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Start Free Trial Cancel anytime. Uploaded by RicardoDavidAleMo. Date uploaded Nov 24, Did you find this document useful? Is this content inappropriate? Report this Document. Flag for Inappropriate Content. Download Now. Related titles. Carousel Previous Carousel Next. Solucionario de Calculo Diferencial e Integral - Granville.
Jump to Page. Search inside document. A rational number is a repeating or terminating decimal. In Exercises , define the function piecewise and sketch the graph. Check by plotting. Values not in the implicit damain are bolded.
See Exercises , a versus b. Pind formulas for f o. Sketch the graphs of f, 7, and f og. Maximum Product of a set of positive numbers of constant. See Ex. Exercises 1. A person's brain weight 6 Ib is directly proportional to his body weight w Ib, and a person weighing Ib has a 4 Ib broin a b w.
The period p se of a pendulum i directly proportional to the square root of the mumber 2 of feet in ils length, and au 8 ft pendulum has a 2 see period. The period is 1 sec, 4. C 2 dollars ie the cost of shipping 2 tb. Cla , y z cents is the cost of mailing ounces. AL A lot with walkways 22 fe ide at the frowt and hack and 15 ft nt the sides is to contain a 13, 7 building. The field is Choose 6. Ino doing, we use the following theorem, a consequence of Definition 1.
Buorcises 1. G z dollats is the admission for age x years. Sec Solutions 9 and The domain of f is [—5,5]. It fellows from statements 2 and 3 that if 0. The following theorems concerning the continuity of a function follaw from Definities 1. Often they can be used to determine if e function is continuous at a number.
By observing where there is a bron jn the graph, determine fsumber at which the function is diteantinuous; and show why Defsition 1. Hence condition i of Definition Pose 4 Moi. We show how Definition 1. Thus, condition i of Definition 1. Hence f is diseontinuous at 0. Henee condition i of Definition 1.
Thus Jimg z does not exist. Does the discontinuity appear to he removable? If so, how should f be redefined to remove it? Hence, f is continuous for all real numbers. Tence f is continuods on its domain: al real numbers except 3.
Hence f is continuous at numbers other than 2. Mence F is continuous at 3. At what waluss af 2 i? Br Theorem 1. F9 i continuous on [-3, 4. Bp Theorem 1. Sy Theorem 1. Sy Theoret 1. By Theorem 1. Hence by Theorem 1. Hence by Theorem. Band 2. In Exercises , find the domain of the funetion.
Because a rational funetion is continuous om ite and g is a rational function, g is continuous om any interval that does not inelude 2. Beeause F is a rational function, F is continu: en its domsin. See Bx4 In Exercises , dues the intermnediate-valuc theorem hold for the function j, interval a, ] and constant K? Furthermore, f is continuons on [0,8]. The figure shows the graph of f and the line 8.
Wut Bin not in Because 9 is continuous on [0. Ie aye 1. Let t2—m, Then arte. Then e tim. Because sin r is continuous for all 2 and 29 is continuous for all , the composition sin 29 ia continuons for all 9, and so is the product Vg? Because sin , and hence kin , and cos 0 arr continuous for all 9, then by Theorem 1.
See Bx td. Heeause ? Oy rota 1 eee ne an 7 3 i a See the figure. S -2 end 1 do not exist. Uenee condition of Definition 2. There are breaks in the graph at —] and 1. Henee condition 3 of Definition 1. Theve is a brenk in the graph at 3. Thus, there is an essential diseontinuity at 1. Thus, there is am essential discontinuity ak 2, 1s.
Therefore the discon F: ee 3 In Exercises , f ix diseontinnovs at a. We nationalize the numeratar. BY Theorem 1.
For J to be continuous at -3, For f to be continuons at 3. J is continuous on 90,3 U 5, Since 7 is continuous aaly on 1. Because Hea al? The converse of Theorem 2.
If function fs differentiable at a point, then the graph of the function must be smooth at that point, Furthermore, if a fonction is Aifteentinble i a print, then the tnrgnt line to Une graph of the fonction at that point fot be vertical.
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