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### Use the method in Example 3 to find (a) the slope…

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SB
Pennsylvania State University

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Problem 7

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and (b) an equation of the tangent line at $P .$
$$y=x^{2}-3, \quad P(2,1)$$

a. 4
b. $y=4 x-7$

## Discussion

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## Video Transcript

Hello, everyone. Now that does. Look at an example problem. To find the slope off given car given point B and also finding the question off a tangent line. Be so no knowledge is looking given example. Learn now it is no down. The information that is given were given with equation off a curve y equals X squared minus three and we're also given. But the point b to come. A one Knowledge is no down what is to be found. So we need to find the slope off the given go act being that is this point. And no, we also need to find the equation off the tangent. The bosses, too, the boy be. Neither does find chances to these questions. Owner doesn't okay, how to solve or how to find the slope of the given go like so busting spurs. We can just we have received the given point b do come a one. Does it actually lie on the Given CO y equals X squared minus street so we can just substitute. We can just substitute y equals one and X equals tool in Equation one, and we find that one is equal to the square minus tree, which is equal to one and sends the equation. One is satisfied so we can say that point B lies on the give and go. So no, we don't. We need to find the slope of the give and go. And how do we find that the slope of the given girl is found using instantaneous slope, which is phone using the first diverted and valued the given point. So to start off with that, we need to first vehicle first, familiarize arsenals with a few standard that ever does and a few properties of defense off the first delivery. So now let us look at one off the very important properties to the reality, so D by d. X off e ffx less BG affects can be leading us in go de vie DX affects plus being don't anybody x Oh gee, if ICS and under the standard derivative that we need to familiarize Oscars is D by D. X, both exp over in This is e Quito and do Explorer and minus one. And on a did standard that ever that we need to familiarize asked his DVD X off a constant that does not depend on eggs is equal, though. But this can be returned as into D by D X off export are zero and we can use the standard of evidence and we find Do you idiots off Z quick go into zero until Explorer my his line are This is just equal to zero So do you buy DX A very or a constant zero. So using these properties in the standard derivatives we can just find the divi by d X and a valued ridiculous point. So this is to be found so we can start off y equals X squared minus three on we can different you one with respect X on board sites. So read egg de vie by D X equals two eggs my zero And this needs to be evaluated at the to cover one and this is equipped for so slow off the given girl be Do come a one is equal for No no, it does look good. Hope this on the second question, let us perform the equation off the tangent I to be to solve this question, we need to familiarize ourselves that the standard equation off line that is why equals MX plus C this and is slope off the line, See, is why you descent. So, using this, we can start off to find the equation on the engine using the first offensive there is a slope of Tokyo and cover it. Be using the same slow for the tangent we can write Slow equals not. Yes, it is. Why? Because I'm expresses so slow because M equals four and, uh, to find by intercept were given it the point b to come a one. And we know that beetle come a one lies on the line. Right? So the equation 10 with Donna's y equals for X plus c And if you substitute be, do come a one And why? Because for explosive you can substitute be do come a one and y equals for expert for explicit Because the point b lies on the line So we can just substitute y equals one and X equals two and VIII goes for its proceed So we get one equals foreign control. Let's see, uh, we can just transports foreign capital and decided and C equals one minus aid. This is equal to minus seven. So we have found equation on a tangent line by equals four x minus side. Therefore, equation off the tangent line that be is given by y equals forex. Minus seat are y equals for X minus seven. So overall, we have learned coatings. So the first thing we learned is the instantaneous slope can be found using the first elevator divide by D. X alluding at a given point. No, because it was to come. A one in general can be X come alive. Okay. And we can no down reality. That is Deve Ideas off E ff IX. Yes, be gox. This can return as e d by D. X off ever fix plus B Do you buy dicks till six? And, uh, we will smoulder down the standard that ever gonna be by its exposure and And this is equitable. And it'll explorer and minus one. Andi by deitz off a constant that does not depend on X is you know, and stand early. Question off line is why equals? I m explicit this and is a slope off the line. And, uh, C is see is nothing but the Y intercept. Great. My intercept love tonight. So these are the few tins that we learned from this example. Thank you.