1. ACTIVIDAD GENERAL
1.3. ANÁLISIS DE OFERTA
1.3.2. Sector Industria
43. r( ' =
ti- 2j 3 k t + t 0 0
44. r( § =cos(3 'i sin(3t'j +tah(3 )kt 0 ?r
45. r( � eti e-tj � k o 0
In each of Exercises 46-48, a parametrized space curve is given. Calculate the moving frame (
NB) at the point a point Po on the curve parameterized by r. Find a Cartesian equation for the normal plane to the curve at Po
49. r( )
51. r(t) = cos(t)i - sin(t)j
+
tk Po = (-1, 0, 7r)
In each of Exercises 52-54, verify that the given para
metrization p is an arc length parametrization. Then calculate T(s), N(s), and �(s). parameterized by r, calculate the radius of curvature and the center of curvature at r(to).
55. r(t) = e-1i - e'j
+
v'2tk to = 0 56. r(t) = t2i- tj +
tk to = 1 57. r(t) = (1 - 2t)i+
t2j - tk t0 = 0In each of Exercises 58-60, calculate the curvature at each point of the graph of y = f(x).
58. f(x) =x4 59. f(x) =tr - e-x 60. f(x) =
3.x2 - 4x
In each of Exercises 61-63, a vector-valued function r is given. Calculate the curvature at r(t) of the planar curve that r parameterizes. decomposition a(t) = arT(t)
+
aNN(t) without calculating T(t) and N(t).73. r(t) = (t2 - t)i
+
(t2+
t)j - tk 74. r(t) = e-21i+
e2'j+
3tk 75. r(t) = (t - l)i+
(t+
l)j+
t2kIn each of Exercises 76-78, write the Cartesian equations of the ellipse that is described in words.
76. foci at
(4,0)
and (-4
,0), major axis6
77. center (0, 0), major axis 8, minor axis
6,
foci on the x-axis 78. foci(2, 6), (2,
10), eccentricity3/4
On the night of November 11, 1572, Tycho Brahe, a young Danish nobleman with a hobbyist's interest in astronomy, cast his eyes toward the constellation Cassiopeia. To his astonishment, he sighted a new star, one that was much brighter than any other. Tycho was well aware that the appearance of a new star in the firmament, a nova as he called it, was extraordinary. He characterized his discovery as "the greatest wonder that has ever shown itself in the whole of nature since the beginning of the world." At the very least, Tycho understood that the spectacle he witnessed contra
dicted the ancient astronomy of Aristotle.
We now know that Tycho observed a cataclysmic explosion that is called a type I supernova. This celes
tial event is the final evolutionary stage of a white dwarf star. It occurs when the core of the star collapses,
Tycho Brahe was born in his family's ancestral seat, Knutstorp Castle, in 1546. At the age of two he was abducted by a paternal uncle and childless aunt, who brought him up. Tycho studied at the University of Copenhagen and at several renowned universities in the Germanic territories. It was at Basel that he learned the new theories of Copernicus, which he did not be seen in several subsequent portraits.)
After his return to the Kingdom of Denmark, Tycho's social standing afforded him the leisure and wealth to pursue his interest in astronomy. In time, the King of Denmark, Frederick II, offered Tycho a choice of several fiefdoms. Tycho demurred. He wrote to a spotted the island of Hven. He granted it to Tycho and
the island in 1576. For the next twenty-one years, to honor his father's pledge to allow Tycho's children to inherit Hven. (Because Tycho had married a com
moner, his children did not have automatic rights of inheritance.) Tycho abandoned Hven for Copenhagen, where he started over. When Christian had him dis ponder the mechanics of the solar system. He published his first book on the planets, the Mysterium Cosmo
graphicum, in 1596. Though this work is more flight-of
fancy than science, it reveals Kepler's early interest in the three mathematical questions that he would even
tually answer with his laws of planetary motion.
By nature, Kepler was more a mathematician than a stargazer. He did not have the instruments of Tycho, and his eyesight was poor. Tycho, to whom Kepler sent a copy of his book, commented that the observations of Copernicus on which Kepler relied were not sufficiently accurate to be the basis of the conclusions Kepler reached. Moreover, Tycho did not think highly of Kepler's inclination to theorize. Tycho chided Kepler that the force behind the motion of the planets could only be Tycho did not share with other scholars. The problem was how to gain access. Referring to Tycho's wealth of data, Kepler wrote "Like most rich men Tycho does not know how to make proper use of his riches. Therefore one must take pains to wring his treasures from him."
As it happened, fate brought the two men together.
At the same time royal hostility was driving Tycho from his native Denmark, religious persecution was driving Kepler from his post in Graz. Tycho made his way to Prague, the capital of the Holy Roman Empire, in 1599.
Kepler arrived shortly afterwards. Although Tycho maintained a tight grip on his secrets at first, he even
tually allowed Kepler to work on the orbit of Mars. Its eccentricity had caused great difficulties for all the circular motion theories that Ptolemy, Copernicus, and Tycho had put forth.
In 1601, Kepler became Tycho's salaried assistant.
Only a few days later, Tycho attended a banquet where, according to Kepler's account, he "drank a little over
generously and experienced pressure on his bladder.
He felt less concern for the state of his health than for etiquette," which required guests to remain seated at the dinner table. On the basis of this document, Tycho's death has been traditionally attributed to a burst bladder. Other reports, originating soon after Tycho's death, raised the suspicion of heavy metal poisoning.
The modern verdict is that Tycho died of kidney failure brought on by enlargement of the prostate.
Tycho was laid to rest in the famous T:Yn Church of Prague. During the religious turmoil of the 1620s, many graves were desecrated. According to legend, Tycho's corpse was removed from its burial site during one of those desecrations. In 1901, the 300th anniversary of his interment, Tycho's crypt was refurbished. That restoration provided an opportunity to investigate the contents of the crypt. It was opened and the grave
robbing story debunked: The male skeleton found there had a cranial wound that was consistent with the injury Tycho suffered during his sword fight. In the late 20th Mathematician. By compensating Tycho's heirs, the emperor was able to ensure that Tycho's observa
tions were placed at Kepler's disposal. Those mea
surements were all Kepler had to work with-there were no established physical theories such as gravita
tion to guide him, no advanced mathematical tools such as analytic geometry and calculus to aid in calculation,
Genesis & Development 857
no tools such as logarithms to ease the burden of numerical computation. After four years of intense mathematical labor, Kepler discovered the first two of the planetary laws that now carry his name. He pub
lished them in his Astronomia Nova of 1609.
For the most part, Kepler left remarkably candid accounts of the steps that precipitated his discoveries.
In the case of his third law, however, Kepler was unu
entist has remarked, Kepler's standard biography has passages that can bring its readers to tears. Poverty, religious persecution, war, smallpox, typhus, plague
Kepler's life was an unrelenting struggle filled with hardship and sorrow. In his 58th Year, he had a pre
monition of death and composed his own epitaph:
I used to measure the heavens, now I measure the shadows of the Earth.
Although my mind was heaven-bound, the shadow of my body lies here.
A few months after penning this distich, Kepler took ill and died. The peace that eluded Kepler in life, eluded his mortal remains as well. Scarcely a few years passed before war ravaged the churchyard of St.
Peter's, Regensburg, obliterating every trace of Kepler's burial site.
Sir Isaac Newton
The idea of a gravitational force originated in the early 16th century. By the last half of the 17th century, the foremost question of natural philosophy had become, How could Kepler's laws of planetary motion be derived from a theory of gravitation? The first step was to deduce the form of the gravity law. This was a problem with which Kepler wrestled unsuccessfully for thirty years. Ironically, his third law became one of the two keys that together were used to unlock the secrets of gravity. Christiaan Huygens (1629-1695) provided the
second when, in 1659, he published the law of cen
trifugal force for uniform circular motion.
If a body of mass m moves with constant speed v
terbalances the planet's centrifugal force. Now imagine planetary motion in an elliptic orbit with semi-major substitute this formula for v into Huygens's formula for centrifugal force, we find
IFI
=
mv2 = 4�m. _!__r k r2
The fall of an apple caused Newton to reflect on the nature of gravity, but it was by these considerations that he deduced, in 1666, the law of gravity. Independently, Sir Edmond Halley, Sir Christopher Wren, and Robert Hooke came to the same conclusion several years later.
The next step was to show that the inverse square law implies Kepler's laws. During a visit to Cambridge in 1684, Halley asked Newton if he knew the curve that an inverse square law would entail. Without hesitation, Newton answered that it would be an ellipse. Three months later, he sent Halley a short manuscript that derived the three laws of Kepler from the inverse square law of gravitation. For the next fifteen months, Newton went into seclusion, devoting himself to setting
out a general science of dynamics that included the Universal Law of Gravitation as well as Kepler's laws.
The result was Newton's Principia, the most important scientific treatise ever written.
Publication was always an anxious process for New
ton. In 1672, he was forced to defend a paper on optics against several criticisms, most notably from Robert Hooke, who managed to question both Newton's con
clusions and Newton's priority. Newton found the busi
ness tiresome. Shortly thereafter, he declined to publish a more complete treatment of optics, remarking that with further use of the press "I shall not enjoy mY former serene liberty." Many years later, when Newton looked back on the ensuing period of silence, he explained, "I began for the sake of a quiet life to decline corre
spondencies by Letters about Mathematical & Philoso
phical matters finding [that they] tend to disputes and controversies."
The Principia interrupted Newton's quiet life, embroiling him in a new dispute with his old nemesis, Robert Hooke. As the Principia neared completion, Hooke began to stir things up. Halley, who was over
seeing the publication of the Principia, communicated the problem to Newton: "Mr Hook has some preten
sions upon the invention of ye rule of the decrease of Gravity . . . He sais you had the notion from him ...
Mr Hook seems to expect you should make some mention of him in the preface." Newton was aggra
vated: "Philosophy is such an impertinently litigious Lady that a man had as good be engaged in Law suits as have to do with her. I found it so formerly and now I no sooner come near her again but she gives me warning."
Newton responded to the baseless charge of plagiarism by deleting some references to Hooke that were already in the manuscript. In private correspondence, he referred to his antagonist as an "ignoramus."
After completing the Principia, Newton rapidly lost interest in mathematical research. A number of docu
ments leave no doubt that he suffered a complete mental breakdown in 1693. The cause remains uncer
tain, but mercury poisoning (resulting from carefree handling of substances in chemical experiments) is a plausible candidate. Samples of Newton's hair, taken from four preserved locks, were analyzed in 1979.
Although elevated levels of mercury, antimony, arsenic, gold, and lead were measured, a conclusive diagnosis is not possible 300 Years after the illness.
What is not in question is that in the remaining thirty
four years of his life, Newton's scientific activity was
largely confined to polishing the exposition of earlier work. He retired from academic life in 1696, serving at the Mint until his death in 1727.
In 1820, the apple tree that set Newton to think about the nature of gravity succumbed to disease and was felled. Scions were taken and the line lives on, both in England and the United States. The fruit is pear
shaped and, it is said, without flavor.
Many unexpected details concerning Newton's life came to light in the 20th century. Newton's interests in alchemy, theology, and biblical chronology had long been known. However, the extent of those interests
Genesis & Development 859
remained hidden until 1936 when a large Portion of his estate was put up for auction. Of the volumes in Newton's personal library, 3% and 7% pertained to physics and mathematics, respectively, whereas 8% and 27.5% concerned alchemy and theology. Of Newton's personal papers, about 1 million words were devoted to scientific subjects compared to 2 million words on alchemy, theology, and chronology. These revelations led the great economist John Maynard Keynes (1883-1946) to say of Newton, "Not the first of the age of Reason. He was the last of the magicians."
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