Math 1080 - 1 Assignments
Term Project
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- Math 1080 - 1 Term Paper Due April 12, 2003 Treibergs - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Here are some suggestions for your Semester Project, to write a short paper about some specific topic of calculus. Calculus includes all kinds of infinite processes: differentiation, the computation of a limit, the computation of an infinite sum, or integration. Your paper could be on an important development, breakthrough or theorem. You could do some experiment or research. Or you could write on a mathematical model or an application. Think of a topic that might make a good science fair poster project. Your paper should include a quick mathematical description of the calculus involved, as well as a discussion of its history and importance. Please talk to me and get my approval for any project other than those suggested. Three or four pages plus displays and references should suffice, but say enough to make your argument. Explain the importance and history. Explain the mathematics as appropriate by giving formulas, examples, experiments, diagrams, and computer runs. Provide detailed references, particularly for internet sources. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Examples of topics might be: @ Driving Experiment. We have talked about how to deduce the average velocities from position vs. time data. Take a 20 minute ride and record actual data. You should record both mileage to the nearest tenth of a mile every minute. You should also record the speedometer reading at that instant. Your analysis should include the computation of average velocities, as we did with imaginary data during the first weeks. You should also compute of the area under the velocity curve to show how it is related to the distance travelled. (See me for more details.) @ Archimedes determination of Pi. How do computers calculate as many digits of Pi as anyone ever would want? What is the current worlds record for the number of digits? Archimedes was among the first to develop a method that could give an approximation of Pi to any desired accuracy. From his analysis, the Greeks used the approximation that is familiar from highschool Pi is approximately 22/7. Archimedes ``method of exhaustion'' is very simple. He approximated the inner and outer areas by regular polygons. As the number of sides increases, the area of the circle is more closely approximated. In fact, simple formulas give the areas as the number of sides is doubled. (Knowing a little trigonometry might be useful. [D], [Y]) @ Xeno's Paradoxes. The Pythagorean philosohers belived that nature is simple, that all things ultimatly boil down to finite numbers. Atoms are a finite size and time comes in packets. One of their most outspoken critics was the Eleatic philosopher Xeno, 450 B.C., who belived in the continuity of time, and who devised some thought experiments that seemed paradoxical seen from the Pythagorean view. One of his paradoxes, "Achilles and the Tortoise," is a story of Achilles who spies a tortoise on the road and runs after it. Achilles runs to the point where he first spotted the tortoise, but during that time, the tortoise has moved along to a new point. Then Achilles runs to the new point, but the tortoise moves along to another point. This process continues on and on but Achilles never manages to catch up to the tortoise. How do you resolve the paradox? (The mathematics you'll need might involve is a little bit about infinite sums. That is, you will have to make sense of expressions like 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... A reference among many is [B1].) @ Newton vs Leibniz: the Fundamental Theorem of Calculus. The fundamental theorem of calculus relates the two major branches of calculus, the differential (computation of slopes by approxinmating by difference quotients) and integral (computation of areas under curves by approximating by sums of rectangles.) Briefly stated, it says that is given a function y=f(x) defined for a < x < b , then the area in the region between the lines x=a , x=b , the x-axis and the curve y=f(x) (the integral) is exactly F(b)-F(a) where F is a function whose derivative is f , i.e.,) F'(x)=f(x) (the antiderivative.) In symbols, b / | (F.T.C.) | f(x) dx = F(b) - F(a). | / a Newton viewed the derivation dynamically whereas Leibniz was more geometric. It took several attempts before Newton was able to provide a explanation without using imprecise "infinitessimally small quantities." Although people could find slopes and areas earlier, both Newton and Leibniz were the first to see the F.T.C. clearly, that one processs is the inverse of the other. Did Leibniz see an early manuscript of Newton's or did they work independently? Did they both find inspitration in the same sources? Or have historians invented a controversy on nationalistic grounds? (this issue is discussed in several histories, such as [B2], [C], [K], [S]) @ Buffon's Needle Problem, or how to find Pi experimentally. Georges Leclerc, Compte de Buffon (1707-1788) posed the following problem, which is in a subject now known as geometric probability. The experimental determination of Pi from his formula, is an example of a "Monte Carlo method." Suppose that a needle of length L is dropped randomly onto a wooden floor covered by parallel boards of width d > L , what is the probability that the needle will touch a crack? The answer is 2 L (1.) P = ----- Pi d If you were to drop a needle on the floor N times, and the number of times that the needle touches a crack is n then P is about n / N. It follows that N L Pi ~ 2 --- n d Do the experiment a few hundered times and compute your estimate. If you know programming, you could ask the computer to run through lots of trials. More generally, you could discuss integration using Monte Carlo techniques. (The easiest derivation of formula (1) is using an integral [Be]. Without it, it's a little more involved [D].) @ Euler's Number. e occurs in a banker's formula for how much A dollars is worth after T years if the interest at rate r is compounded k times a year. As the number k increases without limit, the limiting amount is k lim A (1 + Tr/k) = A exp(T r). k -> infinity It turns out that e ~ 2.718281828 . e is an amazing number. The only function f(x) which is its own derivative ( f'(x)=f(x) ) is f(x)=Aexp(x). An infinte summation formula for the number can also be given e = 1+1/1 1/2 + 1/6+1/24 + 1/120 + ... + 1/n! + ... Whatever you describe, you should explain the meaning of the limit or the infinite sum or whatever math is involved. ( [D], [Y]) @ The theory of games. Game theory is about strategies for obtaining optimal outcomes in "games" modelling deals or negotiations. One of the most famous is called the Prisoner's Dilemma. Two suspects are captured on suspicion of having committed a crime together. The prisoners are isolated from each other and are each offered the following deal: if neither offers evidence against the other then they will both be sentenced to three years jailtime. If one offers evidence against the other but the other does not, then the one giving evidence will get one year, the other will get ten years. But if both offer evidece against each other, then both get five years. If the first prisoner doesn't squeal, he might get ten years if the other also squeals. But if he does squeal, then he could get five years if the other squeals. So what to does the prisoner do? The dilemma is that each prisoners cannot make a good choice between two options without knowing what the other prisoner will do. If each acts unilaterally, then the situation might turn out badly for both. But if they cooperate, then it may not be as bad. John Nash's beautiful thesis was about cooperative equilibria in game theory. [N]. @ Steiner's argument that the circle has the largest area among planar figures with given length. Steiner's idea is to replace a given figure G in the plane by the figure S(G) symmetrized about a line L . Say if L is the horizontal axis, then the symmetrized figure is gotten by sliding each section of the figure vertically until its midpoint lines up on L . (see fig. 1) Integration shows that the symmetrized figure S(G) has the same area as G but has a shorter boundary length |bS(G)| <= |bG|. By choosing lines all in different directions, the sequence of symmetrized figures G_1= G, G_2=S(G_1), G_3=S(G_2), G_4=S(G_3),..., G_{n+1)=S(G_n),..., get rounder and rounder and converge to a circle C whose area is the same as the area of G . If L_n is the length of bG_n then since each symmetrization decreases the length, the length of bG is |bG| = L = L_1 >= L_2 >= L_3 >=...\ge L_n >= ... >= \sqrt{4 Pi A} where A is the area of G and \sqrt{4 Pi A) is the length of the boudary of the circle C whose area is A . 4 Pi A <= L^2 is called the isoperimetric inequality. It is also true, that if G is any figure such that 4 Pi A=L^2 then it must be the circle. [Bl] @ Computer algebra systems. MAPLEŠ is a computer algebra system has enormous capacity to do calculus problems and graph. Sketch the development of MACSYMA or MATHEMATICA or any similar system. Get an account and demonstrate its powers. @ The Bell Curve, The reliability of exit polls has to do with the number of people being surveyed. The nub is Laplace's bell curve theorem, the law of large numbers. [Y] @ Modelling competitive species. Many models in biology and physics have to do with differential equations. For example, why are the populations of wolves and rabbits on Isle Royale oscillatory functions of time? The predator-prey model postulates that the rate at which some variables grow (say as the growth rate of the population of wolves x' ) depends on the variables themselves (such as x' = c_1 x + c_2 x y where c_i are positive constants. Here the wolf population is positively affected by both the presence of lots of eligible mates and the presence of lots of rabbits y to eat. Meanwhile the growth of rabbits y' = c_3 y - c_4 xy is positively influenced by lots of other rabbits around but detrimentally influenced by all those wolves eating rabbits.) Some differential equations can be easily solved using some algebraic tricks and integration. Knowledge of exponentials and logs is essential for this project.) @ Fractals. Fractals are sets in the line or the plane that have fractional dimension. There is a huge general theory, but some of the starting notions can be seen using infinite sums. For example, one of the most famous fractals is the Koch snowflake curve. Start with an equilatreral triangle. To each edge of length L , attach an equilateral triangle of length L/3 at the middle. After doing this to a triangle you get thhe Star of David. Doing it again gives a spiny Star of Davis with 48 edges of length L/9 . Repeat over and over. The resulting region has a finite area which you get by summing the areas of all the little triangles. The length is increased by a factor of 4/3 each step, thus the limiting star has an infinitely long boundary if it were one dimensional. It turns out to have dimension between one and two. Fractals have been popularized by movies\dots anyone see the Star Trek Movie with the Genesis Project? [Y] @ Soap bubbles. Calculus can be used to describe the shape of natural objects. A soap bubble has the smallest possible area among closed surfaces that enclose a given volume. The equation that governs soap films and bubbles says that the inward force of the membrane at all points is exactly balanced by the pressure inside the bubble, which is constant. However, the inward force of the membrane depends on the curvature of the surface. Describe how to measure the curvature of the surface, and some of its properties. "Minimal Surfaces" is a mathematical subject that belongs to the larger area "differential geometry." It may involve heavy duty differential calculus. However, there are popular accounts, as well. [By], [M] - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - @ References [Be] P. Beckmann, A History of Pi, St. Martin's Press, New York, 1971. [Bl] W. Blaschke, Kreis und Kugel, Chelsea, New York, 1949, Originally published by Teubner, Leipzig, 1916 [By] C. Boys, Soap Bubbles, Their Colors and Forces which Mold Them, Dover Publ. Inc., New York, 1959, Originally published in 1911 [B1] C. Boyer, A History of Mathematics, Princeton University Press, Princeton, , 1985., Originally published by John Wiley & Sons, New York, 1968. [B2] C. Boyer, The History of the Calculus and its Conceptuual Development, Dover, New York, , 1959, Originally published by Hafner Publishing Co., New York, 1949. [C] R. Calinger, Classiccs of Mathematics, Prentice Hall,, 1995, Englewood Cliffs, N.~J. [D] H. Dorrie, Triumph der Mathematik: Hundert beruhmpte Probleme aus zwei Jahrtausenden mathematischer Kultur, Physika-Verlag, Wurzburg, 1958, English transl. of 5th ed., 100 Great Problems of Elementary Mathematics, their Historyand Solution, Dover, 1965, New York, 184--188 [Fo] E. Fourrey, Curiosit\'es G\'eom\'etriques, Librairie Vuibert, Paris, 1938 [K] M. Kline, Mathematical Though from Ancient to Modern Times, Oxford University Press, New York, 1972 [M] F. Morgan, Calculus Light, {\smc 2nd ed.), A. K. Peters, Ltd., Wellesley, 1997, p. 66 [N] J. Newman, The World of Mathematics, Simon \& Schuster, New York, 1956 [P] G. Polya, Mathematics of Plausible Reasoning, Princeton U. Press, Princeton, 1954, Volume I: Induction and Analogy in Mathematics [S] D. Struik, A Consise History of Mathematics, Dover, New York, 1967 [TG] S. Thompson & M. Gardner, Calculus Made Easy, St. Martin's Press, New York, 1998, Originally puplished by Macmillan, London, 1910 [Y] R. Young, Excursions in Calculus: An Interplay of the Continuous and the Discrete, Dolciani Mathematical Expositions No. 13, Mathematical Association of America, Hartford, 1992 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Homework Problems
Math 1080 - 1 Eleventh Homework Assignment Due April 18, 2003 Treibergs -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- Reminder: Term projects are due, Friday, April 18, 2003. You are welcome to discuss your projects with me if you have any questions about the mathematics or the suitability of your topic. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Read Chapter 10 of Sawyer's "What is Calculus About?" and the handout from Hughes-Hallet et. al. "Calculus." Answer FIVE of the following problems. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= (1.) Suppose you invest $ 1000 at an interest of 7% per year, compunded yearly. How much is in your account after eight years? What would it be if the interest is compunded semiannually? quarterly? monthly? continuously? (2.) A bank account is earning interest at 6% per year compounded continuosly. [See 58[6] on the handout.] (a.) By what percentage has the bank balance in the account increased over one year? (This is the effective annual yield.) (b.) How long does it take the balance to double? [Hint: the natural logarithm is the inverse operation to e^x= exp(x). Thus if you know e^x=a then apply ln to both sides to get x=ln(e^x)=ln(a). For example if e^(2y) = 1.3 then 2y = ln(1.3) = .262364 or y = .131182 . Similarly exp is the inverse operation to the logarithm. Thus if ln(x)=3 then apply exp to both sides to get x= exp( ln(x) ) = exp(3) = e^3=20.0855.] (c.) Assuming the interest rate is i , find a formula giving the doubling time in terms of the interest rate. (3.) Compute (a.) 1 + 3 + 9 + 27 + 81 + ... + 177147 (b.) 1 /16 - 1 / 32 + 1 / 64 - 1 / 128 + ... (c.) infinity ----- \ n ) 5 (.1 ) / ----- n = 4 (4.) Express the repeating decimal fraction 1.07107107107... as a rational number (a ratio m/n where m,n are whole numbers.) (5.) Suppose that you save money by putting $500 a year into a savings account that earns 7% per year compounded annually. How much would be in your account right after the 20th deposit? (6.) Suppose that the owners of the Jazz wished to give Karl Malone a new contract starting next year for $15 million a year for the next five years. How much money would the owners of the Jazz have to deposit on the day of the signing in order to cover all of the payments, assuming that the account earns interest at 6% compounded annually throughout the period of the contract? What would the present value of the contract be if $15 million a year was paid forever? Assume that the interest is 6% per year compounded annually. (7.) The following fractal set is called the "Sierpinski Carpet." Starting from the unit square, divide it into nine equal squares and then remove the middle one, leaving eight squares. From these eight, remove the middle square from each of them. The Sierpinski Carpet is the set that's left if this process is continued infinitely. What is the area of the Sierpinski Carpet? [Hint: figure out how many square are removed at the n th stage and how much area is removed. Then add up the total area removed and subtract it from the original area.] -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 Tenth Homework Assignment Due April 11, 2003 Treibergs Read Chapter 9 of Sawyer's "What is Calculus About?" Answer the following problems. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (1.) This question is based on Figure 58. Suppose that F(t) denotes the area of the region bounded by the curves y=0, x=t and x=y. Find F(t) geometrically. Draw a diagram that shows that the total area F(3+h) consists of the areas of the triangle F(3) plus the trapezoid bounded by the curves y=0, x=t, x=t+h and y=x. Determine the area of the trapezoid geometrically. Check that it equals the difference of the areas of the bigger triangle minus the smaller Area(trapezoid) = F(3+h) - F(3). Then find F'(t), the limit of F(3 + h) - F(3) --------------- h as h tends to zero. Because the area of the trapezoid is about h times the height of the right edge, show that this limit is the same as the height of the right edge of the triangle. (2.) This time, lets do the same computation, except we will not have a convenient formula for the areas involved. Suppose that F(t) denotes the area of the region above the line y=0, between x=1 and x=t but below the curve xy=1. Find F(3) by doing the following. On a LARGE graph, carefully plot the curve xy=1 between 1 and 3. Count the number of squares of the graph paper that lie beneath the curve. You will have to add in some fractional squares. Do the best that you can. Then figure out the area of ONE of the little squares and multiply to get F(3). (If you really must know, the answer is the natural logarithm ln(3), which is on your calculator.) Draw a diagram that shows that the area F(3+h) consists of the area of the region whose area is F(3) plus the curvilinear trapezoid bounded by the curves y=0, x=t, x=t + h and xy=1. Determine the approximate area of the trapezoid geometrically. Since Area(trapezoid) = F(3 + h) - F(3), use your approximation to find F'(3) as the limits of F(3+h)-F(3) Approximation of Area(trapezoid) h x height of edge ----------- ~ -------------------------------- ~ ------------------ h h h as h tends to zero. Because the area of the trapezoid is about h times the height of the right edge, show that this limit is the same as the height of the right edge of the region. (3.) In this problem we compute the volume of a right circular cone with height a and radius of base b, following the discussion starting on p. 90 about computing the volume of a sphere. We paste together a little pile of paper disks, whose radius gets larger and larger as t moves from 0 to a, the ``height'' of the cone. If b is the radius of the base, what is the radius PR of the disk at time t? (It is PR^2 = 1 - t^2 for the unit sphere.) What is V'? What is V(t)? What is V(a), the volume of the cone? -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 Ninth Homework Assignment Due April 4, 2003 Treibergs - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - In this homework assignment we'll analyze the timing data we obtained in the Zig Peacock's physics lab. Our experiments replicate the ones done by Galileo, who deduced his laws of motion and gravitation from such data. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Ramp Experiment. A steel ball was rolled down an aluminum trough on a beam that could be adjusted to various different inclinations. Five timers would start when the electromagnet holding the ball was released and each would stop and indicate the time when the ball passed that point. The starting elevation was 1.055m. above the floor for each ramp. The last timer was 5.000m from the start. (c=5.000m in the diagram.) For the first ramp, the ending timer was .596m. above the floor (so h=1.055-.596 m.) We made three runs for each ramp. First Ramp Experiment. The ending timer is .596m above the floor. Distance from start(in meters) Elapsed times(in seconds) ------------------------------ ------------------------- .20 .97 .98 1.00 .80 2.02 2.04 2.14 1.80 3.08 3.09 3.10 3.20 3.95 3.95 3.96 5.00 5.08 5.08 5.05 Second Ramp Experiment. The ending timer is .694m above the floor. Distance from start(in meters) Elapsed times(in seconds) ------------------------------ ------------------------- .20 1.11 1.11 1.11 .80 2.26 2.28 2.31 1.80 3.37 3.40 3.62 3.20 4.48 4.51 4.55 5.00 5.96 5.71 5.78 Third Ramp Experiment. The ending timer is .409m above the floor. Distance from start(in meters) Elapsed times(in seconds) ------------------------------ ------------------------- .20 .85 .85 .85 .80 1.84 1.73 1.75 1.80 2.67 2.54 2.56 3.20 3.38 3.37 3.40 5.00 4.38 4.34 4.39 Dropping Ball Experiment A switch caused an electromagnet to release a steel ball and start a clock. The clock stopped when the ball dropped through an electric eye. The ball was timed three times from various heights. Distance of fall(in meters) Elapsed times(in seconds) --------------------------- ------------------------- .500 .328 .326 .327 1.000 .464 .470 .469 1.500 .571 .568 .570 2.000 .652 .655 .654 2.400 .701 .703 .712 For each angle setting of the ramp and for the dropping experiment, do the following: (1.) Plot carefully the distance dropped versus the time. Sketch an approximate curve that interpolates the data to all times. (2.) Compute the average velocities over each interval. Plot the average velocity versus the time. Take the midpoint of the time interval for the time. Sketch the curve through your points that interpolates the velocity at all times (it should be a strainght line!) Determine the slope of the line graphically. (3.) Using the midpoints from Problem (2), compute the average accelerations from midpoint to midpoint. (This is something new. We haven't taken ``second differences'' before.) These numbers should be close to constant, corresponding to the acceleration in the ramp directions or down. Average to get your estimate for the constant acceleration. How close is the estimate of g to the actual value? (Express as a percenttage error.) (4.) For each interval, using your sketched curves in Problem (1), determine how far below the starting point the ball is at the midpoint times. Plot on the same graph velocity vs. distance below the starting point. Use a different color for each ramp. Galileo observed that that all of these points lie on the same curve, irregardless of the angle of the ramp. In other words, for all ramps, the speed of the ball depends on the distance below the starting point. He used this observation for the evidence that the acceleration down the ramp is proportional to the ramp component of the acceleration vector. In symbols, if the slope of the ramp is v / h, where v is the vertical drop and h is the horizontal run and c^2 = h^2 + v^2 is the square of the hypotenuse, then the acceleration should be proportional to h / c . (So if the slope above horizontal is larger, then the acceleration in the ramp direction is larger. In terms of trigonometric formulas, if the angle of the ramp is t then tan t = v / h and acceleration is proportional to sin t = v /c. Thus if g is the downward acceleration, then expect that a= g v / c is the acceleration in the ramp direction. The downward acceleration g is split into the part parallel to the ramp a plus the part perpendicular to the ramp.) Do your data corroborate Gallileo's observations?
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 Eighth Homework Assignment Due March 28, 2003 Treibergs Read chapter 8 from Sawyer's What is Calculus About? Answer the following questions which are related to this reading. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. (1.) Find functions of t that satisfy the following formulas. These are differentiation formulas in reverse. Guess a function, differentiate it to see if it works, and adjust your guess and try again. Repeat until you hit upon the solution. (There is a system behind all this madness! can you figure it out?) {a.) s' = 3 t^2 ; {b.) s' = 1 + t + t^2 ; {c.) s'' = 2 - t ; {d.) s'= -32 t ; {e.) s' = (1 + t^2)^{-3/2} . (2.) Find a curve in the plane with the property that for each point P , the tangent line PT is perpendicular (makes an angle of 90^\circ ) to the line OPR (see the diagram on page 85.) You may not be able to express this problem symbolically, but at least you can argue geometrically. (3.) You may not be able to name the five functions, but at least by reasoning geometrically, you will be able to deduce their shape. Sketch all of them on the same graph over the interval -2 < x < 2. The first function should go through the point (0,1) and satisfy the equation y' = -y. The others should satisfy the same equation and go through a different one of the points (0,1/2), (0,0), (0,2), (0,-1). (4.) Using trial and error, find a nonzero function y = f(x) that satisfies the following equations. ( f(x)=0 satisfies all of the differential equations!) {a.) y' = 2 sqrt(y) ; {b.) y' = 4 y / x ; {c.) y' = 2 y/(x+1) and y=3 at x=0 . -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 Seventh Homeweork Assignment Due March 7, 2003 Treibergs - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Read chapter 7 from Sawyer's " What is Calculus About?" and think about the problem on p. 76, but don't hand it in. Answer the following questions which are related to this reading. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - You should be thinking about your SEMESTER PROJECT, to write a short paper about some specific topic of calculus. Calculus includes all kinds of infinite processes: differentiation, the computation of a limit, the computation of an infinite sum, or integration. Your paper could be on an important development, breakthrough or theorem. Or it could be on a mathematical model or an application. Your paper should include a quick mathematical description of the calculus involved, as well as a discussion of its history and importance. A list of suggested topics will be provided shortly. Just about anything that interests you might do. Please talk to me and get my approval for any project other than those suggested. I might be able to point you references. Examples of topics might be: Xeno's Paradoxes. Archimedes determination of the area of a circle. Contrasting Newton's and Leibnitz's development of the Fundamental Theorem of Calculus: did they work independently or did Leibnitz copy an early manuscript of Newton's? Buffon's needle problem, or how to find \pi experimentally. Euler's infinte summation formula for the number e, which is the base of the natural logarithm and occurs in a banker's formula for continuous compounding of interest. Weierstass's objections to nineteenth century logic and his rigorous reformulation of limits and derivatives. Nash equilibria and the theory of games. Steiner's argument that the circle has the largest area among planar figures with given length. The development of computer algebra systems, such as MAPLE, and a demonstration of their capacity to do calculus problems and graph. The reliability of exit polls and Laplace's bell curve theorem. Modelling competitive species: why are the populations of wolves and rabbits on Isle Royale oscillatory functions of time? Fractals. The mathematics of soap bubbles........ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (1.) Compute y' and y'' for the following functions: 3 2 (a.) y = x - 4x + 5x - 6, (b.) y = sqrt{1+x^2}, x (c.) y = -------. 1 + x (2.) Sketch the graph. Besides all the usual information (where y is increasing/decreasing, maxima, minima, intercepts, blowups and asymptotes) find where y''>0 (concave up) and where y''<0 (concave down.) 1 y = ------ 2 1 + x (3.) For this problem you only need to sketch such a curve, you don't have to find a formula! Find a continuous function y = f(x) that (a.) tends to -\infty as x\to-\infty; (b.) increases when x <1, 2 < x < 4 and 5 < x and decreses when 1 <x <2 and 4 < x < 5; (c.) has y''>0 in the ranges x<-2, 1.5 < x < 3 and 4.5 < x < 5.5 and has y'' < 0 in the intervals -2 < x <1.5, 3 < x < 4.5 and 5.5 < x; (d.) f(0) = 0; (e.) the graph stays below the height y = 10, (i.e., f(x) < 10 for all x.) (4.) Suppose that the position s of our car (in miles) at time t (in hours) is given by the following formulas. Find the velocity v and the acceleration a as functions of time. (a.) s = 75t + 100, (b.) s = sqrt[(10+20t)(10+50t)], 1 (c.) s= ---. t -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 Sixth Homeweork Assignment Due February 28, 2003 Treibergs Review the Product, Quotient and Chain Rules for differentiation from the supplement from Anshel and Goldfeld, or from any other handy calculus text. Answer the following questions which are related to this reading. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. (1.) Compute the following derivatives in two ways. One way is by multiplying out first, and then differentiating. The other is to use the product or chain rule first. Multiply out your answer to show that both methods agree. 3 8 6 (a.) f(x) = (x + 1) (x + x ) 4 2 3 (b.) g(x) = (x + x ) (2.) Using the Product, Quotient and Chain rules (possibly several times for each problem,) find the derivatives of the followintg functions. 4 (1/3) (a.) h(x) = (1 + x ) 2 x (b.) i(x) = --------------- 2 4 4 + x + x + x 3 x + 4 (c.) j(x) = ---------- 2 (5 x + 6) (d.) k(x) = sqrt(1 + sqrt(1 + sqrt(x))) 2 1 + x (e.) l(x) = ---------------- 2 sqrt(1 + x + x ) (3.) Sketch the graph. (Find the interesting points: maxima, minima, axis intercepts.) x y = -------- 2 {1 + x } (4.) Find the point on the line 4y-3x=100 which is nearest to the origin in two ways: (a.) using geometry; (b.) using calculus.
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 Fifth Homeweork Assignment Due February 21, 2003 Treibergs Reread pp. 69--70 from Sawyer's "What is Calculus About?" Answer the following questions which are related to this reading. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. Do FIVE of the following problems. (1.) Among all pairs of real numbers whose difference is 36, find a pair whose product is minimum. (2.) What is the largest rectangular area that can be fenced in with 240 ft. of fencing? (3.) A piece of cardboard 10 inches on a side is to be made into a box with no top by by cutting out squares from the corners and folding up the sides. What size square should be cut out to obtain the largest volume box? (4.) What is the smallest square that can be placed in a given square so that each corner of the small square touches a side of the larger square? (5.) Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder (a.) whose volume is a maximum; (b.) whose lateral area is a maximum; (c.) whose total area is a maximum. (6.) The accountant of a shirt manufacturing firm runs a cost analysis of producing a single shirt. After six months of careful data collecting, she finds that within the range of producing 1000 to 2000 shirts per day, the costs are as follows: The material and labor costs $5 per shirt and the fixed cost (rent, insurance, etc.) are $500 per day. But this production schedule causes an an added cost of 2 cents times the square root of the number of shirts produced. This is due to an increased employee error and machine breakdowns. How many shirts should be manufactured each day to minimize the cost per shirt? How much will each shirt cost? This firm can sell 2000 shirts at $7 each. For each 10 cents increase in price, it sells 20 less shirts. What price will maximize revenue? (7.) A wire of length 24 inches is to be cut into two pieces. One piece is to be bent into a closed circle, the other into four sided square. How long should the circular and the square parts be so that the sum of the areas of the two figures is maximum? minimum?
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 Fourth Homework Assignment Due February 14, 2003 Treibergs Read chapter 6 from Sawyer's "What is Calculus About?" Make sure that you can do exercises on pages 59 and 68 but do not hand in these problems. Answer the following questions which are related to those from the text. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. (1.) Sketch the following curves. Compute the derivative y'. Use easy information from both y and y' to deduce the important features of the curve. In each case, locate the interesting points, such as where the curve crosses the x- and y-axes, and where the minimum and maximum points are. Label where the curve is flat (y'=0), where it is increasing (y'>0) and where it is decreasing (y'<0.) State what each clue tells you about the shape of the graph. (a.) (p. 68[4]) y=x^2-2x-8. (b.) (p. 68[5]) y=x^3-6x^2. (c.) (Read p. 68[8] for a similar question.) y=-x-x^{-1}. (2.) In this question, we consider the three dimensional analog of the Steiner formulas for thickenings of convex sets in the plane. (a.) Look up the formulas for the volume V(r) and surface area S(r) of the sphere in three space with radius r. Check that V'(r)=S(r). Write a formula for S(r+h) and V(r+h) in terms of S(r), A(r) and h. (b.) Let G denote a rectangular box in three space with height h, length l width w. Then the volume of the box is V=hlw and the surface area is S=2hl + 2hw + 2lw, which is the sum of the areas of the front and back, two sides, top and bottom. Now let G_r be the r-thickening of the box. That is, G_r consists of all points in space that are in the box or are within a distance r of the box (front and side views are sketched. It looks like a larger rectangular box, but with rounded edges and corners.) Figure out the volume V(r) and the surface area S(r) of G_r. (Steiner's formulas.) Show that V'(r)=S(r). -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Math 1080 - 1 Third Homework Assignment Due January 31, 2003 Treibergs Read chapter 4 & 5 from Sawyer's What is Calculus About? Make sure that you can do exercises on pages 33, 45 and 47 but do not hand in these problems. Answer the following questions which are related to those from the text. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. ***********The first MIDTERM EXAM is Friday, Jan. 31, 2003.*********** (1.) Using the formulas for derivatives, find s' when a. 4 s = 15 t b. 2 s = 66 + 40 t - 16 t c. {1/3} 1 5 s = 6 t + sqrt(t) + ------- + t sqrt(t) sqrt(t) d. {2.22} {.11} {-.23} s = t + 3.4 t - 3 t (2) Using a calculator, make a table showing t vs. s=t^{0.7} where t takes the values 0.0, 0 .1, 0.2, ..., 1.9, 2.0. Then using the computer or on full page of graph paper, plot the graph of s=t^{0.7} as t ranges between 0 and 2 . (Don't just draw a freehand grid on notebook paper: this exercise comes out better when carefully done!) Using a straightedge, draw the tangent lines to your graph at the points (0.6,(0.6)^{0.7}) and (1.4,(1.4)^{0.7}) . (Make the lines as long as you can, not puny like fig.16.) By measuring the rise and the run, estimate the slopes of your two tangent lines. (You can count grid squares or use a ruler.) Now, use the formula from calculus to find s' as a function of t . Compute the actual s' when t=0.6 and t=1.4 . Compare both methods and comment. (3) These questions are based on the discussion on p. 48. According to Gallileo and Newton, if you throw a stone upwards, then its height above the ground s in feet after t seconds is given by the formula (**) 2 s = a + b t - 16 t where a and b are constants. (Explanations required here!) a. If you are standing at a height of h_0 feet and throw the stone upward with velocity v_0 at time t=0 , what should a and b be in terms of h_0 and v_0 ? (What are s and s' according to formula (**) when t=0 ?) b. If you stand on a 200 ft. high building and throw the stone upwards with velocity 112 ft. per second, how fast is the stone moving after 1 second? after 3 seconds? after 4 seconds? When is the stone at its highest point? What is the upward velocity s' when the stone is at the highest point? How high is the stone at its highest point? c. When does the stone from (b.) hit the ground? How fast is it moving when it hits the ground? -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 Second Homework Assignment Due January 24, 2003 Treibergs Read chapter 3 from Sawyer's "What is Calculus About?" Make sure that you can do exercises on pages 29 and 30 but do not hand in these problems. Answer the following questions which are related to those from the text. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. (1.) Suppose that the position of an arrow shot into the air satisfies the law s = 500 - 16 t^2 where s is the height in feet and t is the time in seconds. Using the procedure outlined in this section, find the instantaneous velocity s' when t=5 seconds. (Sketch the flight path and the tangent line when t=5 . Guess what the answer might be. Show that in this case, the velocity over the intervals 4.9 to 5 gives a slope greater than s' , whereas the velocity computed over the interval 5 to 5.1 gives value less than s' --watch out for inequalities between negative numbers! Compute the difference quotients to estimate s' for the pairs of intervals 4.9 to 5 to 5.1, 4.99 to 5 to 5.01, 4.999 to 5 to 5.001 and 4.9999 to 5 to 5.0001. What is s' ?) (2.) Use the algebraic procedure from p. 28 discussing the law s=t^2 for a more general law. Find s' when t=a where s is given by the law s = c t^2 + dt + e where c, d, e are constants. For example, in the first question, c = -16 , d = 0 and e = 500 . Check that your formula agrees with your answer from (1.) [Hint: the answer is s'=2ca+d . You will have to make a table showing what s is when t=a and when t=a+h . Compute the velocity over this interval. Decide what limiting value s' the difference quotients approach as h approaches zero.] (3.) In this question we study another method to compute s' . Let's consider the law s = t^2 again, and find s' when t = 3 . But this time, suppose that we consider the "entered differences." That means we consider the velocity on the interval from t = 2.9 to t = 3.1. Then on the interval from t = 2.99 to t= 3.01 , t = 2.999 to t = 3.001. Finally do it algebraically on the interval t = a - h to t = a + h . What does the centered difference quotient approach as h approaches zero? Draw a diagram to convince yourself that the velocity over these centered intervals tends to s' . (4.) Look up the formula for the surface area S of the sphere in terms of its radius t . Assume that the earth is a perfect sphere of radius 4000 miles. (The earth is really an oblate spheroid with seas, mountains, valleys and other surface irregularities.) Suppose that a global rainfall increases the radius at an instantaneous rate of one foot per hour, how fast is the surface area increasing in square feet per hour? (Find S' when t = 4000 miles. Be careful about units.) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Math 1080 - 1 First Homework Assignment Due January 17, 2003 Treibergs Read chapters 1 and 2 from Sawyer's What is Calculus About? Make sure that you can do exercises 20[1,2,3] and 21[2,7,8.12] (page[problem number]) but do not hand in these problems. Answer the following questions which are related to those from the text. Please make your work self-contained and complete. Your answer should include formulas, diagrams and short explanations. (1.) Starting from the zero mile marker, suppose that you drive at a steady velocity of 70 miles pe hour. Find the law (write a formula) which gives the position of the car at any time. What is the velocity at any time? Where is the car after three hours? (2.) Suppose that the position of three cars is given by the formulas (a.) s=35t+50, (b.) s=50t+35, (c.) s=-40t+125 where $t$ is the elapsed time in hours and $s$ is the position on the road according to the mile marker. Find the velocities $s'$ in each case. On the same graph, sketch illustrations of the motions, like figures 5,6,8, showing the position of the car for times from zero to three hours. Figure out when the cars meet each other along the road. (3.) Pick two towns in Utah which are at least 100 miles apart. Choose a highway route from the first town to the second. Choose at least five intermediate points, and find the distances between these consecutive points. (e.g., Use a road map, atlas, distance table or the internet.) Make up a reasonable timetable for travelling along this route, noting the time that you would be at each of the locations. Then compute the velocity for each leg of your trip. Make sure that all legs are not travelled at the same velocity. Tabulate your results. Suggest what factors, like rush hour, road condition, mountains, gas stops, condition of the driver or whatever else, might account for the differences in the velocities for the legs of your trip. On graph paper, make a large graph showing the position vs. the time. Discuss how slower and faster legs may be seen from your graph. Finally, comment on how realistic your numbers really are. (Here are the first few entries from my example from class. Of course, you should each have a different example!) Location Time (Hrs.) Position (Miles) Velocity this leg (MPH) -------- ----------- ---------------- ----------------------- UU .00 0 * * * Intersection I80 & I15 .25 5 5 - 0 --------- = 20 .25 - .00 Point of the Mtn. .75 24 24 - 5 -------- = 38 .75-.25 Provo 1.50 43 43 - 24 -------- = 38 1.50-.75 etc. * * * * * * * * * -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=