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]]>I am very directive – do these problems by hand, use WolfAl for these problems. But, I am teaching a different audience from you – I’m at a state college; if they are in my class, they are engineering majors. So, I might do a bit more handholding than you are comfortable with.

I kind of like the complex part of the solution thing in WA. I think it shows students there is more math ahead in Complex Analysis; they have to be smart enough to ignore it when it doesn’t apply. Something else I do in Calc II is put an elliptic integral into WA, which it of course recognizes, and they get an answer like F(theta, phi). This makes the point that even if WA gives you the answer, you might not know enough Math at this point to successfully interpret it. (WA also occasionally uses the DiLogarithm function in an answer, which I don’t know much about myself – but it makes the same point.)

I worry about instructors who teach this course the same way they did 30 years ago. If I am going to err, I would rather err on the side of using technology.

I really enjoy your blog Vince – based on the topics you choose to talk about, I suspect our teaching styles are similar.

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]]>Of course they can try to graph both forms of the answers and see if they are the same, but does not prove they are the same.

In general, how should a student handle the situation when their answer is different from that provided by Alpha, or for that matter, from the solutions manual/back of the book answers? I think this is the critical question.

So, I’m not sure it’s just “noodling.” I think it’s more a matter of being able to use technology intelligently .

And as you mention Alpha, it’s especially critical, since Alpha often provides complex parts of solutions. I find students have a difficult time with this – they are not sure when it is relevant.

I don’t claim to have the answer. But when a student is attempting a homework problem, how would they know if you would do it by hand or not? I claim that a typical undergraduate would not know how to discern this.

So a question for you – how do you guide students regarding whether they should use Alpha (or some equivalent), or whether they should perform the calculation by hand? I do not think this is an easy question to answer….

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]]>Your rewriting the logistic function in terms of tanh is interesting, but I wonder if it doesn’t seem to students as algebraic noodling. And isn’t the shape of the curve, as related to what happens to the population what really matters?

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]]>Re:hyperbolic substitutions in Calc II, there is a book available on Google books for free called Integration Using Trigonometric and Imaginary Substitutions. It teaches substitutions that we now routinely do using hyperbolic functions using imaginary substitutions, e.g. x = i*sin(theta). It overlooks a lot of complex analysis complications, but is really interesting if you have never come across it. I bet it would be right up your alley.

Looking forward to part 2. If you have any rules of thumb as to which Calc Ii trig substitution integrals are actualy easier using hyperbolic substitution that you teach students, I hope you will share them. I’m teaching this stuff myself this summer.

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