Good Afternoon Everyone!
First of all, welcome back students, teachers, staff, and administrators, I’ve enjoyed the summer off but am looking forward to getting back into the swing of things.
This semester is one of the heaviest semesters I’ve had with 180 students divided among 4 classes over 3 preps. I am teaching two sections of Calculus I (Math 140), one section of Calculus with Engineering Technology Applications (Math 210), and Matrices (Math 220).
Okay. I’ve got my coffee and a set of completed weeks worth of notes… let’s get down to discussing them. Today I am going to reflect on my first week of Calculus I.
My first week of calculus notes are a little different than most as I hardly spend any time at all reviewing. I view Calculus I as big interconnected mess of three relatively simple concepts: The Infinitesimal & Infinite, Slope, and Area. When I say interconnected, I don’t mean the way it is typically presented
Limits -> Derivatives -> Integrals
I motivate the need for derivatives, and a lot of background on derivatives before I even say the words “limit” or “derivative”. I do this by exploring students’ prior knowledge of slope. A discussion we had at some point during the first day went something like this:
Remember your first time taking algebra, when you were first learning about lines… You spent all this time studying slope, and at any given instance you could give a handful of definitions of slope:
m=rise / run = (y_2-y_1)/(x_2-x_1)=Average Rate of Change = etc.
Then quadratics happened, then polynomials, rational functions, exponential functions, logarithmic functions, trigonometric functions, etc..
And what never comes back up?
SLOPE.
Why not? Those functions have slope too, it’s just that their slopes are not always constant. Finding a way to describe the slope of any function is OVERARCHING GOAL NUMBER 1 in Calculus I.
From here we have a brief refresher about slopes of lines, move straight into secant lines, from secant lines, we develop the need for some freedom by briefly discussing tangent lines to describe the slope of a function, and hence the difference quotient. We work through this development with the aid of Desmos.
Consider for just a second having two points in the plane, say (-2,5) and (3,7). With only these two points, we can find the slope (and equation) of the line passing through them, this however is not very interesting to me as it was something my students and myself learned how to do in Algebra I. Next we start thinking about a function, picking two points on the graph of the function helps start our conversation about secant (and tangent lines), but it is not enough to get what we are looking for. Then we slowly build from there, to get to a tangent line, we need some control over the second point, so we move over Delta x (h in Desmos), now we can move the second point around on our graph. Noticing that as h gets smaller and smaller, we get closer and closer to a tangent line, but if we set h=0, our line vanishes because we can’t have division by 0. But this gets the ball rolling, next we say that there was nothing at all important about our initial point, so we make that point arbitrary and now we have an interactive version of the difference quotient.
As we work through these, I continually remind the students that we cannot divide by zero, but we can make h (or Delta x) as close to zero as we want (even negative at that!). I use this idea to segue into a formal discussion on limits, then it’s smooth sailing from there.
Or is it?
TL;DR:
My first weeks of Calc I goes a little something like
Slope -> Secant Lines -> Tangent Lines & Their Problems -> Difference Quotients -> Limits -> Derivatives
I try my best to make my presentation of The Derivative as underwhelming as possible.
Gabe