Review Sheet for MATH 124 at UA 2
Review Sheet for MATH 124 at UA 2
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WileyPLU s MATH 124129 5th Ed WileyPLUS Home Tlel Contact us Loout HughesHallett Calculus Single Variable 5e Chapter 1 A Library of Functions 39 v Reading content D 11 Functions and Change D 12 Exponential Functions D 13 New Functions from Old D 14 Logarithmic Functions D 15 Trigonometric Functions D 16 Powers Polynomials and Rational Functions u I 18 Limits DChapter Summary D Review Exercises and Problems for Chapter One DCheck Your Understanding D Projects for Chapter One El p Student Solutions Manual p Student Study Guide gt Graphing Calculator Manual p Focus on Theory p Web Quizzes 17 Introduction to Continuity This section introduces the idea of continuity on an interval and at a point This leads to the concept of limit which is investigated in the next section Continuity of a Function on an Interval Graphical Viewpoint Roughly speaking a function is said to be continuous on an interval if its graph has no breaks jumps or holes in that interval Continuity is important because as we shall see continuous functions have many desirable properties For example to locate the zeros of a function we often look for intervals where the function changes sign In the case of the function f x 3x3 x2 2x l for instance we expectL1 to find a zero between 0 and l becausef0 l andf1 3 See Figure E To be sure thatf x has a zero there we need to know that the graph of the function has no breaks or jumps in it Otherwise the graph could jump across the xaxis changing sign but not creating a zero For example f x lx has opposite signs at x l and x 1 but no zeros for l S x S 1 because ofthe break at x 0 See Figure To be certain that a function has a zero in an interval on which it changes sign we need to know that the function is defined and continuous in that interval Figure 175 The graph offx 3x3 x2 2x 1 leClDocuments20and2OSettingsmthDesktopindex uni him 1 of88262009 63356 AM WileyPLU s Figure 176 No zero althoughf1 andf1 have opposite signs A continuous function has a graph which can be drawn without lifting the pencil from the paper Example The lnction f x 3x3 x2 2x 1 is continuous on any interval See Figure 175 Example The lnction f x lx is not de ned at x 0 It is continuous on any interval not containing the origin See Figure 176 Example Suppose px is the price of mailing a firstclass letter weighing x ounces It costs 41 for one ounce or less 58 between the first and second ounces and so on So the graph in Figure is a series of steps This lnction is not continuous on any open interval containing a positive integer because the graph jumps at these points ycenisi 39 MI T5 39 5 Viv I 4 rquot C 39 moundss 1 i2 3 Figure 177 Cost ofmailing a letter Which Functions are Continuous Requiring a function to be continuous on an interval is not asking very much as any function whose graph is an unbroken curve over the interval is continuous For example exponential functions polynomials and the sine and cosine are continuous on every interval Rational functions are continuous on any interval in which their denominators are not zero Functions created by adding multiplying or composing continuous functions are also continuous leClDocuments20andZOSettingsmthDesktopindex uni him 2 of88262009 63356 AM WileyPLU s The Intermediate Value Theorem Continuity tells us about the values taken by a function In particular a continuous function cannot skip values For example the function in the next example must have a zero because its graph cannot skip over the x axis Example 1 What do the values in Table 118 tell you about the zeros of fx cos x 2x2 Table 118 x fx 0 100 02 090 04 060 06 011 08 058 10 l46 Solution Since f x is the difference of two continuous functions it is continuous We conclude that f x has at least one zero in the interval 06 lt x lt 08 since f x changes from positive to negative on that interval The graph of f x in Figure suggests that there is only one zero in the interval 0 S x S l but we cannot be sure of this from the graph or the table of values 1 ex flr39 cm 1 21139 L Ir 12 H llJiHllubquot 1 I 39 Figure 178 Zeros occur where the graph of a continuous function crosses the horizontal axis leClDocuments20andZOSettingsmthDesktopindex uni hlm 3 of88262009 63356 AM WileyPLU s In the previous example we concluded that f x cos x 2x2 has a zero between x 0 and x 1 because f x is positive at x 0 and negative at x 1 More generally an intuitive notion of continuity tells us that as we follow the graph of a continuous function f from some point a f 61 to another point b f b then f takes on all intermediate values between f a andfb See Figure This is Theorem 11 Intermediate Value Theorem Suppose f is continuous on a closed interval 61 b If k is any number between f a and f b then there is at least one number 0 in 61 b such that f c k 3 ill flb JII i l i i i i i i 4 i 1 T I L IIJ Figure 179 The Intermediate Value Theorem The Intermediate Value Theorem depends on the formal de nition of continuity given in Section See also wwwwileycomcolle0e hugheshallett The key idea is to nd successively smaller subintervals of a b on which f changes from less than kto more than k These subintervals converge on the number c Continuity of a Function at a Point Numerical Viewpoint A function is continuous if nearby values of the independent variable give nearby values of the function In practical work continuity is important because it means that small errors in the independent variable lead to small errors in the value of the function Example Suppose that f x x2 and that we want to compute f TE Knowing f is continuous tells us that taking x 314 should give a good approximation to f TE and that we can get as accurate an approximation to f TE as we want by using enough decimals of TE Example If px is the cost of mailing a letter weighing x ounces then p 099 pl 41 whereas pl01 58 because as soon as we get over 1 ounce the price jumps up to 58 So a small difference in the weight of a letter can lead to a signi cant difference in its mailing cost leClDocuments20andZOSettingsmthDesktopindex uni him 4 of88262009 63356 AM WileyPLU s Hence p is not continuous at x 1 In other words if f x is continuous at x c the values of f x approach f c as x approaches c In Section Q we discuss the concept of a limit which allows us to de ne more precisely what it means for the values off x to approach f c as x approaches c Example 2 Investigate the continuity of f x x2 at x 2 Solution From Table 119 it appears that the values of f x x2 approach f 2 4 as x approaches 2 Thus f appears to be continuous at x 2 Continuity at a point describes behavior of a function near a point as well as at the point Table 119 Values ofx2 nearx 2 x 19 199 1999 2001 201 21 x2 361 396 3996 4004 404 441 Exercises and Problems for Section 17 Exercises Are the functions in Exercises 1 2 g 4 Q Q 1 9 and 10 continuous on the given intervals 1 2xx23 on1 1 2 2x9r1 on 1 1 3 1 on1 1 4 1 on 0 3 leClDocuments20an1ZOSettingsmthDesktopindex uni him 5 of88262009 63356 AM WileyPLU s In Exercises Q Q Q and show that there is a number c with 0 S 0 S1 such thatfc 0 11 fx x3 x2l 1239 fx ex 3x 13 fx x cosx 14 fx 2quot lx 15 Are the following functions continuous Explain a f I 1 l J L7 1 l 7 raw 2 e v 7 Problems 16 Which of the following are continuous functions of time a The quantity of gas in the tank of a car on a journey between New York and Boston b The number of students enrolled in a class during a semester c The age of the oldest person alive 17 An electrical circuit switches instantaneously from a 6 volt battery to a 12 volt battery 7 seconds after being turned on Graph the battery voltage against time Give formulas for the function represented by your graph What can you say about the continuity of this function 18 A car is coasting down a hill at a constant speed A truck collides with the rear of the car causing it to lurch ahead Graph the car39s speed from atime shortly before impact to a time shortly after impact Graph the distance from the top of the hill for this time period What can you say about the continuity of each of these functions 19 Find k so that the following function is continuous on any interval leClDocuments20andZOSettingsmthDesktopindex uni mm 6 of88262009 63356 AM WileyPLU s 21 Is the following function continuous on 1 l 22 If possible choose k so that the following function is continuous on any interval Jquot g 3 m J Y r T i L 1 L j j IR 3 2 z 23 Find k so that the following function is continuous on any interval g 1111 1 v g 24 Discuss the continuity of the function g graphed in Figure 180 and defined as follows sin U Elia I 1x 1 i U l l I quot for 9 U 11 I 94 x a t f Figure 180 2539 a What does a graph of y ex and y 4 x2 tell you about the solutions to the equation ex 4 x2 b Evaluatefx ex x2 4 atx 4 3 2 1 0 1 2 3 4 In which intervals do the solutions to ex 4 x2 lie 26 Let px be a cubic polynomial with p5 lt 0 p10 gt 0 and p12 lt 0 What can you say about the number and location of zeros of px 27 Sketch the graphs of three different functions that are continuous on 0 S x S l and that have the values given in the table The rst function is to have exactly one zero in 0 l the second is to have at least two zeros in the interval 06 08 and the third is to have at least two zeros in the interval 0 06 x 0 02 04 06 08 10 f 100 090 060 011 058 146 x leCDocuments20and2OSettingsmthDesktopindex uni mm 7 of88262009 63356 AM WileyPLU s 28 a Sketch the graph of a continuous function f with all of the following properties i f0 2 ii f x is decreasing for 0 S x S 3 iii fx is increasing for 3 lt x S 5 iv f x is decreasing for x gt 5 V fx gt9asx gtltgtltgt b Is it possible that the graph of f is concave down for all x gt 6 Explain 29 A 06 ml dose of a drug is injected into a patient steadily for half a second At the end of this time the quantity Q of the drug in the body starts to decay exponentially at a continuous rate of 02 per second Using formulas express Q as a continuous function of time t in seconds 30 Use a computer or calculator to sketch the functions y xl sin and if 2 L93 x for k 1246810 In each case find the smallest positive solution of the equation yx zkx Now define a new lnction f by f 13 Smallest positive soluticm of ylfpgj zk xj Explain why the function f k is not continuous on the interval 0 S k S 10 leClDocuments20andZOSettingsmthDesktopindex uni mm 8 of88262009 63356 AM
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