set3 - Lecture 7 A nal note on inverse trigonometric...

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Lecture 7 A fnal note on inverse trigonometric Functions Someone in class asked the question, “Why is the inverse sine function, sin - 1 , also called the arcsin function?” The answer is to be found in the Fgure below. O Q x y θ P (1 , 0) ( x, y ) 1 The circle has radius r = 1. A point ( x,y ) that lies on the circle deFnes an angle θ such that sin θ = y θ = sin - 1 y cos θ = x θ = cos - 1 x. (1) The length of the arc PQ is given by arclength PQ = = θ. (2) ±rom the relations in (1), arclength PQ = sin - 1 y = cos - 1 x. (3) In this way, we see how the inverse trig functions are related to arc length. Limit oF a Function (Relevant section from Stewart, Sixth Edition: Section 2.2, p. 88) One of the purposes of this course is to investigate the concept of the limit of a function a little deeper. You are familiar with the following mathematical expression, lim x a f ( x ) = L, (4) which is read as follows, 41
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“The limit of f ( x ), as x approaches a , is equal to L ”. According to Stewart (Sixth Edition), Section 2.2, p. 88, this statement means that “we can make the value of f ( x ) arbitrarily close to L (as close to L as we like) by taking x to be suFciently close to a (on either side of a ), but not equal to a .” This can be taken as a working de±nition of “limit.” There are, of course, a number of questions to be raised regarding this de±nition: 1. What is meant by “arbitrarily close” to L ? 2. What is meant by “suFciently close” to a ? In order to have a clear and unambigous de±nition, in other words, a mathematical defnition , we’ll have to translate the above ideas of “closeness” into mathematical expressions. This will be done a couple of lectures from now. ²or the moment, let us consider some motivating examples and examine a more “intuitive” approach to limits – something with which you are familiar from your earlier course(s) in Calculus. Example 1: We consider the function f ( x ) = x 2 x + 2 for values of x close to 2. This example was treated by Stewart (p. 88), but we present an abbreviated version of his table below: x f ( x ) x f ( x ) 1.995 3.985025 2.005 4.015025 1.999 3.997001 2.001 4.003001 As x approaches 2 from the left and the right, f ( x ) appears to be approaching the value 4. As such, it seems that we might be able to write that lim x 2 ( x 2 x + 2) = 4 . (5) But are we sure of this? Shouldn’t we at least try to go a little closer, keeping in mind that we are not allowed to consider the point x = 4? If we go a little closer, we obtain the following values: x f ( x ) x f ( x ) 1.99990 3.999700 2.00010 4.000300 1.99999 3.999970 2.00001 4.000030 42
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Yes, it certainly looks as if f ( x ) is approaching the value 4 as x gets closer to 2. (The error of ± 3 times powers of ten is quite intriguing, and can be explained – more on this later.) But would this experimental evidence hold up in a “mathematical court of law”? A good lawyer for the prosecution
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set3 - Lecture 7 A nal note on inverse trigonometric...

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