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FUNCTIONS, LIMITS, AND CONTINUITY
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(b) Illustrate the result in (a) graphically by constructing the graphs of y tan x and y x and locating their points of intersection. (c) Determine the value of the smallest positive root of tan x x. Ans. c 4.49 approximately 3.89. 3.90. 3.91. 3.92. Prove that the only real solution of sin x x is x 0. (a) Prove that cos x cosh x 1 0 has in nitely many real roots. (b) Prove that for large values of x the roots approximate those of cos x 0. x2 sin 1=x Prove that lim 0. x!0 sin x Suppose f x is continuous at x x0 and assume f x0 > 0. Prove that there exists an interval x0 h; x0 h , where h > 0, in which f x > 0. (See Theorem 5, page 47.) [Hint: Show that we can make j f x f x0 j < 1 f x0 . Then show that f x A f x0 j f x f x0 j > 1 f x0 > 0.] 2 2 (a) Prove Theorem 10, Page 48, for the greatest lower bound m (see Problem 3.34). (b) Prove Theorem 9, Page 48, and explain its relationship to Theorem 10.
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THE CONCEPT AND DEFINITION OF A DERIVATIVE Concepts that shape the course of mathematics are few and far between. The derivative, the fundamental element of the di erential calculus, is such a concept. That branch of mathematics called analysis, of which advanced calculus is a part, is the end result. There were two problems that led to the discovery of the derivative. The older one of de ning and representing the tangent line to a curve at one of its points had concerned early Greek philosophers. The other problem of representing the instantaneous velocity of an object whose motion was not constant was much more a problem of the seventeenth century. At the end of that century, these problems and their relationship were resolved. As is usually the case, many mathematicians contributed, but it was Isaac Newton and Gottfried Wilhelm Leibniz who independently put together organized bodies of thought upon which others could build. The tangent problem provides a visual interpretation of the derivative and can be brought to mind no matter what the complexity of a particular application. It leads to the de nition of the derivative as the limit of a di erence quotient in the following way. (See Fig. 4-1.)
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Let Po x0 be a point on the graph of y f x . Let P x be a nearby point on this same graph of the function f . Then the line through these two points is called a secant line. Its slope, ms , is the di erence quotient f x f x0 y x x0 x 65
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where x and y are called the increments in x and y, respectively. ms where h x x0 x. See Fig. 4-2.
y B y
Also this slope may be written
f x0 h f x0 h
Q P _ S f (x0 + h) f (x0) N R
P y = f (x) f (x0)
h = Dx
M x0 x0 + h b x x0 x
Fig. 4-2
Fig. 4-3
We can imagine a sequence of lines formed as h ! 0. It is the limiting line of this sequence that is the natural one to be the tangent line to the graph at P0 . To make this mode of reasoning precise, the limit (when it exists), is formed as follows: f 0 x lim f x0 h f x0 h!0 h
As indicated, this limit is given the name f 0 x0 . It is called the derivative of the function f at its domain value x0 . If this limit can be formed at each point of a subdomain of the domain of f , then f is said to be di erentiable on that subdomain and a new function f 0 has been constructed. This limit concept was not understood until the middle of the nineteenth century. A simple example illustrates the conceptual problem that faced mathematicians from 1700 until that time. Let the graph of f be the parabola y x2 , then a little algebraic manipulation yields ms 2x0 h h2 2x0 h h
Newton, Leibniz, and their contemporaries simply let h 0 and said that 2x0 was the slope of the tangent line at P0 . However, this raises the ghost of a 0 form in the middle term. True understanding of 0 the calculus is in the comprehension of how the introduction of something new (the derivative, i.e., the limit of a di erence quotient) resolves this dilemma. Note 1: The creation of new functions from di erence quotients is not limited to f 0 . If, starting with f 0 , the limit of the di erence quotient exists, then f 00 may be constructed and so on and so on. Note 2: Since the continuity of a function is such a strong property, one might think that di erentiability followed. This is not necessarily true, as is illustrated in Fig. 4-3. The following theorem puts the matter in proper perspective: Theorem: If f is di erentiable at a domain value, then it is continuous at that value.