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Locate the discontinuities of the function and illustrate by graphing.

$ y = \ln (\tan^2 x) $

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Campbell University

Oregon State University

Harvey Mudd College

University of Nottingham

natural log of ancient squared Oh Becks. And we want to find now that dysfunction log of changing squared of X is discontinuous when X equals what numbers, what values? Okay, when is this function log of tangent squared effects. Going to be discontinuous? Let's take a look at dysfunction. We're taking this is a composite function. We're taking the log of tangent squared effects. Tangent squared of X means detention of X squared. So whatever the tangent of X comes out to be tangent squared effects is that value squared? Now tangent itself has some discontinuities. So funk values of X where tangent is discontinuous are going to be values of X where this entire function is discontinuous. If tangent of X has a discontinuity and a log of tangent squared effects is going to have a discontinuity. Uh So we have to look at when tangent has discontinuities. Uh Then after we investigate that, we're not done yet just finding the values of X where tangent is discontinuous is not, it's half the battle. But it's it's not the entire problem, then we have to look at the natural algorithm function. Natural log is only defined. You can only take the natural log of positive numbers. Now, tangent squared effects means you're taking whatever tangent of X is whether it's positive or negative or zero and you're squaring it while tangent of X is positive then when you square it's going to stay positive. No problem. You can take the log of a positive number, let's say tangent of X came out to be negative. Well, when you square it negative times a negative or positive and that's okay because uh you know, you can take the log of a positive number. But what if the tangent of X came out to be zero? The tangent of X 10. The in tangent squared of X means zero squared which is zero. You cannot take the log of zero. So we have two things we need to look for. We have to look for. What values of X will make tangent discontinuous. And we have to look for which values of X will make tangent equal to zero Because the tangent of X0. Uh Then you're going to be attempting to take the log of zero which does not exist. So that's going to be another dis continuity. Alright. When does tangent have discontinuities? Well, Qianjin has discontinuities at one pi over two, three pi over two, five pi over two, Basically any odd number times pi over two. And this also goes for negative odd numbers. So negative one pie over to. Uh It will be when X is negative one pi over two. Tangent is going to be discontinuous when X's negative three pi over to change and we'll have it just continuity. When X is negative five pi over two. Tianjin will have just continuity. So uh discontinuities for tangent are X equals positive or negative one pi over two, positive or negative, three pi over two positive or negative, five pi over two and so on. So these are values where these are values of X. Where attention will have this continuity. And if tangents having a discontinuity then this entire function is going to have a discontinuity. Uh So the log of tangent squared of X will be discontinuous when X equals any of these X X values but we're not done. Remember tangent squared of X will be greater than or equal to zero. It's the places where tangent squared of X. Or more specifically tangent of X zero that are going to cause us problems. If tangent of X comes out to be 00 square to zero and you cannot take the log of zero. So now we're going to find the X values where tangent is zero. And these X values that make tangent zero will be discontinuities for this function. Okay, tension will be zero when X is a multiple of high. So when X equals zero. One pie, two pi three pipe. And this also goes for negative one pie. So positive or negative one pie, positive, two pi or negative, two pi positive or negative, three pie. Any integer positive or negative times pi will be a location where tangent is zero. So when X is any of these values, tangent will be zero. Tangent squared effects will be zero and you cannot take the log of zero. Um That will be a dis continuity for this composite function. So log of tangent squared of X is discontinuous. When X equals any of these and any of these. Now, let's take a look at that. All right. We're going to graph the natural log of tangent squared events. Remember we're expecting discontinuities at any integer uh times high or any odd number. Uh I need your times pi over two. So we expect discontinuities at zero. Hi over two pi three pi over 22 pi five pi over 23 pie. Let's take a look. So here is the graph of the log of tangent squared of X. Let's look for the discontinuities. All right. This continuity of zero. This continuity at pi over to this continuity at pi. This continuity at three pi over to this continuity at two pi. This continuity of pi pi over two. So you can see that at any of these X values the log of change and squared effects will have a discontinuity

Temple University