وحدة limits and continuity الرياضيات منهج انجليزي الصف الثاني عشر متقدم
When you enter a darkened room, your eyes adjust to the reduced level of light by increasing the size of your pupils, allowing more light to enter the eyes and making objects around you easier to see. By contrast, when you enter a brightly lit room, your pupils contract, reducing the amount of light entering the eyes, as too much light would overload your visual system
study such mechanisms by performing experiments and trying to find a mathematical description of the results. In this case, you might want to represent the size. Of the pupils as a function Of the amount Of light present. Two basic characteristics Of such a mathe- matical model would be
. As the amount of light (x) increases, the pupil sin (V) decreases down to a minimum value p; and
. As the amount Of light (x) decreases, the pupil sptiptzpte (y) increases up to a maximum value P
There are many functions with these two but one possible graph of such a function is shown in Figure 2.1. (See example 3.11 for more.) In this chapter, we develop the concept Of limit, which can used to describe properties such as those listed above. The limit is the fundamental notion Of calculus and serves as the thread that binds together virtually all of the calculus you are about to study. An investment in carefully studying limits now will have very significant payoffs throughout the remainder Of your calculus and byond
A Brief Preview of Calculus: Tangent Lines and the Length of a Curve
In this section, we approach the boundary between precalculus mathematics and the calculus by investigating several important problems requiring the use Of calculus. Recall that the slope of a straight line is the change in y divided by the change in x. This fraction is the same regardless of which two points you use to compute the slope. For example, the points (O, I), (l, 4) and (3, 10) all lie on the line y = 3x + I. The slope Of 3 can be obtained from any two Of the Boints. For instance
In the calculus, we generalize this problem to find the of a curve at a point For instance, suppose we wanted to find the slope of the curve y = x2 + 1 at the point (l, 2). You might think Of picking a second point on the parabola, say (2, 5). The slope of the line through these two points (called a secant line; see Figure 2.2a) is easy enough to compute. We have
However, using the points (O, 1) and (1, 2), we get a different slope (see Figure 2.2b)
In general, the slopes of secant lines joining different points on a curve are not the same, as seen in Figures 2.2a and 2.2b
so, in general, what do we mean by the Of a curve at a point? The answer can be visualized by graphically zooming in on the specified point. In the present case, morning in tight on the point (1, 2), you should get a graph something like the one in Figure 2.3, which looks very much like a straight line. In fact, the more you zoom in, the straighter the curve appears to be. so, here's the stratew: pick several points on the parabola, each closer to the point (1, 2) than the previous one. Compute the slopes
of the lines through (1, 2) and each of the points. The closer the second point gets to (l, 2), the closer the computed slope is to the answer you seek
For example, the point (1.5, 3.25) is on the parabola fairly close to (1, 2). The slope of the line joining these points is
The point (1.1, 2.21) is even closer to (1, 2). The slope of the secant line joining
EXERCISES 2.4
WRITING EXERCISES
I. Think about the following "real-life" functions, each of which is a function of the independent variable time: the height of a falling object, the amount of money in a bank account, the cholesterol level of a the amount of a certain chemical present in a test tube and a machine's most recent measurement Of the cholesterol level Of a per- son. Which of these are continuous functions? Explain your answers
, Whether a process is continuous or not is not always clear- cut. When you watch television or a movie, the action seems to be continuous. This is an optical illusion, since both movies and television consist of individual "snapshots" that are played back at many frames per second. Where does the illusion of continuous motion come from ? Given that the average person blinks several times per minute, is our perception of the world actually continuous
. When you sketch the graph of the parabola y x2 with pencil or is vour sketch (at the molecular level) actually the graph of a continuous function ? Is your calculator or com- puter's graph actually the graph of a continuous function Do we ever have problems correctly interpreting a graph due to these limitations
. For each of the graphs in Figures 2.22a-2.22d, describe (with an example) what the formula forf(x) might look like to produce the given graph
In exercises 1—14, determine wheref is continuous. If possible, extend f as in example 4.2 to a new function that is continuous on a larger domain
TODAY IN MATHEMATICS
Michael Freedman (1951- An American mathematician who first solved one of the most farnous problems in mathematics. the four- dimensional Poincare conjecture. A winner of the Fields Medal, the mathematical equivalent of the Nobel Prize Freedman says. "Much of the power of mathematics comes from combining insights from seemingly different branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such. it may applied to any branch of knowledge: Freedman finds mathematics to be an open field for research, saying that - It isn't necessary to be an old hand in an area to make a contribution
TODAY IN MATHEMATICS
Paul (1916- 2006) A Hungarian-born mathematician who earned a reputation as one of the best mathematical writers ever. For Halmos calculus did not come easily. with understanding coming in a flash of inspiration only after a bng period of hard work "l remember standing at the blackboard in Room 213 of the mathematics building with Warren Ambrose and suddenly I understood epsilons I understood what limits were and all of that stuff that people had been drilling into me became clear I could prove the That afternoon I became a mathematician