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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
“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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 20

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online