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Beginnings of the Calculus Maxima and Minima EXTREMA a Sample Situation Beginnings of the Calculus Maxima and Minima EXTREMA A Sample Situation • Find two numbers that add to 10 and whose product is b. • The two numbers are and , and their product is . • So the equation modeling the situation is . • Now, which value(s) of x make b the largest? 30 28 Fermat 26 24 22 • Fermat noted that each 20 18 had a corresponding 16 value , and that 14 12 these two values got 10 8 closer to one another at 6 the maximum. 4 2 • In fact, at the -15 -10 -5 5 10 15 20 -2 maximum, the two -4 -6 values are the same. -8 -10 Fermat’s Method • Fermat reasoned thusly: For two values and , and will usually be distinctly different. But near the maximum, the difference will be very small, almost imperceptible. • Thus, to find maxima, Fermat equated and , realizing they weren’t really equal, but close to it (he called it adequality). Fermat’s Method After equating and , he simplified by dividing through by , then let , and finally solved for the value . This is equivalent to setting → ….which seems oddly familiar. Fermat’s Method‐ Example • To maximize the polynomial , set . This is equivalent to . Then: and cancelling the , we get and letting , . So, , and . Fermat’s Method • Eventually, instead of looking at two x‐values a and b, Fermat looked at a and a+E, where E was a small number. After (ad)equating p(a) and p(a+E), he would divide by E and then let E = 0. Of course, this is equivalent to setting → Fermat’s Method ‐ Notes • This works well for polynomials, since the constant term cancels and there is always a way to factor out the (a‐b) or E and cancel it. TANGENTS Fermat’s Method D (x,y) E (x+e, f(x+e)) • In finding the tangent to a curve f, Fermat let the y point of tangency be x e (x,y) and the horizontal A O B C distance from the point t of tangency to the point where the tangent intersected the axis be t. Fermat’s Method • He then moved a small D (x,y) E (x+e, f(x+e)) distance e from the point of tangency to y x e another point on the A O B C curve, . t • This gave him two triangles, ΔABD and ΔACE. Fermat knew they weren’t exactly similar, but they were close…. Fermat’s Method E (x+e, f(x+e)) • So it was almost true D (x,y) that y x e A O B C ‐‐ another “adequality.” t But simplifying this, including dividing through by e and letting , yields an equation relating t and x that leads to the tangent line. Fermat’s Method –Example • Let’s suppose . In this case, we get the “adequality” . Simplifying, we get : . We cube both sides to get: , or Cancelling like terms, we get , and dividing by e we have: Fermat’s Method –Example From , we let to get: or, since , . Now, given our original picture, the slope of the line is , or , or . D (x,y) E (x+e, f(x+e)) Of course, this is the derivative y of our function x e A O B C . t Descartes’ Method for Tangents • Descartes found the normal to a curve, from which the tangent could be found. • Descartes’ method started with the fact that the radius of a circle is always normal to the circle. • So the idea is to find a circle that is tangent to the curve at the desired point, and use its radius. • In practice this was a lot of work, as the example in the book (p. 512) of finding the normal to demonstrates. Descartes’ Method 6 4 2 5 10 -2 -4 Descartes’ Method • It really was much more complicated than Fermat’s method –as, I believe, Fermat happily pointed out. Hudde and Sluse • We won’t go over the mathematics here, but both Hudde and Sluse found general rules for determining tangents for polynomial functions. Hudde also applied his methods to finding extreme values. Isaac Barrow • Isaac Barrow was a teacher of Newton at Cambridge, and stepped aside as Lucasian Chair in deference to Newton when he returned to Cambridge to teach. Among other things, he described a procedure for finding tangent lines. Barrow’s Method • We want to find the tangent to the curve at 4 the point P. We assume 3 2 P the line is the a Q tangent line, and let Q 1 e R be a point on the curve T 2 N 4 M 6 near P. Now, note that -1 the triangles and are very nearly similar; more so as Q gets closer to P. Barrow’s Method Thus, as Q is closer to P, it 4 is more nearly true that 3 . Now, let 2 P a Q 1 e R T If P has coordinates , 2 N 4 M 6 then Q has coordinates -1 . Barrow’s Method We substitute the coordinates of these two 4 points into the equation of 3 2 P the curve and since e and a a Q are small, we ignore squares 1 e R and higher powers of both. T 2 N 4 M 6 Simplifying will often allow -1 for finding the ratio , which is the slope of the tangent line if e and a are almost 0. Barrow’s Method‐ Example For the curve we would plug in : , or expanding, we get: Ignoring squares and higher powers of e and a, we get: . Now, Remember that , and we get: Barrow’s Method –Example This becomes: Which is the slope of the tangent line at (x, y). Barrow’s Method • This is a general method that works not only for polynomials in x, but for polynomials in x and y. ‐‐ sort of like implicit differentiation, almost. Isaac Barrow • It is told of Barrow that in his youth he was so troublesome at home that his father was heard to pray that should God decide to take one of his children, he could best spare Isaac. He was also noted for his strength and courage, and once when traveling in the East he saved the ship by his own prowess from capture by pirates. Tangents and Extrema – Summing Up • From what we have seen, we know that many procedures were developed to find tangents and extreme values for a variety of functions, including fairly general polynomials. It was known, for instance, that the slope of the tangent line to was given by . • There are other similar rules we have not mentioned. • So, a lot of “calculus” was known before Newton and Leibniz came on the scene. AREAS AND VOLUMES Kepler • With his wine casks, Kepler accomplished a kind of integration to determine the volumes of certain solids. Cavalieri • Bonaventura Cavalieri considered himself a disciple of Galileo. He published Geometria Indivisibilibus Continuorum in 1635. He thought of plane regions as made up of an infinite set of parallel segments (“all the lines”), and solids as made up of an infinite set of parallel plane sections (slices). Cavalieri • Cavalieri’s Principle, which is still useful today, states that “the volumes of two objects are equal if the areas of their corresponding cross‐sections are in all cases equal.” Cavalieri • Cavalieri’s Principle works for lines, as well: Cavalieri • Using his principle of “all the lines/squares,” Cavalieri was able to accomplish the integration of all “higher parabolas:” Fermat, Pascal, Roberval, Torricelli • Between these folks, the area under both “higher parabolas” and “higher hyperbolas” was established. Fermat used clever ways of cutting up the area under these curves, and essentially the same kinds of techniques we now learn in first semester calculus when we study Riemann sums. John Wallis • Published Arithmetica Infinitorum in 1655, which systematized and extended methods of Cavalieri and Descartes; explained the meanings of zero, negative, and fractional exponents; wrote ; used infinite series and products. John Wallis • Wallis found an approximation for ∙∙∙∙∙∙∙⋯ by attempting to find the area of ∙∙∙∙∙∙∙⋯ a quarter circle by integration. That is, he ⁄ tried to find . What he needed was Newton’s generalized binomial theorem, but he didn’t have that. John Wallis • So, he computed the sequence 1, 1, 1, 1, ⋯ and so on, obtaining the numerical sequence 1, 2/3, 8/15. 16/35, . He then determined an algebraic formula that would give him that sequence for and finally used interpolation to obtain the numerical value for . Pretty clever. John Wallis • So now they could pretty much do for all rational values of k except ‐1. • Between Gregory of St. Vincent, Alfonso Antonio de Sarasa, and Nicolaus Mercator, that was pretty much taken care of as well. .
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