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Curve Sketching with Derivatives - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Sketching a curve from knowledge of the signs of the first and second derivatives is a useful way to find the approximate shape of a function's graph. When curve sketching making a sign chart of the derivatives is an easy way to spot possible inflection points and to find relative maxima and minima, which are both key in sketching the path of a function.

I want to talk about curve sketching, we have a method here for curve sketching. I have 3 basic steps that I'm going to tend to go through on all my curve sketching problems. If you're given a problem that asks you to sketch y equals f of x the first thing you're going to need to do if you're using Calculus to sketch these curves is take the first 2 derivatives. Take f prime and f double prime, and make a sign chart for the 2 derivatives, that'll help you analyze where it's increasing, decreasing, concave up or concave down. And then plot special points local max and min, inflection points and intercepts if they're easy to plot. These points will help guide you when you're drawing your curve.
Let's take a look at a quick example, you might see something like this in homework graph y equals f of x given that f is continuous and it satisfies the requirements of this table. So I have a table that tells me whether f prime and f double prime are positive or negative 0 or undefined. That's what these little codes here mean, and I've got intervals x less than 5, x=5 etcetera. I'm just going to go through this table really quickly and draw a little shape for each of these intervals for example x less than 5 if f prime is positive that means that f is increasing and if f double prime is negative, it's concave down. So I'm going to draw a little curve that's increasing and concave down something like that at x=5 f prime is 0 and f double prime is negative. So that indicates it's concave down but it has a horizontal tangent I'll just draw like a little local maximum that's what is going to happen there.
And then here I've got f prime is negative and f double prime is negative, it's decreasing concave down so I'll draw something like that. At x=8 both derivatives are undefined so let me come back to that, and then here at x greater than 8 f prime is positive, f double prime is negative so f should be increasing and concave down. Increasing and concave down again looks like this, now when I piece these 2 things together for x=8 I'm going to get the only thing I can get really is some kind of corner or cusp like so. So that's how the pieces are going to fit together. So let me draw a quick sketch and about the third point, plotting points here all I have are x coordinates of some key points and so I can't get too specific about the actual locations of the points. So what I'm going to do as much as I can and this is a pretty open ended problem there are a lot of different possible answers.
But at x=5 I'm going to put my local maximum keeping in mind that because the first derivative is 0 there's a horizontal tangent here so I want to keep that in mind but I want it to be increasing to the left and decreasing to the right. And I'm going to have it decrease and concave down all the way down to x=8 so something like that. And then at x=8 we've got our little corner or cusp and then it's going to increase and be concave down afterwards. Something like that, so pretty easy if you look at your sign chart you analyze the behavior based on whether it's increasing concave down or concave up.
There are really only are 4 possibilities when you combine increasing or decreasing concave up and concave down and so you're going to have 4 sort of prototypical shapes that you're going to be working from but once you have your sign charts made and you have these little arcs drawn you'll know what your graph is going to look like pretty closely.

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