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Snapchat's total users can be modeled by S(t) million Market research determines that users_ where is months after the start of 2018. S" (18) S" (t) ...

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Snapchat's total users can be modeled by S(t) million Market research determines that users_ where is months after the start of 2018. S" (18) S" (t) < 0 for - and S" (t) 0 for > 18(a) This means that for the months after the start of 2018 leading up to month 18 of users. the rate change of totalreached critical pointincreaseddecreasedneither increased nor decreasednone of the above(b) This means that 18 months after (he start of 2018 the number of users reachespoint of

Snapchat's total users can be modeled by S(t) million Market research determines that users_ where is months after the start of 2018. S" (18) S" (t) < 0 for - and S" (t) 0 for > 18 (a) This means that for the months after the start of 2018 leading up to month 18 of users. the rate change of total reached critical point increased decreased neither increased nor decreased none of the above (b) This means that 18 months after (he start of 2018 the number of users reaches point of inflection point of diminishing returns minima maxima none ol (he above



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The number of daily active Snap chat users $S$ was 46 million in January 2014 and grew linearly to 94 million by July 2015 . It then continued to grow linearly to 160 million by January 2017 . (Source: Recode.) (a) Write a formula for a piece wise-linear function $S(x)$ that models these data, where $x$ represents the number of months after January 2014 . (b) Sketch a graph of $y=S(x) .$ Is $S$ a continuous function on the interval $[0,36] ?$ (c) Interpret the rates of change in $S$.

For this problem, we are told that the number of smartphone users and penetration in the United States continues to grow steadily. The number of users in millions from 2011 through 2015 is projected to grow at the rate of RFT equals 14.3 for T between zero and 4 million per year. The number of users in 2011 or T equals zero was 90.1 million. Were then asked to find an expression given the projected number of smartphone users in year T. And what is the estimated estimated number of smartphone users in 2015. So what we want to do here is take the anti derivative. I'm not sure if you've seen this notation by this point in the book, but this is the notation for anti derivative integral. We want to first of all take the anti derivative of r f T. So the anti derivative r f t is just going to be the anti derivative of 14.3, Which is going to be 14.3 T plus a constant. Now we need to determine that constant. So let's say that Since we're talking about a number of users, let's call this NFT. We know that end of zero Is equal to 90.1 since we're making everything in millions. And then that in turn would be 14.3 times zero plus C. So we get that 90.1 must equal our C then tells us that end of T It's going to equal 14.3 T Plus 90.1. Now, we are lastly asked what is the estimated number of smartphone users in 2015? So that is going to be end of four. So it will be four times 14.3 Plus 90.1. I'm going to pause and calculate that off screen. So that comes out to 147.3. Okay.

For parts is we need to find the average growth rate for the giving. Two intervals. The number off phones sold can be modelled by equations. See off T in two 114.9 into the race to the power 0.345 t for the first interval. 2000 and 7 to 2000 and nine in the year 2000 and seven. The number of cell phones sold can represented by sea off zero. Putting in the values we get 114 0.9 into E raised to the power 0.35 into zero, which is calculated to be 114.9 for the year 2000 and nine. We put two in place off T C to see off. Two becomes 114.9 into E raised to the Power Mind 0.345 into two, which is calculated to be go to to 29 points or 779 were in mind that thes and values are in millions. To change the doctor to find the rate of change. Delta P over Delta T we subtract the number of phone sold in year 2000 and nine from the number phone sold in the year 2000 and seven and also subtract the time period to minus zero. Bum Putting in the values we get. 57.89 million for the second interval, 2000 and 12 to 2000 and 14. For the year 2000 and 12 we have C five equals 114.9 into E rest of power 0.345 into five, which is evaluated guys 6 44.88 where, as for the U 2 2014 we have see off seven equals 114.9 series to government 0.345 into seven, which is evaluated as 12 85.705 million. To find the rate for to find the growth rate. Delta Pete by Delta T for the second interval, we surprised, see of seven minus from CIA five minus seven. Divide minus five. Putting in the values we get. 12 85.705 minus 6 44.88 divided by two, which makes our answer to be 320.41 to 5 million. The group parade is great Her for the interval. 2012 through 2014. For Part B, we need to find the instantaneous growth rate to find instantaneous growth rate reversing To find the derivative off the function, see off P to evaluate the door obeyed every find the inner and outer functions. The inter function is u equals 0.345 t The ultra function y equals you D U by D. T is 0.345 d y by D. U is clearest to the power You C dash Empty can be written as 114.9 into Do I I. D. You into D. U by D T, which gets US 39.64 into erase to the power 0.345 into T in order to find an instantaneous growth rate in the year 2000 and eight given by T equals one, we get C dash up. One equals 39 points. Explore into erased to the power 390.345 into one, which can be evaluated as 55.97 million for the year 2000 and 13 t becomes equal to six as given in the problem, so C Dash of six can be evaluated as 39.64 into eat 0.345 into sex equals 314.14 million. The instantaneous grow three is great to refer. The year 2013 for part C. By looking at the graph, the growth rate is greater with every passing year. We can plot the graph off the given function and the door of a tive, which is actually the growth rate by using a graphing utility.

All right. When they give you a rate problem like this R. T. Is equal to 14.3. Uh What their I guess the original problem for, we'll call it number uh in years uh is doing the integral of that. So, if we asked you for the integral with respect to time uh is add one to the X. Moment Earth in this case will be 14.3 T. Because the derivative of this 14.3 T. Would just give you 14.3. But you need to make sure that you write a plus C. Because they also give you this when time is zero, Whatever year that is 2011? Uh There's 90.1 million. Uh I guess users so plug in zero for tea. Well, zero times anything is zero plus. He must give you a total of 90.1 million. So therefore zero plus anything is is just that thing. So C must equal 90.1 million. Um So that's how we get that equation. When I substitute this back in and again, this is the year 2011, 2011 is the year zero. So if we want the year 2015, that's four years later, we want to figure out what end of four is, and we can do that by plugging in four in for tea. And when I do that, I get an answer of 147 point uh three million. There you go.

The town of 50,000 people. We know the number of people at time T who have the flu is N. F. T. is about 10,000 over one plus 99,000. Actually 9,999 each of the negative t. So we see that if we keep zooming out it eventually reaches this maximum point For about 10,000. And that's where it's in a town of 50,000 people. We know that the flu is spread by one person who has in the teeth of zero. So we want to find the rate of change of the number of infected people. So that's going to be an crime factor and prime of T. And we see that based on this function it's going to be increasing from zero to up until it reaches the maximum point right here. I reached that maximum point at about 9.21 these and then we see it decreases from then on out. So it's gonna be the final results.


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