Intro To Learning Calculus

I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education. Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs lead to resistant germs (survival of the fittest). Calculus is similarly enlightening. They are. Unfortunately, calculus can epitomize what’s wrong with math education. It really shouldn’t be this way. Math, art, and ideas I’ve learned something from school: Math isn’t the hard part of math; motivation is. Teachers focused more on publishing/perishing than teachingSelf-fulfilling prophecies that math is difficult, boring, unpopular or “not your subject”Textbooks and curriculums more concerned with profits and test results than insight ‘A Mathematician’s Lament’ [pdf] is an excellent essay on this issue that resonated with many people: Poetry is similar. Feisty, are we? Yowza!

Derivatives of Trig Functions Trigonometric functions are useful in our practical lives in diverse areas such as astronomy, physics, surveying, carpentry etc. How can we find the derivatives of the trigonometric functions? Our starting point is the following limit: Using the derivative language, this limit means that . To see why, it is enough to rewrite the expression involving the cosine as But , so we have This limit equals and thus In fact, we may use these limits to find the derivative of and at any point x=a. So which implies So we have proved that exists and Similarly, we obtain that exists and that Since , and are all quotients of the functions , we can compute their derivatives with the help of the quotient rule: It is quite interesting to see the close relationship between (and also between From the above results we get These two results are very useful in solving some differential equations. Example 1. . So using the product rule, we get which implies, using trigonometric identities, Exercise 1. at the point Answer. Exercise 2.

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