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HARVARD MATH 1A - math1a_2011

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Math 1A: introduction to functions and calculus Oliver Knill, 2011Lecture 1: What is Calculus?Calculus is a powerful tool to describe our world. It formalizes the process of taking differencesand taking sums. Both are natural operations. Differences measure change, sums measure howthings accumulate. We are interested for example in the total amount of precipi t at i on in Bostonover a year but we are also interested in how the temperature does change over time. The processof taking differences is in a limit called derivative. The process of taking sums is in t h e lim i t cal l edintegral. These two processes are related in an intimate way. In this first lectur e, we want to lookat these two processes in a discrete setup first, where functions are evaluated only on integers. Wewill call the process of taking d i ffer en ce s a derivative and the process of takin g sums as integral.Start with the sequence of integers1, 2, 3, 4, ... .We say f(1) = 1, f(2) = 2, f(3) = 3 et c and call f a function. It assigns to a number a number .It assigns for example to the number 100 the result f(100) = 100. Now we add these number s up.The sum of the first n numbers i s calledSf(n) = f(1) + f(2) + f(3) + ... + f (n) .In our case we get1, 3, 6, 10, 15, ...It defin es a new function g which sati sfies g(1) = 1, g(2) = 3, g(2) = 6 etc. The new numbers areknown as the triangular numbers. From the function g we can get f back by taking difference:Dg(n) = g(n) − g(n − 1) = f(n) .For ex am p l e Dg(5) = g(5) − g(4) = 15 − 10 = 5 and this is indeed f(5).Finding a formula for the sum Sf is not so easy. The young mathematician Karl-FriedrichGauss realized as a 7 year old ki d when giv i n g the ta sk to sum up the first 100 numbers thatit is the same as adding up 50 times 101 which is 5050. Gauss found g(n) = n(n + 1)/2 . Hedid that by pairing things up. To add up 1 + 2 + 3 + . . . + 10 for example we can wr it e this as(1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) leading to n/2 terms of n + 1 if n is even. Takingdifferences again is easier Dg(n) = (n + 1)n/2 − n(n − 1)/2 = n = f(n).Lets add up the new sequence again and comput e h = Sg. We get t h e sequence1, 4, 10, 20, 35, ...These numbers are called the tetrahedral numbers because one use h(n) marbles to build atetrahedron of side length n. For example, we need h(4) = 20 gol f balls for example to build atetrahedron of side length 4. The formula wh i ch holds for h is h(n) = n(n + 1)(n + 2)/6 . Wesee that summing the differences gives the function in the same way as differencing th e sum:SDf(n) = f(n) − f(0), DSf ( n) = f(n)Don’t worry yet, if this is too abstract. We will come back to it again and again. But this isan arithmetic version of the fundamental theorem of calculus which we will explore in thiscourse. The process of addin g up numbers will lead to the integralRx0f(x) dx. The process oftaking differences will lead to th e derivativeddxf(x). One of the high lights of this cou rs e is t ounderstand the fundamental theorem of calculus:Rx0ddtf(t) dt = f(x) − f(0),ddxRx0f(t) dt = f(x)and see why it is such a fantastic result. You see formally that it fits the result for differ ence andsum. A major goal of this course will be to understand the fundamental theorem resu lt and seeits use. But we have packed the essence of the theorem in the above version with S and D. It isa version which will lead us.1 Problem : Given the sequence 1, 1, 2, 3, 5, 8, 13, 21, . . . which satisfies the rule f(x) = f(x −1) + f(x − 2). It defines a functio n on the positive integers. For example , f(6) = 8. Whatis the fu n ct i on g = Df, if we assume f(0) = 0? Solution: We take the difference betweensuccessive numbers and get the sequence of numbers1, 0, 1, 1, 2, 3, 5, 8, ...After 2 entries, the same sequence appears again . We can also deduce d i re ct ly from the aboverecursion that f has the property thatDf(x) = f(x − 2 ) . It is called the Fibonnaccisequence, a sequence of grea t fame.2 Problem : Take the same function f given by the sequence 1, 1, 2, 3, 5, 8, 13, 21, ... but nowcompute the function h(n) = Sf(n) obtained by summin g the first n numbers u p . It givesthe sequence 1, 2, 4, 7, 12, 20, 33, .... What sequence is that?Solution: Because Df(x) = f(x − 2) we have f (x) − f(0) = SDf(x) = Sf(x − 2) sothat Sf(x) = f(x + 2) − f(2). Su m m i n g the Fibonnacci sequence produces the Fibonnaccisequence shifted to the left with f(2) = 1 is subtracted. It has been relatively easy to findthe sum, because we knew what the difference operation did. This example shows:We can study differe n ces to understa n d sums.The next problem illustrates this too:3 Problem : Find the next term in the sequence2 6 12 20 30 42 56 72 90 110 132 . Solution: Take differences2 6 12 20 30 42 56 72 90 110 1322 4 6 8 10 12 14 16 18 20 222 2 2 2 2 2 2 2 2 2 20 0 0 0 0 0 0 0 0 0 0.Now we can add an additional number, starting from the bottom and working us up.2 6 12 20 30 42 56 72 90 110 1321562 4 6 8 10 12 14 16 18 20 22 242 2 2 2 2 2 2 2 2 2 2 20 0 0 0 0 0 0 0 0 0 0 0In the rest of this hour, we t al k about some applied and not so applied problems which involvecalculus.Homework1 We have defined Sf(n) = f(1) + f(2) + ... + f(n) and Df(n) = f(n) − f (n − 1) and seenf(n) = 1 we have g(n) = Sf(n) = n .f(n) = n we have g(n) = Sf(n) = n(n + 1)/2.f(n) = n(n + 1)/2 we have g(n) = Sf(n) = n(n + 1)(n + 2)/6.Guess a formula g(n) = Sf(n) forf(n) = n(n + 1)(n + 2)/6 and verify using algebraicmanipulation th at it satisfies Dg(n) = f(n). Can you see a pattern?2 Fin d th e next t er m in the sequence 3, 12, 33, 72, 135, 228, 357, 528, 747, 1020, 1353.... To doso, compute successive …


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