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Trig, the Exponential Function, and Complex Numbers

Trig and exponential functions appear in virtually every lesson in the courses. There really aren't special sections dedicated to isolating them to study their properties and to study identities they satisfy. There aren't sections entitled, "Derivatives of sine, cosine and tangent". These things just happen, and students get used to the functions through experience. We give trig identities to the students through Euler's formula, which is used throughout, and derived via Taylor series. Once one admits that i is safe to use, lots of things become possible. Once one pursues the idea of constant percentage growth rate, the exponential function, its differential equation, growth and decay problems, and even investment problems become straighforward.

Here's a Literacy Sheet problem on logs and the complex exponential function.

What is the value of e^ip ?
How does the value of e^ip force Log[-1]=ip ?
Give the value of Log[-4] = Log[4] + Log[-1].

We frequently don't know ahead of time what things cause difficulties for students. One big example is the use of different variables, and what functional expressions mean. Here's one for you to ponder.

Given a positive number t, factor the derivative of x^t e^-x to explain why the curve y = x^t e^-x first goes up as a advances from 0 and grows until x reaches a point x₀ after which the curve goes down. Find the exact value of the turning point x₀ in terms of that t.

That darned t is a puzzle to lots and lots of folks.