Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 30. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Continuity and the Intermediate and Extreme Value Theorems 271 If f is continuous on the closed and bounded interval a fe and a A f fe B then somewhere in the interval f attains every value between A and B. In particular if a continuous function changes sign on an interval it must be zero somewhere on that interval. If a function f is continuous on a closed interval a fe then f takes on both a maximum high and a minimum low value on a fe .8 For f to attain the maximum value of M on a fe means that there is a number c in a fe such that f c M and f x M for all x in a fe . Analogously for f to attain the minimum value of m on a fe means that there is a number c in a fe such that f c m and f x m for all x in a fe . a continuous function on a closed interval 1 a continuous function on a open interval ii a discontinuous function on a closed interval iii Figure Studyinging parts ii and iii in Figure should convince you that both conditions the continuity of f and the interval being closed are necessary in order for the statement to hold. 8 For a a proof of either of these theorems look in a more theoretical calculus book. 272 CHAPTER 7 The Theoretical Backbone Limits and Continuity Note that even if x fc where k is a constant the statement holds. Given the definitions of maximum and minimum presented above if x k on a then for every x e a x k is both a maximum value and a minimum value. Principles for Working with Limits and Their Implications for Derivatives The following general principles can be deduced from the definition of limits. Suppose x L1 and g x L2 where L1 and L2 are finite. Then 1. x g x L1 L2 The limit of a sum difference is the sum difference of the limits. 2. x g x L1 L2 The limit of a product is the product of the limits in particular g x may be constant lum. . fc x fcLi . 3. limx a i L1 provided L2 0. The limit of a quotient is the quotient of the limits provided the denominator has a nonzero limit . 4. If h is
