Notes on Calculus by Dinakar Ramakrishnan 253-37 Caltech Pasadena, CA 91125 Fall 2001 1 Contents 0 Logical Background 2 0.1Sets........................................ 2 0.2Functions..................................... 3 0.3Cardinality.................................... 3 0.4EquivalenceRelations............................... 4 1 Real and Complex Numbers 6 1.1DesiredProperties................................ 6 1.2 Natural Numbers, Well Ordering, and Induction . 8 1.3Integers...................................... 10 1.4RationalNumbers................................. 11 1.5OrderedFields.................................. 13 1.6RealNumbers................................... 14 1.7AbsoluteValue.................................. 18 1.8ComplexNumbers................................ 19 2 Sequences and Series 22 2.1Convergenceofsequences............................. 22 2.2Cauchy’scriterion................................. 26 2.3ConstructionofRealNumbersrevisited..................... 27 2.4Infiniteseries................................... 29 2.5TestsforConvergence............................... 31 2.6Alternatingseries................................. 33 3 Basics of Integration 36 3.1 Open, closed and compact sets in R ....................... 36 3.2 Integrals of bounded functions . 39 3.3 Integrability of monotone functions . 42 b 3.4 Computation of xsdx .............................. 43 a 3.5 Example of a non-integrable, bounded function . 45 3.6Propertiesofintegrals.............................. 46 3.7 The integral of xm revisited,andpolynomials................. 48 4 Continuous functions, Integrability 51 4.1LimitsandContinuity.............................. 51 4.2Sometheoremsoncontinuousfunctions..................... 55 4.3 Integrability of continuous functions . 57 4.4Trigonometricfunctions............................. 58 4.5Functionswithdiscontinuities.......................... 62 1 5 Improper Integrals, Areas, Polar Coordinates, Volumes 64 5.1ImproperIntegrals................................ 64 5.2Areas........................................ 67 5.3Polarcoordinates................................. 69 5.4Volumes...................................... 71 5.5Theintegraltestforinfiniteseries........................ 73 6 Differentiation, Properties, Tangents, Extrema 76 6.1Derivatives..................................... 76 6.2Rulesofdifferentiation,consequences...................... 79 6.3Proofsoftherules................................ 82 6.4Tangents...................................... 84 6.5Extremaofdifferentiablefunctions....................... 85 6.6Themeanvaluetheorem............................. 86 7 The Fundamental Theorems of Calculus, Methods of Integration 89 7.1 The fundamental theorems . 89 7.2Theindefiniteintegral.............................. 92 7.3Integrationbysubstitution............................ 92 7.4Integrationbyparts................................ 95 2 6 Differentiation, Properties, Tangents, Extrema 6.1 Derivatives Let a be a real number and f a function defined on an interval around a. One says that f is differentiable at a iff the following limit exists: f(a + h) − f(a) L : = lim . h→0 h When the limit exists, we will set f (a)=L. df We will also denote it sometimes by dx (a). Consider, for example, the case of a linear function f(x)=mx + c, whose graph is the line of slope m, passing through the point (0,c). Since f(x + h)= m(x + h)+c = f(x)+mh,wehaveatanypointa in R, f(x + h) − f(x) mh = = m, h h which is independent of h. Hence it has the limit m as h approaches 0. Thus f is differentiable at x = a with f (a)beingtheslopem. In particular, when m = 0, the function f is just the constant function x → c,andthe derivative is 0. The next simple example toconsideris the quadratic function f(x)=αx2 + βx + γ, with α, β, γ in R.Then f(x + h) − f(x) (2αx + β)h + αh2 = =2α + β + αh, h h which has the limit 2αx + β as h tends to0. Thus df =2αx + β. dx This is L:eibnitz’s notation, and it means that for any a, df f (a)= (a)=2αa + β. dx 76 In particular, the squaring function f(x)=x2 is differentiable everywhere with deriva- tive 2x. This is a superspecial case of the following important result on the power function xt. Proposition 6.1.1 For any real number t, consider the function f(x)=xt. Then f is differentiable at any point a in R,with f (a)=tat−1. We will prove this in four stages. The first step is the following: Proof when t is a positive integer m.Leta, h ∈ R. By the binomial theorem, m m(m − 1) (a + h)m = am−jhj = am + mam−1h + am−2h2 + ...hm. 2 j=0 In other words, (a + h)m − am = mam−1 + hg(a, h), h where m m(m − 1) m(m − 1)(m − 2) g(a, h)= am−jhj−2 = am−2 + am−3h + ... + hm−2. 2 2 j=2 m(m−1) m−2 Then g(x, 0) = 2 a ,andsohg(x, h)goesto0ash goes to 0. Consequently, (a + h)m − am lim = mam−1, h→0 h as asserted. Here is the next step. Proof when t =1/n with n a positive integer.Foranya ∈ R, note that h = zn − wn if z =(a + h)1/n,w= a1/n. Thus (a + h)1/n − a1/n z − w = . h zn − wn But we have the factorization zn − wn =(z − w)(zn−1 + zn−2w + zn−3w2 + ...+ zwn−2 + wn−1), 77 which can be verified by direct computation. Thus (a + h)1/n − a1/n z − w 1 lim = lim = lim −1 −2 −2 −1 . h→0 h h→0 zn − wn h→0 zn + zn w + ...+ zwn + wn Since the function x → x1/n is continuous, the limit of ziwj as h goes to 0 is simply wi+j. We then obtain 1/n 1/n (a + h) − a 1 1 1 −1 n lim = −1 = a . h→0 h nwn n which proves the assertion of Proposition 6.1.1 for t =1/n. We will come back to this proposition in the next section after deriving the product rule. Proposition 6.1.2 The functions f(x) = sin x and g(x)=cosx are differentiable at any point a in R,and f (a)=cosaandg(a)=− sin a. Proof. Recall the addition formula sin(x + y) = sin x cos y +cosx sin y. Therefore sin(a + h) − sin a cos h − 1 sin h =sina +cosa . h h h We have already seen that sin h cos h − 1 lim = 1 and lim =0. h→0 h h→0 h Consequently, sin(a + h) − sin a lim =cosa, h→0 h as asserted. Similarly, the addition formula for the cosine function, namely cos(x + y)=cosx cos y − sin x sin y, implies cos(a + h) − cos a cos h − 1 sin h =cosa − sin a . h h h The expression on the right has the limit − sin a as h goes to 0. Done. One can check that if f is differentiable at a, f must necessarily be continuous at a, but it is not sufficient. Indeed, the function f(x)=|x| is continuous everywhere, but it is not integrable at x = 0. This is because for h>0, f(h) − f(0) h − 0 = =1, h h 78 while for h<0, f(h) − f(0) −h − 0 = = −1. h h f(h)−f(0) − So h is 1, resp. 1, as h approaches 0 from the right, resp. left. So there is no unique limit and f is not differentiable at 0. Note, however, that f is differentiable everywhere else. Let us end this section by looking at a couple of more examples. The first one below is not differentiable at x =0: x sin 1 , if x =0 f(x)= x 0, if x =0 Indeed, for any h =0, f(h) − f(0) h sin 1 − 0 1 = h =sin , h h h 1 − and sin h has nolimit as h goes to 0; it fluctuates wildly between 1and1. The second example is x2 sin 1 , if x =0 g(x)= x 0, if x =0 In this case g(h) − g(0) 1 = h sin , h h which approaches 0 as h goes to 0. So g(x) is differentiable at x =0with g(0) = 0. 6.2 Rules of differentiation, consequences The twobasic results here are the following: Theorem 6.2.1 Let f,g be differentiable functions at some a ∈ R. Then we have: (i) (Linearity) For all α, β in R, the function αf + βg is differentiable at a,with (αf + βg)(a)=αf (a)+βg(a). (ii) (Product rule) The product function fg is differentiable at a,with (fg)(a)=f (a)g(a)+f(a)g(a). 79 (iii) (Quotient rule)Ifg(x) is non-zero in an interval around a, then the ratio f/g is differentiable at a,with f f (a)g(a) − f(a)g(a) (a)= . g g(a)2 Theorem 6.2.2 (Chain rule) Let f be a differentiable function at some a in R, g a differentiable function at f(a). Then the composite function g ◦ f is differentiable at a,with (g ◦ f)(a)=g(f(a)) · f (a). Before giving proofs of these assertions, which will be done in the next section, let us note that as a consequence, rational functions, trigonometric functions,andvarious combinations of them are differentiable wherever they are defined.For example, the function x72 − 21x3 +sin9(x4 +43x) − cos3(x3 − 5) f(x)= √ 12 sin x − 29x4 − 4 is differentiable at any x where the denominator is non-zero. A simpler, and more commonly occurring example is g(x)=tanx. Since tan x =sinx/ cos x, we get by applying the quotient rule and Proposition 6.1.2, d (cos x)(cos x) − (sin x)(− sin x) 1 (6.2.3a) tan x = = =sec2 x. dx cos2 x cos2 x Here we have used the fact that sin2 x +cos2 x = 1, and the formula is valid at any x where cos x is non-zero, i.e., at any real number x not equal to an odd integer multiple of π/2. Similarly we have d (6.2.3b) cot x = −2x, dx d (6.2.3c) sec x =secx tan x, dx and d (6.2.3d) x = −x cot x, dx for all x where the function is defined. Now we will complete the proof of Proposition 6.1.1. In the previous section we proved the formula for the functions f(x)=xm and g(x)=x1/n for integers m, n with m ≥ 0, n>0. The next step is to look at the function h(x)=xm/n, 80 for a rational number m/n,withm, n > 0.