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Problem 71 Hard Difficulty

The Fresnel function $ S $ was defined in Example 3 and graphed in Figures 7 and 8.

(a) At what values of $ x $ does this function have local maximum values?
(b) On what intervals is the function concave upward?
(c) Use a graph to solve the following equation correct to two decimal places:
$$ \displaystyle \int^x_0 \sin (\pi t^2/2) \, dt = 0.2 $$

Answer

a. $S^{\prime}(x)=0$
$s^{\prime \prime}(x)<0$
b. $S^{\prime}(x)=\sin \left(\frac{\pi x^{2}}{2}\right), S^{\prime \prime}(x)=\pi x \cos \left(\frac{\pi x^{2}}{2}\right)$
c.

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Video Transcript

So this function on your textbook is defined by basically the formula is given here. So this is how the sa Becks, so local maximum means way Want the first derivative, Toby zero on the second derivative should be negative. No, let's compute the first derivative by fundamental theorem of calculus. That should be side pi X square over too. And second derivative should be that the ribs health is so no co sign pi X squared over two. If I was not that by beauty pie pie that the rip through inside which is pious ix so local maximum We want this to be local Max. We want this side pious square over two to be zero Whoever signs something Ciro on, uh, that there's something should be interred your times pi So this it was integer times pi This city knows integer so that keeps us cans of the pie. So whatever acts can be returned to us square it off, plus or minus square root of two K In a case in future, that's a critical point. But we also long the second derivative to be toupees Toby Toby Negative. So osho me the Prague in this plus or minus square root of two K. Oh, we are high times X, which is this times pi times the coastline Marcos I pi x square, which is two k pi or two's a cosa que pie thiss Toby Negative number. So what is this negative number off? So this is always positive. So that's basically a co sign. This this one ofthe case even thiss description one and con case are this Keeps you that one. So So this means are when Casey even this keeps you What? So you want to take the negative side that one case hot. You want the whole thing to be inactive, you have to take the process from so then that this part gives you this and all the hacks a hostess case Inter jer and satisfied this Basically it's either an active even in the drawer. A positive are the future. Then it gives us a local maximum. This looks a little bit weird, but that's idea and conclave upward means seven to Ruth. You positive so I mean, we want to solve this A pious constantly does not change the sign. Pai explain Over too positive. So is eater both positive or post active. So this part's easy. Hello. If you are cold time, something was possible. Something's attempted So coz I'm basically if you goes for so it looks like this for co sign This part will be positive that this is three pi over too. This probably be negative This power connective negative three pi over too excess right Keep often a sign for each pie So basically you used this graph and figure out when pi X squared over two is each interval. I'Ll leave the details to us by Massey but that's idea basically at the end you want X to be quite a recall when X is greater or equal zero you want coastline to Pisa. You want this quantity. You want the pi X squared over two to be inside the partner, this's plus and you are one axis negative. You want this part to be inside a region that is negative and now deeps equation solving part to you You can use the graphic calculator to do that