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Problem 67 Hard Difficulty

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = \dfrac{x^3 + 4}{x^2} $

Answer

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Video Transcript

our domain. Here's, um, ex cannot be zero because of the denominator. We have an intercept at, um negative cube root of 40 But let's just estimate so that we can see our graph. So negative 1.60 is our estimate of the intercept. No symmetry. Um, for Assam toes, we know that X equals zero is one. As we provisions stated the denominator of a rational function Can I be zero so x can so extended zero Therefore, X equals your preserver to glass into So let's look at slant asking tote. Now, um, we can do polynomial division. So this is the numerator and this is the denominator. Okay, how many times can x squared go into X cubed crew left with X X Times X squared would normally be excused, but we're gonna switch the sign so that we can cancel out the first term, bring down the four so we could no longer do polynomial division. Therefore, like will's ex or quotient is our explains asking tote or oblique Aspen took Okay. Next, we can do our first derivative test to find, um, increasing and decreasing intervals or Max's Inman's so white prime is X cubed minus feet, kosher, over execute set that equal to zero critical point at X equals two. And by the way, if we substitute f of two, we will get through just so we know the lie value. So we're testing to. And just so we know of two or three, it's gonna be decreasing on this interval and then increasing here so that this is a minimum value local minimum value at 23 Oh, and remember this was using the first derivative test. So this is f prime, and then we can do second rivet a test. So why double prime is 24 over X to the fourth, but that's inconclusive. We cannot use it. So we can't with Akane Cavity, but we can use the first river to test. Okay, so now we can make our rough. So say, this is one. This is two kisses. Negative one. Negative too. Is, as one says to 123 okay. And Allison told at X equals zero, which is we have X equals zero and y equals X. It's this diagonal line. Okay, we haven't intercepted negative 1.60 which is approximately here is is a rough sketch, and we have a local minimum value at 23 So two three would be about here. This is just a rough sketch. So are rough approaches, but never touches our Assam totes. So at 23 it's gonna go like this and like that. And why is that? Because, um it's gonna be decreasing in an increasing, decreasing and then increasing. And then over here, we know that's gonna go up from the axis and approach, but never touch her, ask him to. And then the same year, no approach, but never touch until heads the axis. And this is a rough sketch of the graph.