وحدة Preliminaries of calculus الرياضيات منهج انجليزي الصف الثاني عشر متقدم
In this chapter, We present a collection of familiar topics, primarily those that We con- Sider essential for the study Of calculus. While we do not intend this chapter to be a comprehensive review Of precalculus mathematics, we have tried to hit the highlights and provide you with some standard notation and language that we will use throughout the text
As it grows, a chambered nautilus creates a spiral shell. Behind this beautiful geometry is a surprising amount Of mathematics. The nautilus grows in such a way that. the overall proportions of its shell remain constant. That is, if you draw a rectangle to circumscribe the shell, the ratio of height to width of the rectangle remains nearly constant
There are several ways to represent this property mathematically. In polar coordi- nates, we study logarithmic spirals that have the property that the angle of growth is constant, corresponding to the constant proportions of a nautilus shell. Using basic geometry, you can divide the circumscribing rectangle into a sequence of squares as in the figure. The relative sizes of the squares form the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, ... , where each number in the sequence is the sum Of the preceding two numbers
The Fibonacci sequence has an amazing list of interesting properties. (Search on the Internet to see what we mean!) Numbers in the sequence have a surprising habit Of showing up in nature, such as the number of petals on a lily (3), buttercup (5), marigold (13), and pyrethrum (34). Although we have a very simple description of how to generate the Fibonacci sequence, think about how you might describe it as a function. A plot of the first several numbers in the sequence (shown in Figure 1.1 on the following page) should give you the impression of a graph curving up, perhaps a or an exponen- tial curve
Two aspects Of this problem are important themes throughout the calculus. One of these is the importance of looking for patterns to help us describe the world. A second theme is the interplay between graphs and functions. By connecting the techniques of algebra with the visual images provided by graphs, you will significantly improve your ability to solve real- world problems mathematically
Equations of Lines
The federal government conducts a nationwide census every 10 years to determine the population. Population data for several recent decades are shown in the accompanying One culty with analyzing these data is that the numbers are so large. This prob — lem is remedied by transforming the data. We can simplify the year data by defining x to be the number of years since 1960, that 1960 corresponds to x 0, 1970 cor- resBnds to x 10 and so on. The population data can simplified by rounding the Rational Functions numbers to the nearest million. The transformed data are shown in the table and a scatter plot of these data points is shown in Figure 1.11. The B'ints in Figure 1-11 may to form a straight line. (Use a ruler and if you agree.) To determine whether the points are, in fact, on the same line (such B'ints are called colinear), we might consider the population growth in each of the indicated decades. From 1960 to 1970, the growth was 24 million. (That is, to move from the first B)int to the second, you increase x by 10 and increase y by 24.) Likewise, from 1970 to 1980, the growth was 24 million. However, from 1980 to the growth was only 22 million. Since the rate of growth is not constant, the data Bints do not fall on a line. This argument involves the familiar concept of slope
Referring to Figure 1 .12b (where the line has positive slope), notice that for any four Bints A, B, D and E on the line, the two right triangles SABC and ADEF are similar Recall that for similar triangles, the ratios of corresponding sides must be the same. In this case, this says tha
. Piano tuners sometimes start by striking a tuning fork and then the corresB) nding piano key. If the tuning fork and piano note each have frequency 8, then the resulting sound is sin 8t + sin St. Graph this. If the piano is slightly out- of- tune at frequency 8. I, the resulting sound is sin 8t + sin 8.1t Graph this and explain how the piano tuner can hear the small difference in frequency
EXPLORATORY EXERCISES
. In his book and video series The RingofTruth, physicist Philip Morrison an experiment to estimate the circum - ference of the earth. In Nebraska, he measured the angle to a bright star in the sky, then drove 370 mi due south into Kansas and measured the new angle to the star. Some geom- etry shows that the difference in angles, about 5.02 equals the angle from the center Of the earth to the two lcxations in Nebraska and Kansas. If the earth is spherical (it's not) and the circumference Of the B'rtion Of the circle
measured out by 5.020 is 370 mi. estimate the circumference of the earth. This was based on a similar exper- iment bv the ancient Greek scientist Eratosthenes. The an- cient Greeks and the Spaniards of Columbus' dav knew that the earth was round, they just disagreed about the circum- ference. Columbus argued for a figure about half of the ac- tual value, since a ship couldn't survive on the water long enough to navigate the true distance
. An Oil tank with circular cross sections lies on its side. A stick is inserted in a hole at the top and used to measure the depth d of oil in the tank. Based on this nwasurement, the goal is to compute the percentage of oil left in the tank
TO simplify calculations. supBse the circle is a unit circle with center at (0, 0). Sketch radii extending from the origin
TODAY IN MATHEMATICS
Kim Rossmo (1955- ) A Canadian criminologist who developed the Criminal Geographic Targeting algorithm that indicates the most probable area of residence for serial murderers. rapists and other criminals. Rossmo served 21 years with the Vancouver Police Department. His mentors were Professors Paul and Patricia Brantingham of Simon Fraser University. The Brantinghams developed Crime Pattern Theory, which predicts crime locations from where criminals live. work and play. Rossmo inverted their model and used the crime sites to determine where the criminal most likely lives. The premiere episode of the television drama Numbers was based on Rossmo's work