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Question

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Answers

Let $f(x, y)=\left(x^{2}+y^{2}\right)^{2 / 3} .$ Show that $$ f_{x}(x, y)=\left\{\begin{array}{cl} \frac{4 x}{3\left(x^{2}+y^{2}\right)^{1 / 3}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{array}\right.$$ IThis problem, due to Don Cohen, appeared in Mathematics and Computer Education, Vol. $25, \text { No. } 2,1991, \text { p. } 179 .]$

Table, armed of value, awful tanks and f Thanks. And then for the executions up all night. And then we get triple night. Execute yours upon 99 It could you 39 900 f Thanks and job on 999 Then f expected trip on 999 way and grew one more night. And then we get a treat on 9999 And it will e think close a junior one from the right. So one point, Zozo. One day we get a form from those of one. Now in we can't thing examples the one against form bones of the one and to my welders Here we with the one point the one going off upon the one and one point. One day we're gonna focus on one. So we see him. We have two directions. Near the first tension will be X separate Judah. So one would be somewhere here and exhibited to one here from the left and doesn't will be expected to one from the right. So both of them therefore. Then we see the vellum, the function f X and getting rid coast to develop. I'm the far and also here and goes getting requested a far and distant obliged under the limit on a function well, excuse to one and were in Coachella very far.

So in this question, we want to use the table of values toe, help us evaluate the limit as X goes toe one of f of X and, ah, What that means is we want to take a look at the X values that approach one and see what their Y values turn into. So as we move toward 0.9999 and one point there is there, is there one? We see that the Y values or F of X values are getting closer to four, and that is Thea answer to the problem.

The amount and billions of dollars spent by the top 15 financial us institutions on information technology, offshore outsourcing is approximated by A.F.T. Okay Is equal to 0.92. Yeah. I am. T Plus one. What is your .61? And we're gonna be looking at that between zero and 4. So let's just uh narrow this down Between zero and 4. And then we're also gonna have to make this a lot smaller, Perhaps negative 1- one. So zooming out, we see that this is our function right here. T equals zero is going to correspond in 2000 and four. So we want to show that A. Is going to be increasing and the way that we do that is by multiplying this here times this will get the A. Prime of T. Is greater than zero. So that means that this function is going to be increasing. So that means that the amount of money being spent is increasing over time.

Uh Let's see here we have this is the amount of in billions of dollars spent by the top 15 U. S. Financial institutions on I. T. Uh offshore outsourcing is approximated by dysfunction here. Uh T. Is measured in years and t. Zero corresponds to uh 2004. So they want us to show that a. Is increasing on uh on that range and interpret your results. So we can obviously plot it and we can see that indeed it is increasing. We can also show that um that we can take the derivative and so the derivative that this is this and this is always positive. So our function this is always increasing even if we went out further. So this is always increasing because this is always positive now uh show that it's con cave down. Uh Well to do that we need to take a second derivative which I can be here. Let's see here um When we take a second derivative. Um And that's gonna give us minus 0.219. All over one plus t. To the 1.39. And so we can see that that is always negative. So this thing is always concave down. Um And what does that mean? That means that well the rate of change is decreasing. So maybe you know we're not there probably kind of exaggerating things. We're saturating, you know how much we can outsource. Um So that the rate of change is decreasing. Uh But but it's still growing


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