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Calculus

Fundamental Theorem of Calculus: a Tale of Two Parts

(A new question of the week) A recent question about the application of the Fundamental Theorem of Calculus provided an opportunity to clarify what the theorem means in practice, and specifically how the two parts are and are not related. Misunderstandings like these are probably more common than many instructors realize! We’ll also glance at …

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Euler’s Formula: Complex Numbers as Exponents

Last week we explored how the polar form of complex numbers gives multiplication a simple geometric meaning. Here we’ll go one more step, and express polar form exponentially, which makes DeMoivre’s theorem trivial, and gives us a simple notation to replace “cis”.

Limit of sin(x)/x

Last week we looked at some recent questions about limits, where we focused first on what limits are, in terms of graphs or tables, and then on finding them by algebraic simplification. This week, we’ll look at two old questions about a trigonometric limit that can’t be determined that way: sin(x)/x, as x approaches zero.