it as values of x so when x is equal to 1 we will get the rate at which the total receipts are changing after 1 year so we would have let's mark this as t -x right so t -dash -1 t dash of 1 would be minus 240 divided by 1 plus 4 squared plus 240 divided by 1 plus 240 divided by 1 plus 4 so this would simplify to minus 240 divided by 1 plus 4 is 5 5 squared is 25 plus 240 divided by 5 and now we can multiply 5 on both sides of this fraction and we will have minus 240 divided by 25 plus when we multiply 240 with 5 the number that we get is 1 ,200.
06:34
So 1 ,200 divided by 25.
06:37
So since now we have the same denominator, we can run.
06:41
1 ,200 minus 240 divided by 25 is the answer...