Inflection Points, Concavity & Curve Sketching
We've been talking a lot lately about ways of using derivatives to analyze the shape and slope of graphs, and inflection points are the final piece of that puzzle. So now you've got all the tools you need to sketch the graphs of functions including extrema, intercepts, asymptotes... Oh wait, you say you don't remember all that stuff? Because it was last year? Well you're in luck! Below are the videos covering each of those things, along with links back to the chapters they came from all over the site! You're welcome.
The thing I won't be able to help you with is figuring out exactly how your particular teacher wants you to show your work on these curve sketching problems. I tutor calculus students from schools all over West Los Angeles, high school and college, and I've never seen two teachers who do it the same way. They ask different problems with different steps; they use different types of functions; and worst of all, they all have totally different ways of keeping track of all the critical values and test points. Some use tables, some use number lines, some use crazy grid things... So to ace these you'll need to figure out how your teacher wants it all written out, in addition to the usual practice practice practice.
Inflection Points, Concavity & Second Derivative Test
These follow a very similar process to finding maxima and minima, except instead of setting the first derivative to zero and solving, to find inflection points you set the second derivative equal to zero. Why would you want to do this? Because they'll make you do it. But what are they? Watch and find out!
Curve Sketching Examples
In this video I work a couple of full examples of curve sketching -- one polynomial, one rational function. These problems are monsters, including maxima and minima, asymptotes, inflection points, and of course a graph!
Sketching Parent Graphs from Derivative Graphs
Normally, they're going to ask you to sketch derivative graphs based on the parent. I've gotten emailed from time to time about the opposite direction, though, so here goes. WARNING: When you move in the "up" direction, going from derivatives to the parent function, you are either told a point to start from (initial condition) or you're just guessing. But the principles are useful, so here they are.
The videos below are just copied here from elsewhere, for your convenience.
From Graphing Rational Functions, here are some asymptote review videos:
Vertical Asymptotes & Holes
The most important thing about rational functions is that you're never ever allowed to divide by zero (those values aren't in domain); hence, the first thing we'll do in every problem is set the denominator equal to zero. Which zero is a vertical asymptote and which is a hole? What's a hole? And how can an asymptote be friendly or unfriendly? Stay tuned.
Using Limits To Find Horizontal Asymptotes
Yep, that's what we'll do. We'll also briefly review the pre-calc way of finding horizontal asymptotes of rational functions, first covered way back when in Pre-Calc.
Slant Asymptotes (a.k.a. Oblique Asymptotes)
On the down side, these involve polynomial long division. On the up side, many teachers don't even cover this sub-topic, so check your syllabus! (If you don't remember polynomial long division, check out my polynomial division chapter.)
From Relative Maxima & Minima, earlier in Calculus:
Relative Maxima & Minima
In this video I start off explaining the difference between relative and absolute extrema. Then we get into the nuts and bolts of how to find relative extrema (maximums and minimums) using the first derivative test, and how not to get burned by common trick questions.
And finally from the Functions chapter in Algebra 2, a video on finding intercepts of functions:
Finding X and Y Intercepts
Turns out finding intercepts of a function is done the same way whether you're working with lines or high-power polynomial nightmares. In this chapter, naturally, we'll be working with the easier stuff, though we'll keep revisiting the topic (as will your teacher) for years to come.