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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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