Second Derivatives and Beyond
The goal of this section is to be able to go from a formula of a function to an accurate graph of that function. We use the first and second derivatives to help find exact points on the graph and to determine the overall shape(s) of the graph.
There are three steps to drawing a graph.
- Find dots (intercepts, critical points, inflection points).
- Find shapes.
- Play connect-the-dots with the shapes.
Make sure you're happy with each step by itself. The following examples put all the steps together.
Let f (x) = xex. Sketch a graph of f (x). Label all intercepts, critical points, and inflection points.
We'll go through the three steps.
• Find dots (intercepts, critical points, inflection points).
To find the y-intercept we plug in 0 for x and see what we find:
f (0) = 0e0 = 0.
The y-intercept is (0, 0). This is also the only x-intercept since the only time xex can be zero is if x is zero.
Now we find the critical points. From this example we know that there's a critical point at x = -1. In order to graph this point we need the full coordinates:
f (-1) = -1e-1 ≅ -0.37.
There's a critical point at approximately (-1,-0.37).
Now for inflection points. We know from this exercise that there's an inflection point at x = -2. Since
f (-2) = -2e-2 ≅ -0.27
we have an inflection point at approximately (-2,-0.27).
Step 1 is done. We have points:
• Find shapes.
Set up a numberline, marking all the important points we found:
Now figure out the signs of f, f ' and f " in the intervals between the important points. We know f is negative when x is negative, and positive when x is positive:
We know f ' is negative when x < -1 and positive when x > -1:
Finally, we know f " is negative when x < -2 and positive when x > -2:
Using this information, we can figure out the shape of f over each interval:
• Play connect-the-dots with the shapes.
Since f is negative for all negative values of x, we know the concave down, decreasing shape to the left of x = -2 must stay below the x-axis.
We play connect-the-dots, and find this: