Good point, Dan, and a very nice addition to the exercise.
-----Original Message----- From: Teague, Dan [mailto:firstname.lastname@example.org] Sent: Wednesday, August 29, 2012 6:46 PM To: Alan Lipp; AP Calculus Subject: RE: [ap-calculus] Derivatives of trig functions
I agree with Alan here, except I prefer to use y2 = (y1(x+.0001)-y1(x))/.0001 instead of the Nderiv to reinforce the definition. If y1 = sin(x) and your students don't tell you that y2 is the graph of the cosine function, then... Well, I guess I don't believe it is possible for them not to tell you y2 is the cosine function.
Daniel J. Teague NC School of Science and Mathematics 1219 Broad Street Durham, NC 27705 email@example.com ________________________________________ From: Alan Lipp [firstname.lastname@example.org] Sent: Wednesday, August 29, 2012 1:17 PM To: AP Calculus Subject: RE: [ap-calculus] Derivatives of trig functions
I do derive these formulas, but only after my students have guessed them. There are lots of apps available online now, but I've used a simple application of the TI series of calculators. First graph y1 = f(x) . . . your choice, and then y2 = Nderiv(y1, x, x). This graphs both your starting function and a numerical approximation to the derivative. For the trig functions you can stop here since virtually everyone will guess the correct derivatives. I then have them graph y3 = their guess. If they are correct nothing seems to happen since y3 and y2 overlap. I choose to be skeptical and ask how seeing nothing can prove they are correct.
The kicker is to unhighlight y1, y2, and y3 and graph y4 = y2/y3. If y3 is correct then y4 will appear to be the line y = 1, supporting their guess.
The real advantage of this method is that you can now change y1 to any function at all, graph y2, have them guess y3, and use y4 to confirm or refute their guess. Even when they are wrong y4 can yield important information. For example, graph y1 = x^3. The graph of y2 is a parabola and most students will guess y3 = x^2 (I am assuming they have not yet learned the power rule.) Graph y4 and you will see a horizontal line. A trace, however, shows that y4 = 3, and a little algebra shows that if y2/x^2 = 3 then (x^3)' = 3x^2. My kids manage to guess most of the standard formulas long before I have a chance to prove them so they get a week or more of derivative practice before they see the derivation. I am always surprised that they manage to guess the derivatives of tan(x) and a^x. I like this investigation (which usually takes one class and a follow-up homework) both because it gives my students the opportunity to be creative, and also because they then have an investment in the correct answers which provides motivation for the derivations that follow.
Alan Lipp Williston Northampton School
-----Original Message----- From: Steven Jonak [mailto:email@example.com] Sent: Wednesday, August 29, 2012 8:12 AM To: AP Calculus Subject: [ap-calculus] Derivatives of trig functions