Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.05 Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. Wednesday, April 25, 2005 Main Idea: Expanding you understanding of ideas. Key Words: Dedekind, Cauchy, Real, Rational Goal: Give some perspective on the real numbers. --------------------------------------------------------- Previous assignment: Page 556 Problem 2 Name the conic that has the given equation. Find its vertices and foci, and sketch the graph. 2 y -6 x = 0 2 y = 6 x Parabola vertex is (0,0) focus is ( 3/2,0) -------------------------------------------------- p = 6/4; p1 = ParametricPlot[{y^2/6,y},{y,-5,5}]; p2 = ListPlot[ {{0,0},{p,0},{p,2p},{p,-2p}}, PlotStyle->{RGBColor[1,0,0],PointSize[0.1]}]; p3 = ParametricPlot[{p,y},{y,-2p,2p}, PlotStyle->{RGBColor[1,0,0],Thickness[0.02]}]; p4 = Show[p1,p2,p3,AspectRatio->Automatic, PlotLabel->"P556 P2 y^2-6x = 0"]; Display["2.ps",p4]; ------------------------------------------------------------- Page 556 Problem 6 Name the conic that has the given equation. Find its vertices and foci, and sketch the graph. 2 2 x - 4 y -16 = 0 2 2 x y ---- - ---- = 1 16 4 Hyperbola vertices (4,0) and (-4,0) foci (Sqrt[20],0) and (-Sqrt[20],0) -------------------------------------------------- a = 4; b=2; c=Sqrt[20];h = b^2/a; p1 = ParametricPlot[{ 4 Sqrt[1+y^2/4],y},{y,-5, 0}]; p2 = ParametricPlot[{-4 Sqrt[1+y^2/4],y},{y,-5, 0}]; p3 = ParametricPlot[{ 4 Sqrt[1+y^2/4],y},{y, 0, 5}]; p4 = ParametricPlot[{-4 Sqrt[1+y^2/4],y},{y, 0, 5}]; p5 = ListPlot[ {{a,0}, {-a,0}, {c,0}, {-c,0}, {c,h}, { c,-h},{-c,h},{-c,-h}}, PlotStyle->{RGBColor[1,0,0],PointSize[0.02]} ]; p6 = ParametricPlot[{-a, y},{y,-b,b}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p7 = ParametricPlot[{ a, y},{y,-b,b}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p8 = ParametricPlot[{ x, b},{x,-a,a}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p9 = ParametricPlot[{ x,-b},{x,-a,a}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p10 = ParametricPlot[{-c, y},{y,-h,h}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p11 = ParametricPlot[{ c, y},{y,-h,h}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p12 = Plot[ b/a x,{x,-8,8}, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; p13 = Plot[-b/a x,{x,-8,8}, PlotStyle->{RGBColor[0,1,0],Thickness[0.01]}]; p14 = Show[p13,p12,p11,p10,p9,p8,p7,p6,p5,p4,p3,p2,p1, AspectRatio->Automatic, PlotLabel->"P556 P6 x^2-4y^2-16=0 "]; Display["6.ps",p14]; ------------------------------------------- Page 556 Problem 10 Name the conic that has the given equation. Find its vertices and foci, and sketch the graph. r(2+Cos[theta]) = 3 3 3 1/2 r = ------------------ = ---------------- 2+Cos[theta] 1 + 1/2 Cos[theta] Ellipse with one Focus is at the pole e = 1/2; d = 3 Focus = (0,0) Directrix: x = 3; a-c = 1 c/a = 1/2 a-c = 1 c = 1/2 a a = 2 c = 1 b = Sqrt[3] center = {-c,0}; ------------------------------------------------------ a = 2; c = 1; b = Sqrt[a^2-c^2]; center = {-c,0}; h = b^2/a; p1 = ParametricPlot[ 3/(2+Cos[t]) {Cos[t],Sin[t]},{t,0,2 Pi}]; p2 = ListPlot[ {center+{a,0}, center+{-a,0}, center+{c,0}, center+{-c,0}, center+{c,h}, center+{ c,-h},center+{-c,h},center+{-c,-h}}, PlotStyle->{RGBColor[1,0,0],PointSize[0.02]} ]; p3 = ParametricPlot[center+{-a, y},{y,-b,b}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p4 = ParametricPlot[center+{ a, y},{y,-b,b}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p5 = ParametricPlot[center+{ x, b},{x,-a,a}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p6 = ParametricPlot[center+{ x,-b},{x,-a,a}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p7 = ParametricPlot[center+{ c, y},{y,-h,h}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p8 = ParametricPlot[center+{-c, y},{y,-h,h}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; p9 = Show[p8,p7,p6,p5,p4,p3,p2,p1, AspectRatio->Automatic, PlotLabel->"P556 P10 r = 3/(2+Cos[theta]) "]; Display["10.ps",p9]; -------------------------------------------------------- Page 556 Problem 14 Find the Cartesian equation of the conic with the given properties. Eccentricity 5/3 and vertices (0, 3) and (0,-3) Hyperbola a = 3 c = 5 b = 4 2 2 x y - ------ + ------- = 1 16 9 ------------------------------------------------------- Page 556 Problem 18 Find the Cartesian equation of the conic with the given properties. Hyperbola with vertices (2,0) and (2,6) and eccentricity 10/3 center = (2,3) a = 3 c = 10 b = Sqrt[91] 2 2 (x-2) (y-3) - ---------- + --------- = 1 91 9 ------------------------------------------------------------ Page 556 Problem 22 Transform the given equation to a standard form. 3x^2 - 10 y^2 + 36 x -20 y + 68 = 0 3( x^2 + 12 x ) - 10(y^2 + 2 y ) = -68 3( x^2 + 12 x + 36 ) - 10(y^2 + 2 y + 1 ) = -68 + 108 -10 3(x+6)^2 -10(y+1)^2 = 30 2 2 (x+6) (y+1) ------------ - ------- = 1 10 3 Hyperbola -------------------------------------------------------------- Page 556 Problem 26 Analyze the given polar equation and sketch its graph r = 5/Sin[theta] r Sin[theta] = 5 y = 5 ------------------------------------------------------- Page 556 Problem 30 Analyze the given polar equation and sketch its graph r = 5 - 5 Cos[theta] Cardioid ------------------------------------------------------- Page 556 Problem 34 Analyze the given polar equation and sketch its graph r = 4 Sin[3 theta] Three leaf rose ---------------------------------------------------- Page 556 Problem 38 2 Find a Cartesian equation of the graph of r Cos[2 theta ] = 9 and then sketch the graph. 2 2 2 r ( Cos [theta] - Sin [Theta] ) = 9 2 2 x - y = 9 --------------------------------------------------------------- Page 556 Problem 42 Find the area of the region that is outside the limacon r = 2 + Sin[theta] and inside the circle r = 5 Sin[theta]. ---------------------------------------------------------------------- Integrate[ 1/2 ( (5 Sin[t])^2 - ( 2 + Sin[t])^2 ),{t,Pi/6,5 Pi/6}] a = ParametricPlot[(2+Sin[t]){Cos[t],Sin[t]},{t,0,2 Pi}]; b = ParametricPlot[ 5 Sin[t]{Cos[t],Sin[t]},{t,0,2 Pi}]; c = Plot[t Tan[Pi/6],{t,0,5 Sin[Pi/6] Cos[Pi/6]}]; d = Plot[t Tan[5 Pi/6],{t,0,-5 Sin[Pi/6] Cos[Pi/6]}]; e = Show[a,b,c,d,PlotLabel->"Page 556 Problem 42; r=2+Sin[t], r=5 Sin[t]", AspectRatio->Automatic]; Display["42.ps",e]; ---------------------------------------------------------------------- 2 + Sin[t] = 5 Sin[t] 2 = 4 Sin[t] Sin[t] = 1/2 t = Pi/6 and 5 Pi/6 t = 5 Pi/6 INT 1/2 ( (5 Sin[t])^2 - (2+Sin[t])^2 ) dt = Sqrt[3]+8 Pi/3 t=Pi/6 ====================================================================== New Material: I. What is a number? The easy explanation is a number is something like 36.1256.... where the decimals can go on forever. While this seems true, how do you tell if two such numbers are the same, or how do you tell which is bigger? A. When asked this further question, the answer is that one compares the digits and if the digits are all the same the numbers are equal, and if one finds a place where the numbers differ, the number with the bigger digit is bigger. B. This second statement implies that before we can call something a number, we have to have a way of expressing it in decimal form. One could view this as having a recipe, formula, algorithm to produce as many digits of the number as desired. With this type of definition, one can add, subtract, multiply, and divide most numbers, C. If you know the initial segment of two numbers, you can get a shorter initial segment of the sum, difference, product, or quotient. 1. This is not entirely true because 0.999999999999999999........ +0.000000000000000000........ ---------------------------------- You do not know whether to start writing down a 1.000000000000.... or 0.99999999..... Thus this definition is not adequate. II. The natural numbers are 1, 2, 3, .... We pretty much consider these as self evident. They are the counting numbers. The cave man supposedly had a box of pebbles and when the sheep went out in the morning, he took a pebble out of the box. When they came back in the evening, he put a pebble into the box. If all the pebbles were not in the box, he went out looking for sheep. A. I never did consider this realistic. I am not even sure that anybody counted things until one had quantities of things which were identical. I think the Eskimo does not have five dogs, he has a lead dog, a strong male, a younger male, about to replace the older male, a feisty two year old and a puppy who is along to learn. B. Up to about 5, I think that people know how many there are by grouping. Think of the spots on a die. You know how many spots there are without counting them. C. In the old days people argued about everything. Do the numbers start at one? If you have one object, you do not have a number of them. Thus the first number should be two. D. It was even harder to accept 0 as a number. If you do not have anything, how can you even begin to count them. III. The rational numbers were easier to accept than the negative integers. A. The Egyptians had numbers for 1/n and except for 2/3, they had no other fractions. A big part of learning mathematics back then was to convert the sum of fractions into fractions of the form 1/n. Thus 5/6 was written as 1/2 + 1/3. 7/8 was 1/2+1/4+1/8. Since this way of writing was not unique, they had to have a special technique to tell if two fractions were equal. IV. We think our way of writing numbers is superior to what went before. Here is the official way we think of numbers. A. We start with the positive integers. We know how to add subtract, multiply, and divide with remainder the integers. we want to extend this knowledge to include negative numbers. B. So first, what is a negative number, and how do we add them? If add the symbols 0 and -n and we have the integers, But we do not have a direct way of defining describing their properties. There is a more sophisticated way of creating the negative numbers. 1. To visualize the negative integers, we look at the points of the plane with integral coordinates. We partition them into stripes. Notice that the sum of two stripes point by point form a third stripe. |/ / / / / / / / / / / -3 | / / / / / / / / / / | / / / / / / / / / / | / / / / / / / / / / / |/ / / / / / / / / / / -2 | / / / / / / / / / / | / / / / / / / / / / / |/ / / / / / / / / / / -1 | / / / / / / / / / / / | / / / / / / / / / / / |/__/__/__/__/__/__/__/__/__/__/________ 0 1 2 3 4 5 6 a. Positive integers are ordered pairs of integers (a,b) where a > b. b. Negative integers are ordered pairs of integers (a,b) where a < b. cc 0 is that line were a=b. C. The rational numbers are again done by extending the number system we already have created. The rational numbers are pairs of integers where (a,b) = (c,d) if ad = bc. Think of (a,b) and a/b. 1. The problem with rational numbers is that the same fraction has two different names. Thus 1/2 = 2/4 = 3/6. Thus we think of 1/2 not as (1,2) but as the who set of ordered pairs. --------------------------------------------------------------- D. Now jumping from the rationals to the reals is a very big jump. There are two ways to do this. Cauchy sequences and Dedekind cuts. 1. We start with Cauchy sequences. One accepts that the real numbers are the limits of sequences of rational numbers. But the problem is that we cannot talk about the limit of a sequence unless we have that limit. We have to say that |an - L| goes to zero as n ---> infinity. What we need is a way to distinguish converging sequences without first knowing the limit. The fellow who discovered how to do this was Cauchy. a. The Cauchy criterion for a sequence converging is given here. A series is called Cauchy if given any epsilon, there exists an N such that for any n,m > N |an - am | < epsilon. Cauchy sequences are just the convergent sequences, but the big thing is that Cauchy did not have to already know the limit before he knew the sequence converged. 2. Now to Cauchy, the real numbers are just the Cauchy sequences. He showed that the term by term sum, difference, product, quotient of Cauchy sequences is still a Cauchy sequence. Thus he had a model of the real numbers. E. Dedekind used another strategy. He plugged the holes in the rational number line. If he could divide the rationals into two havess, where each number in one half was less than each number in the other half, he had a Dedekind cut. Thus to represent Sqrt[2] the two haves were: Negative and those positive rationals whose squares were less than 2. Those positive rationals whose squares were greater than 2. Dedekind showed that the sum, difference, product, quotient, of two Dedekind cuts was again a Dedekind cut, and to him, the real numbers were these Dedekind cuts of rationals. ------------------------------------------------------------ V. Again, lurking behind all of this is the original question, What do you mean by "ALL". 1. If one has a Cauchy sequence, presumably one should be able to express the numbers in the sequence. This requires some sort of rule. The "Set" of all Cauchy Sequences sweeps it under the rug because we presumably have all of them. But is "all" just those which can be expressed with a formula in words? Or will we allow other ones. --------------------------------------------------------- 2. The same for Dedekind cuts. Usually one states them as the set of all rationals with some property. Again, we are limited to words. Coming up on Wednesday, the axiom of choice. What is reality? How does your vocabulary induce what you see or remember?