ENTRY SINGLE VARIABLE CALCULUS I Authors: Oliver Knill: 2001 Literature: not yet +------------------------------------------------------------ | Abel's partial summation formula +------------------------------------------------------------ Abel's partial summation formula is a discrete version of the partial integration formula: with A_n = sum_k=1^n a_k one has sum_k=m^n a_k b_k = sum_k=m^n A_k(b_k-b_k+1) + A_n b_n+1 - A_m-1 b_m. +------------------------------------------------------------ | Abel's test +------------------------------------------------------------ Abel's test: if a_n is a bounded monotonic sequence and b_n is a convergent series, then the sum sum_n a_n b_n converges. +------------------------------------------------------------ | absolute value +------------------------------------------------------------ The absolute value of a real number x is denoted by |x| and defined as the maximum of x and -x. We can also write |x|=+sqrtx^2. The absolute value of a complex number z=x+iy is defined as sqrtx^2+y^2. +------------------------------------------------------------ | accumulation point +------------------------------------------------------------ An accumulation point of a sequence a_n of real numbers is a point a which the limit of a subsequence a_n_k of a_n. A sequence a_n converges if and only if there is exactly one accumulation point. Example: The sequence a_n = sin( pi n) has two accumulation points, a=1 and a=-1. The sequence a_n = sin( pi n)/n has only the accumulation point a=0. It converges. +------------------------------------------------------------ | Achilles paradox +------------------------------------------------------------ The Achilles paradox is one of Zenos paradoxon. It argues that motion can not exist: "set up a race between Achilles A and tortoise T. At the initial time t_0=0, A is at the spot s=0 while T is at position s_1=1. Lets assume A runs twice as fast. The reace starts. When A reaches s_1 at time t_1=1, its opponent T has already advanced to a point s_2=1+1/2. Whenever A reaches a point s_k at time t_k, where T has been at time t_k-1, T has already advanced further to location s_k+1. Because an infinite number of timesteps is necessary for A to reach T, it is impossible that A overcomes T." The paradox exploits a misunderstanding of the concept of summation of infinite series. At the finite time t = sum_n=1^infinity (t_n-t_n-1)=2, both A and T will be at the same spot s=lim_n to infinity s_n=2. +------------------------------------------------------------ | addition formulas +------------------------------------------------------------ The addition formulas for trigonometric functions are begineqnarray* cos(a+b) = sin(a) cos(b) + cos(a) sin(b) sin(a+b) = cos(a) cos(b) - sin(a) sin(b) endeqnarray* +------------------------------------------------------------ | alternating series +------------------------------------------------------------ An alternating series is a series in which terms are alternatively positive and negative. An example is sum_n=1^infinity a_n = sum_n=1^infinity (-1)^n/n=-1+1/2-1/3+1/4-.... An alternating series with a_n to 0 converges by the alternating series test. +------------------------------------------------------------ | alternating series test +------------------------------------------------------------ Leibniz's alternating series test assures that an alternating series sum_n a_n with |a_n| to 0 is a convergent series. +------------------------------------------------------------ | acute +------------------------------------------------------------ An angle is acute, if it is smaller than a right angle. For example alpha=pi/3=60^circ is an acute angle. The angle alpha=2pi/3=120^circ is not an acute angle. The right angle alpha=pi/2 = 90^circ does not count as an acute angle. The angle alpha=-pi/6=-30^circ is an acute angle. +------------------------------------------------------------ | antiderivative +------------------------------------------------------------ The antiderivative of a function f is a function F(x) such that the derivative of F is f that is if d/dx F(x) = f(x). The antiderivative is not unique. For example, every function F(x) = cos(x) +C is the antiderivative of f(x)=sin(x). Every function F(x)=x^n+1/(n+1) + C is the antiderivative of f(x)=x^n. +------------------------------------------------------------ | Arithmetic progression +------------------------------------------------------------ Arithmetic progression A sequence of numbers a_n for which b_n=a_n+1-a_n is constant, is called an arithmetic progression. For example, 3,7,11,15,19,... is an arithmetic progression. The sequence 0,1,2,4,5,6,7 is not an arithmetic progression. +------------------------------------------------------------ | arrow paradox +------------------------------------------------------------ The arrow paradox is a classical Zeno paradox with conclusion that motion can not exist: "an object occupies at 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." +------------------------------------------------------------ | asymptotic +------------------------------------------------------------ Two real functions are called asymptotic at a point a if lim_x to a f(x)/g(x) =1. For example, f(x) = sin(x) and g(x) = x are asymptotic at a=0. The point a can also be infinite: for example, f(x)=x and g(x)=sqrtx^2+1 are asymptotic at a=infinity. +------------------------------------------------------------ | Bernstein polynomials +------------------------------------------------------------ The Bernstein polynomials of a continuous function f on the unit interval 0leq x leq 1 are defined as B_n(x) = sum_k=1^n f(k/n) x^k (1-x)^n-k n!/(k!(n-k!). +------------------------------------------------------------ | Binomial coefficients +------------------------------------------------------------ Binomial coefficients The coefficients B(n,k) of the polynomial (x+1)^n for integer n are called Binomial coefficients. Explicitly one has B(n,k) = n!/(k! (n-k)!), where k!=k (k-1)!, 0!=1 is the factorial of k. The function B(n,k) can be defined for any real numbers n,k by writing n!=Gamma(n+1), where Gamma is the Gamma function. 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-x^2/8+.... +------------------------------------------------------------ | Binominal theorem +------------------------------------------------------------ The Binominal theorem tells that for a real number |z|<1 and real number p, one has (1+z)^p=sum_k=0^infinity B(p,k) z^k, where B(p,k) is called the Binomial coefficient. If p is a positive integer, then (1+z)^p is a polynomial. For example: (1+z)^4 = 1+4z+6z^2+4z^3+z^4 . If p is a noninteger or negative, then (1+z)^p is an infinite sum. For example (1+z)^-1/2 = 1-x/2+3x^2/8-5x^3/16+... +------------------------------------------------------------ | bisector +------------------------------------------------------------ A bisector is a straight line that bisects a given angle or a given line segment. For example, the y-axis x=0 in the plane bisects the line segment connecting (-1,0) with (1,0). The line x=y bisects the angle angle(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 -1 and 1 because f is continuous and f(-pi/2)=-1 and f(pi/2)=1. +------------------------------------------------------------ | Fermat principle +------------------------------------------------------------ The Fermat principle tells that if f is a function which is differentiable at z and f(x)>f(z) for all points in an interval (z-a,z+a) with a>0, then f'(z)=0. +------------------------------------------------------------ | fundamental theorem of calculus +------------------------------------------------------------ The fundamental theorem of calculus: if f is a differentiable function on a leq x leq b where a0, we can find n, such that for all k>n,m>n one has |a_m-a_k|0, and the integer n closest to e^2c+1 one has for k>n and m>n |a_m-a_k| leq c. +------------------------------------------------------------ | Cauchy's convergence test +------------------------------------------------------------ Cauchy's convergence test. Given a series sum_k=1^infinity a_k with positive summands a_k. If r=lim_n to infinity a_n^1/n<1 then the series is a convergent series. If r>1, then the series diverges. +------------------------------------------------------------ | continuous +------------------------------------------------------------ A function f is called continuous at a point x if for every open interval V around f(x) there exists an open interval U around x such that f(U) is a subset of V. A function f is continuous in a set Y if it is continous at every point in Y. This definition is equivalent to: for every sequence x_n to x, the sequence f(x_n) converges to f(x). Examples: Any polynomial like x^5+5x^3+3x is continuous on the entire line. The sum and product of continuous functions is continuous. the composition of two continuous functions is continuous. Discontinuities can happen in different ways: the function can become infinite like f(x)=1/x at 0 or tan(x) at x=pi/2, the function can jump like f(x)=sign(x) which is 1 if x>0, -1 if x<0 and 0 if x=0. A function can also become too oscillatory at a point like f(x)=sin(1/x) at x=0. Note that f(x)=x sin(1/x) is continuous on the entire real line. There are functions which are discontinuous at every point. An example is f(x)=1 if x is rational and f(x)=-1 if x is irrational. Note that be restricting the domain of a function, one an make it continuous. For example: f(x)=1/x is continuous on the positive real axes. +------------------------------------------------------------ | converges +------------------------------------------------------------ A function f(x) converges to a value z at x if the function g which agrees with f away from x and satisfies g(x)=z is continouous at x. The value z is called the limit of f at x. For example the function f(x)=(1-x^2)/(1-x) has the limit z=2 at x=1. The function g(x) which is defined bo be f(x) for x neq 1 and g(1)=2 is indeed continuous. One writes z = lim_y to x f(y). One has lim_x to z (f(x) + g(x)) = lim_x to z f(x) + lim_x to z g(x). lim_x to z (f(x) g(x)) = lim_x to z f(x) + g(x). lim_x to z f(g(x)) = f(lim_x to z g(x)). A series a_n is a convergent series, if the partial sum sequence b_n = sum_k=1^n a_k converges to a finite limit a. +------------------------------------------------------------ | absolutely convergent series +------------------------------------------------------------ A series sum_n a_n is called an absolutely convergent series if sum_n |a_n| is a convergent series. +------------------------------------------------------------ | series +------------------------------------------------------------ Summing up a sequence is called a series. An important example is the geometric series 1+1/2+1/4+1/8+1/16+..., which sums up to 2. An other example is the harmonic series 1+1/2+1/3+1/4+1/5+... which has no finite limit. +------------------------------------------------------------ | change of variables +------------------------------------------------------------ The change of variables in integration theory is the formula int f(x) dx = int f(g(u)) g'(u) du if x=g(u). For example, int sqrt1-x^2 dx becomes with x=g(u) = sin(u) and dx = g'(u) du = cos(u) du the integral int cos2(u) du. +------------------------------------------------------------ | differentiable +------------------------------------------------------------ A function f is called differentiable at z if there exists a function g which is continuous at z such that f(x)=f(z)+(x-z) g(x). The derivative of f at z is g(z) and also denoted f'(x). By solving for g(x) and letting x to z one can write g(z) = lim_x to z (f(x)-f(z))/(x-z). The quotient is called the differential quotient. The sum of two at z differentiable functions is differentiable at z and (f+g)'(z)=f'(z)+g'(z). This is called the sum rule. The prduct of two at z differentiable functions is differentiable at z and (f g)'=f'g + f g'. This is called the product rule. The composition of two differentiable functions is differentiable and (f circ g)' = (f' circ g) g'. This is called the chain rule. Functions can be continuous without being differentiable. For example f(x)=|x| is continuous at 0 but not differentiable at 0. There are functions which are continuous everywhere but not differentiable at most points. An example is the Weierstrass function f(x) = sum_k=1^infinity cos(k^2 x)/k^2. +------------------------------------------------------------ | Extended mean value theorem +------------------------------------------------------------ Extended mean value theorem. If f(x) and g(x) are differentiable on the interval (a,b) and are continuous on the closed interval I = a leq x leq b then there exists a point x in I for which f'(x)/g'(x)=(f(b)-f(a))/(g(b)-g(a)) . Proof. Otherwise one would have one of the following two possibilies: f'(x) (g(b)-g(a)) < g'(x) (f(b)-f(a)) for all x in (a,b) or f'(x) (g(b)-g(a)) < g'(x) (f(b)-f(a)) for all x in (a,b). Integration of these expressions using the fundamental theorem of calculus gives (f(b)-f(a)) (g(b)-g(a)) < (g(b)-g(a)) (f(b)-f(a)) or (f(b)-f(a)) (g(b)-g(a)) > (g(b)-g(a)) (f(b)-f(a)) which both are not possible. The special case g(x)=x is called the mean value theorem. +------------------------------------------------------------ | factorial +------------------------------------------------------------ The factorial of a positive integer n is defined recursively by n! = n (n-1)! and 0!=1. For example, 5!=120. The factorial function can be extended to the real line and is then called the Gamma function: n! = Gamma(n+1), where Gamma(z) = int_0^infinity t^z-1 e^-t dt . which is finite everywhere except at z=0,-1,-2,.... +------------------------------------------------------------ | limit +------------------------------------------------------------ The limit of a sequence of numbers a_n is a number a such that a_n converges to a in the following sense: for every c>0 there exists an integer m such that |a_n -a|m. Limits can be defined in any metric space and more generally in any topological space. +------------------------------------------------------------ | maximum-value theorem +------------------------------------------------------------ The maximum-value theorem assures that a continuous function on an interval aC on (a,b). Integration gives using the fundamental theorem of calculus f(x)-f(a)=int_a^x f'(t) dt < C (x-a) or f(x)-f(a)=int_a^x f'(t) dt > C (x-a) especially f(x)-f(a)=int_a^b f'(t) dt < C (b-a) or f(x)-f(a)=int_a^b f'(t) dt > C (b-a) which is a contradiction The mean value theorem is a special case of the extended mean value theorem. +------------------------------------------------------------ | Rolle's theorem +------------------------------------------------------------ Rolle's theorem If f(x) is a continuous function on the interval I = a leq x leq b which is differentiable on the open interval (a,b) and f(a)=f(b), then there exists a point x in (a,b), for which f'(x)=0. Proof. f takes both its maximum and minimum on I. If the maximum is equal to the minimum, then f(x) is constant on I, otherwise, either the minium or the maximum is a point x in (a,b). At that point f'(x)=0. qed. Rolle's theorem is a special case of the mean value theorem +------------------------------------------------------------ | rule of three +------------------------------------------------------------ The rule of three ia a rough rule of thumb when solving calculus problems or teaching calculus: Look at a calculus problem graphically, numerically and analytically. In other words, one should try to understand a calculus problem geometrically, algebraically and computationally. For example, the notion of the derivative of a function of one variable can be understood geometrically as a slope, can be understood through algebraic manipulations like (x^n)' = n x^n-1 or computationally by plugging in numbers or doing things on a computer. +------------------------------------------------------------ | Weierstrass function +------------------------------------------------------------ A Weierstrass function is an example of a function which is continuous but almost nowhere differentiable. An example is f(x) = sum_k=1^infinity cos(k^2 x)/k^2. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 42 entries in this file. COUNT: 42