We have spend the last few lessons taking an algebraic approach to finding the derivative function - mainly using limits and the Power Rule. Today we are going to take a graphical approach to sketch the derivative function given a graph of f(x) but not the equation. The goal is that students can visually inspect a graph and know how the behavior of the original function affects the derivative function.
I give students this worksheet and tell students about the goal for this lesson. Then I ask them to look at #1 and the read the directions. Next, I give them about 5 minutes to brainstorm with their table groups about what the derivative of this function should look like. I like to see what they come up with right off the bat before we start sketching it out together.
After students have discussed for a few minutes, I will bring the class together and we will sketch the derivative together. I usually start by pulling out the ideas from the class to springboard our discussion. In the video below I discuss how I walk through this first example with my students.
Next I have students work on #2 by themselves and I choose a student to present their work on the document camera.
These first two examples really set up students to make some generalizations about the derivative graph and how it relates to the original function. After we go through those, I have students fill in the boxes at the bottom of the first page of the worksheet. In the teacher notes you can see what information I we will discuss as a class.
Another really important aspect that I want to address is the degree of the original function compared to the degree of the derivative function. While it may be obvious to you or me, it can be difficult for students to make that connection. I will have students name the degree of each graph on the first page and then ask if they notice any important relationships. Usually a student or two will notice that the degree of the derivative function is one less than the degree of the original function. We will discuss why that is and connect it back to the Power Rule that we learned in a previous lesson.
Questions #1 and #2 on the worksheet were fairly simple because f(x) was a polynomial function. Questions #3 and #4 really up the ante because they are an absolute value and square root function. I have students work in their table groups and have them attempt to sketch the derivative of each. Seven or eight minutes should be sufficient for them to generate some ideas.
When we share our thinking about these two graphs, there are always some interesting discussions that arise because of the unique nature of these two graphs. Here are some points that I am sure will come up in your class. In the video I discuss a little about each so you can be prepared to address these with your class.
1. The corner of the absolute value graph
2. The equation of the absolute value's derivative
3. The square root function's derivative at x = 0
After these graphs are completed, we have two instances of where the derivative of a function does not exist. In the box at the bottom of the page we will generalize and list all of the places where the derivative does not exist. I usually draw a sketch of each so that students can have a visual representation as well.
Finally, I give students this homework assignment to cover what we went over today.