ENTRY SINGLE VARIABLE CALCULUS I[ENTRY SINGLE VARIABLE CALCULUS I] Authors: Oliver Knill: 2001 Literature: not yetAbel’s partial summation formula[Abel’s partial summation formula] is a discrete version of the partial integration formula: with An=Pnk=1akone hasPnk=makbk=Pnk=mAk(bk− bk+1) + Anbn+1− Am−1bm.Abel’s test[Abel’s test]: if anis a bounded monotonic sequence and bnis a convergent series, then the sumPnanbnconverges.absolute valueThe [absolute value] of a real number x is denoted by |x| and defined as the maximum of x and −x. We canalso write |x| = +√x2. The absolute value of a complex number z = x + iy is defined aspx2+ y2.accumulation pointAn [accumulation point] of a sequence anof real numbers is a point a which the limit of a subsequence ankofan. A sequence anconverges if and only if there is exactly one accumulation point. Example: The sequencean= sin(πn) has two accumulation points, a = 1 and a = −1. The sequence an= sin(πn)/n has only theaccumulation point a = 0. It converges.Achilles paradoxThe [Achilles paradox] is one of Zenos paradoxon. It argues that motion can not exist: ”set up a race betweenAchilles A and tortoise T . At the initial time t0= 0, A is at the spot s = 0 while T is at position s1= 1. Letsassume A runs twice as fast. The reace starts. When A reaches s1at time t1= 1, its opponent T has alreadyadvanced to a point s2= 1 + 1/2. Whenever A reaches a point skat time tk, where T has been at time tk−1,T has already advanced further to location sk+1. Because an infinite number of timesteps is necessary for Ato reach T , it is impossible that A overcomes T .” The paradox exploits a misunderstanding of the concept ofsummation of infinite series. At the finite time t =P∞n=1(tn− tn−1) = 2, both A and T will be at the samespot s = limn→∞sn= 2.addition formulasThe [addition formulas] for trigonometric functions arecos(a + b) = sin(a) cos(b) + cos(a) sin(b)sin(a + b) = cos(a) cos(b) − sin(a) sin(b)alternating seriesAn [alternating series] is a series in which terms are alternatively positive and negative. An example isP∞n=1an=P∞n=1(−1)n/n = −1 + 1/2 − 1/3 + 1/4 − .... An alternating series with an→ 0 converges bythe alternating series test.alternating series testLeibniz’s [alternating series test] assures that an alternating seriesPnanwith |an| → 0 is a convergent series.acuteAn angle is [acute], if it is smaller than a right angle. For example α = π/3 = 60◦is an acute angle. The angleα = 2π/3 = 120◦is not an acute angle. The right angle α = π/2 = 90◦does not count as an acute angle. Theangle α = −π/6 = −30◦is an acute angle.antiderivativeThe [antiderivative] of a function f is a function F (x) such that the derivative of F is f that is if d/dxF (x) =f(x). The antiderivative is not unique. For example, every function F (x) = cos(x) + C is the antiderivative off(x) = sin(x). Every function F (x) = xn+1/(n + 1) + C is the antiderivative of f(x) = xn.Arithmetic progression[Arithmetic progression] A sequence of numbers anfor which bn= an+1−anis constant, is called an arithmeticprogression. For example, 3, 7, 11, 15, 19, ... is an arithmetic progression. The sequence 0, 1, 2, 4, 5, 6, 7 is not anarithmetic progression.arrow paradoxThe [arrow paradox] is a classical Zeno paradox with conclusion that motion can not exist: ”an object occupiesat each time a space equal to itself, but something which occupies a space equal to itself can not move. Therefore,the arrow is always at rest.”asymptoticTwo real functions are called [asymptotic] at a point a if limx→af(x)/g(x) = 1. For example, f(x) = sin(x) andg(x) = x are asymptotic at a = 0. The point a can also be infinite: for example, f(x) = x and g(x) =√x2+ 1are asymptotic at a = ∞.Bernstein polynomialsThe [Bernstein polynomials] of a continuous function f on the unit interval 0 ≤ x ≤ 1 are defined as Bn(x) =Pnk=1f(k/n)xk(1 − x)n−kn!/(k!(n −k!).Binomial coefficients[Binomial coefficients] The coefficients B(n, k) of the polynomial (x + 1)nfor integer n are called Binomialcoefficients. Explicitly one has B(n, k) = n!/(k!(n − k)!), where k! = k(k − 1)!, 0! = 1 is the factorial of k. Thefunction B(n, k) can be defined for any real numbers n, k by writing n! = Γ(n + 1), where Γ is the Gammafunction. If p is a positive real number and k is an integer, one has one has B(p, k) = p(p − 1)...(p − k + 1)/k!.For example, B(1/2, 0) = B(1/2, 1) = 1/2, B(1/2, 2) = −1/8. Indeed, (1 + x)1/2= 1 + x/2 − x2/8 + ....Binominal theoremThe [Binominal theorem] tells that for a real number |z| < 1 and real number p, one has (1 + z)p=P∞k=0B(p, k)zk, where B(p, k) is called the Binomial coefficient. If p is a positive integer, then (1 + z)pisa polynomial. For example:(1 + z)4= 1 + 4z + 6z2+ 4z3+ z4.If p is a noninteger or negative, then (1 + z)pis an infinite sum. For example(1 + z)−1/2= 1 − x/2 + 3x2/8 − 5x3/16 + ...bisectorA [bisector] is a straight line that bisects a given angle or a given line segment. For example, the y-axis x = 0in the plane bisects the line segment connecting (−1, 0) with (1, 0). The line x = y bisects the angle6(CAB)where C = (0, 1), A = (0, 0), B = (1, 0) at the point A.Bolzano’s theorem[Bolzano’s theorem] also called intermediate value theorem says that a continuous function on an interval (a, b)takes each value between f (a) and f(b). For example, the function f(x) = sin(x) takes any value between −1and 1 because f is continuous and f(−π/2) = −1 and f(π/2) = 1.Fermat principleThe [Fermat principle] tells that if f is a function which is differentiable at z and f(x) > f(z) for all points inan interval (z − a, z + a) with a > 0, then f0(z) = 0.fundamental theorem of calculusThe [fundamental theorem of calculus]: if f is a differentiable function on a ≤ x ≤ b where a < b are realnumbers, then f(b) − f(a) =Rbaf0(x) dx.integration rules[integration rules]:•Raf(x) dx = aRf(x) dx.•Rf(x) + g(x) dx =Rf(x) dx +Rg(x) dx.•Rfg dx = fG −Rf0G, where G0= g. This is called integration by parts.intermediate value theoremThe [intermediate value theorem] also called Bolzno theorem assures that a continuous function on an intervala ≤ z ≤ b takes each each value between f(a) and f (b). For example f(x) = cos(x) + cos(3x) + cos(5x) takesany value between [−3, 3] on [0, π] because f(0) = 3 and f(π) = −3.Cauchy’s convergence condition[Cauchy’s convergence condition]: a sequence