Question

Applied Calculus - Carefully sketch the graph of a function f that satisfies all of the conditions given within the following table. Be sure to label your graph compeletly. Specifically label all critical points and inflection points. There are many possible correct graphs. I will attach table in excel document.

Answer 1

What does the derivative tell you? It shows the slope. So instantly you know that for some area in a derivative that it's negative that your original function must be going down. If it's positive, it's going up, and if it's 0 then you know that you've possibly reached the top or bottom of a curve.

What's the second derivative tell you? It tells you concavity. What is that? Just that if a graph is a hill or a scoop. Here's an example:See how the one on the left is concave down and the other is concave up? By finding the second derivative of a function you will find the inflection points at 0. What are inflection points? They're just the points between where concavity transitions between being up or down. In the picture you can see that there's a change in concavity.

Consider the top peak and valley at the bottom there. The first derivative at those two points equals zero because the slope is zero at those points, right? So then it must follow that in the middle of those two points, the first derivative must either be at the highest or lowest number when it becomes nearly vertical. So when that number is the highest, we know that's the inflection point of when the graph turns around on the second derivative. So from this we know that when the second derivative is negative we have concave down and for an area between inflection points where the second derivative is positive, we have concave up, which again is like the right side of the picture where the arrows are pointing up!

I hope that clears up what it all means and why.