I was interrupted a couple of times on the second one. I’ll redo it later.
November 12, 2009
Pre-Calculus Assignment List — Chapter 4
Pre-Calculus
Chapter 4
1. Handout 4210
2. Book 4.2.1 page 297: E: 1, 3, 5, 7, 8, 9, 10, 55 – 60, 61, 62, 64, 65, 66, 69
also… Evaluate .999^.999, .5^.5, .25^.25, .001^.001, Did any of these surprise you? I was.
3. Book 4.2.2 page 297: E: 11 – 20, 22, 23, 29, 31, 43, 44, 51, 53, 72
also… Graph these y=2^x, y=5^x, y=e^x, y=1^x, y=x^x
4. Book 4.3.1 page 310: E: 3, 7, 10, 11, 15, 18, 23 – 36, 47, 48, 49, 50, 87, 89 – 93, 95, 97, 98, 99, 100, 101, 102, 103
5. Book 4.3.2 page 310: E: 61, 67, 73, 75, 79, 81, 86, 88, 96, 111, 112, 115, 116, 118
6. Book 4.4 page 321: E: 17 – 24, 61 – 72
7. Handout 4410
8. Book 4.5.1 page 327: E: 1 – 9, 11, 12, 17 – 32, 45, 46, 47
9. Book 4.5.2 page 327: E: 13, 14, 15, 16, 39, 40, 41, 42, 43, 48, 49, 50, 51, 57, 59
10. Handout 4510
11. Book 4.6 page 335: E: 1 – 10, 11, 13, 15, 17, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 41, 42, 43, 46, 49, 50, 51, 52
12. Book 4.7.1 page 327: E: 1 – 10, 11a, 12a, 17, 19
13. Book 4.7.2 page 327: E: 13, 14, 15, 16, 20, 23, 24, 25
14. Handout 4710
15. Book 4.8 page 355: E: 1, 2, 3, 4, 5, 7, 9, 10, 12
16. Book 4.R page 360: 4, 5, 11, 13, 15, 17, 21, 27, 28, 39, 41, 43, 53 – 72, 74, 75, 78, 79, 81, 82, 84
TEST (2 Days)
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November 11, 2009
Geometry Notes — Proving that Quadrilaterals are Parallelograms
Proving that Quadrilaterals are Parallelograms
1. by definition… If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.
2. If the opposite sides of a quadrilateral are congruent, then it is a parallelogram.
3. If the opposite angles of a quadrilateral are congruent, then it is a parallelogram.
4. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram.
5. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
6. If one pair of opposite sides of a quadrilateral is both congruent and parallel, then it is a parallelogram.
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Math History — Chinese Mathematicians — Father Zu Chongzhi and Son Zu Gengzhi
Zu Gengzhi was a Chinese mathematician. He lived from about 450 to 520 in China. He was the son of another famous Chinese mathematician, Zu Chongzhi. Zu Gengzhi discovered the “Zu Gengzhi’s Principle”, stating that “The volumes of two solids of the same height are equal if the areas of the plane sections at equal heights are the same.” This is same as Cavalieri’s principle, but was discovered about 1100 years earlier.
He is credited with the invention of the astronomical sighting tube, which Shen Kuo and Wei Pu would later improve during the 11th century.
The Zu family was an extremely talented one with successive generations being astronomers with special interests in the calendar. They handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted in China at this time. Zu Gengzhi, in the family tradition, was taught a variety of skills as he grew up. Zu Gengzhi’s greatest achievement was to compute the diameter of a sphere of a given volume. He had demonstrated that a previously accepted formula was incorrect by using various constructions for comparison, but he was unable to derive the correct formula.
His father Zu Chongzhi and Zu Gengzhi wrote a mathematical text entitled Zhui Shu (Method of Interpolation). This book most likely contained astronomical calculation techniques due to the accuracy of his calendars. It most likely contained formulas for the volume of the sphere, cubic equations and the accurate value of pi. Unfortunately, this book didn’t survive to the present day, it has been lost since the Song Dynasty, which began in the middle to late tenth century.
Their mathematical achievements included:
- the Da Ming calendar introduced by the father in 465. The midwinter was the starting point in making the calendar; therefore it was of extremely important to pinpoint the position of the sun on that day. However, the astronomers before Zu all believed the position was fixed, which caused errors in the calendar-making task from the very beginning. To solve the problem, Zu introduced the concept of procession (of the sun) in making the Da Ming Calendar, greatly enhancing the accuracy of calendar computation.
- distinguishing the Sidereal Year and the Tropical Year. The sidereal year is the time taken for the Sun to return to the same position with respect to the stars of the celestial sphere. It is the orbital period of Earth. A tropical year (also known as a solar year) is the length of time that the Sun takes to return to the same position in the cycle of seasons.
- calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today. His estimate is only 46 seconds different from the modern estimate.
- calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today. Using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
- calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
- deriving an approximation of pi that was correct to the seventh decimal place, which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355 ⁄ 113 (detailed approximation) and 22 ⁄ 7 (rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 sided polygon. This was an impressive feat for the time, especially considering that the only device he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. No one discovered more of pi for nearly 1000 years.
- finding the volume of a sphere as 4πr³/3, where r is radius.
- discovering the Cavalieri’s principle, 1000 years before Bonaventura Cavalieri in the West.
Not too shabby. Very talented mathematicians without technology. I guess talent, hard work, and perseverance count for something huh. Outstanding.
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Geometry — Biography — Hippasus
Hippasus of Metapontum, ca. 500 B.C., was a Greek philosopher. He was a disciple of Pythagoras. Hippasus is attributed the discovery of the existence of irrational numbers. More specifically, he is credited with the discovery that the square root of 2 is irrational.
It is believed that he proved the existence of irrational numbers at a time when the Pythagorean belief was that whole numbers and their ratios could describe anything that was geometric. Not only that, they didn’t believe there was a need for any other numbers.
Up until Hippasus’ discovery, the Pythagoreans preached that all numbers could be expressed as the ratio of integers. Despite the validity of his discovery, the Pythagoreans initially treated it as a kind of religious heresy and they either exiled or murdered Hippasus. Legend has it that the discovery was made at sea and that Hippasus’ fellow Pythagoreans threw him overboard.
The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive. They all took oaths to ensure that their discoveries remained with the Pythagorean society. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. An event would happen that would change the very core of their beliefs. Along came Pythagorean Hippasus who discovered that the diagonal of a square whose side was one unit could not be expressed as a whole number or a ratio. Hence, the Pythagorean Theorem which crushed their original beliefs. Thus, they certainly didn’t want Hippasus’ discovery to be revealed and shatter their pride and core beliefs.
In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n not equal to zero. It can be deduced that they also cannot be represented as terminating or repeating decimals.
Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. 3.1415926535897932384626433832795 (and more …)
The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. 2.7182818284590452353602874713527 (and more …)
The square root of 2, also known as Pythagoras’ constant (ironic huh), is 1.41421356237309504880168872420969807856967187537694807317667973799 (and more …)
Statistics Notes — Poisson Distribution Table (partial) — Not Cumulative
This is a partial Poisson Distribution Table. I will use it only to point out that it exist and for one or two examples. I will always use my outstanding graphing utility. Thanks TI.
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Geometry Notes — Parallelograms — Sketchpad Slides
What you should gain from these slides is the repetition of the values.
AB isn’t always equal to 3.875 inches. But AB is always equal to CD.
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