www.YoYoBrain.com - Accelerators for Memory and Learning Questions for mathematics

Category: Default - (161 questions) Define an inverse function reverses the result of the function operation and tells you what you started with or what what the input was to get the value What is the sin 0 Give sin 1 / 2 Give sin root (2) / 2 Give: sin square root (3) / 2 Give: sin 1 What are the trigonometric (angle) sum sin (A + B) = sinA cosB + cosA sinB cos(A + identities (sin(A+B), cost(A+B),tan B) = cos A cos B - sin A sin B tan(A + B) = (tan A + tan B) / (1 - tan A tan B) cos 2a = ???? 1 - (2 sin^2 a) (2 cos^2 a) - 1

What is the Fibonacci series 0, 1, 1, 2, 3, 5, 8 .... after first 2 values each next value is generated by adding the preceding 2 What is the cos 1 Give: cos square root (3) / 2 Give: cos square root (2) / 2 Give: cos 1 / 2 Give: cos 0 Define: isosceles 2 sides that are equal and angles opposite those sides are equal Define: sufficient condition A condition that must be satisfied for a statement to be true and without which the statement cannot be true. Define: abscissa and ordinate abscissa - independent variable (usually x) of a point on the Cartesian plane ordinate - dependent variable coordinate Define: inclusion theorem If limit of x approaching a for f(x) = limit of x approaching a for g(x) = A and f(x) = h(x) = g(x) for all x near a then limit of x approaching a h(x) = A Define: function mathematical relation in which every abscissa corresponds to at most one ordinate Define: minimal surface the shape of least area when bounded by a given closed space ex: shape of soap film between 2 empty circular rings

Define: pi ratio of the diameter to the circumference 3.14.... Define: quaternion name given by W.R. Hamilton to ordered 4 set of real numbers used to represent hypercomplex numbers a + b i + c j + d k Define: one degree (of a circle) 1/360 of a full circle Who is J.W. Gibbs American mathematician (1839 - 1903) and physicists who developed much of vector analysis as we know it today Define: sin and cos in terms of unit circle on sin - is the y value of point cos - is the x cartesian system value of the point Define: Russell's paradox of sets If sets are subject only to some principle of free construction, then what of the set of all sets that are not members of themselves? The set of dogs, which is not a dog Define: tan a sin a / cos a Define: logistics equation correctly models a wide range of situations in which a growing population competes for limited supplies. A good example might be fish in a lake that has limited amount of food. X n+1 = kXn * ( 1 - Xn) The population at step n+1 = some constant * population at step n * ( 1 - population at step n) with population as a% of maximum carrying capacity Define: csc cosecant = 1 / sin Who is Edward Lorenz one of the early pioneers of choas theory - worked evolved out of trying to model weather Define: sec secant 1 / cos Define: geodesic (mathematics) the shortest line between two points on a mathematically defined surface (as a straight line on a plane or an arc of a great circle on a sphere) Define: cot 1 / tan or cos / sin Define: Thomsom lamp Suppose you have a reading lamp with a push-button that switches the light on and off. If the light is "off" when you press the button once, or any odd number of times, the lamp will be "on". Press it any number of even times and it will be "off". A little demon appears and decides he will press the button continually so as to leave the lamp 'on' for 1/2 a minute, then 'off' for 1/4 a minute, 'on' for 1/8 of a minute, 'off' for 1/16 of a minute and so on. He will have pressed the button an infinite number of times after 1 minute. So the big question is: Will the light be 'on' or 'off' after one minute. What does sinh, cosh, and tanh stand for hyperbolic sine hyperbolic cosine hyperbolic tangent Define: Goldbach's conjecture states: Every even integer greater than 2 can be written as the sum of two primes. Expressing a given even number as a sum of two primes is called a Goldbach partition of the number. \ For example, 4 = 2 + 2, 6 = 3 + 3 , 8 = 3 + 5,10 = 3 + 7 = 5 + 5, 12 = 5 + 7, 14 = 3 + 11 = 7 + 7, etc.

Define: irrational number a number that can't be precisely represented as a ratio of 2 whole number ex: square root of 2 Who was Evariste Galois early 19th century French mathematician who laid the foundation for group theory Define: acute and obtuse angle acute angle - less than 90 degreesobtuse angle - greater than 90 degrees Define: logarithm of N Every positive number N can be expressed as a power of 10; we can always find p such that: N = 10^p. We call p the logarithm of N to the base of 10 or the common logarithm of N Define: vertical and supplementary angles when two line intersect: vertical angles are opposite each other supplementary angles are side by side (their angles add to 180 degrees) Define: mantissa and characteristic of the decimal part in logarithm is the mantissa logarithms the part to the left of the decimal is the characteristic

What is the irrational number Phi also known as the ratio created when a line is divided in such a way that the relationship between the larger segment and the entire length is the same as the relationship between the smaller and larger segments. 1.6180339887..... How do you determine the characteristic of a 1.) for a number greater than 1, the logarithm of number N characteristic is positive and is one less than the number of digits before the decimal point 2.) for a number less than 1, the characteristic is negative and is one more than the number of zeros immediately following the decimal point.

Define: the antilog of number p the number whose logarithm is p (reverse of logarithm) Define: log(M*N) log(M) + log(N) Define: log ( M / N) log(M) - log(N) Define: log (M^p) p * log(M) Define: manifold simple numbers in a particular order, removes the idea of actual physical space from coordinate analysis. Makes it simpler to analyze higher dimensions Who is responsible for creation of set theory Georg Cantor Define: cardinal number in set theory Each cardinal number corresponds to sets that have the same size. Cardinal number of 5 is assigned to all sets that have 5 members Define: chromatic number of a surface The number of colors that is sufficient so that regions with common boundary-line segments on a surface are distinguished by different colors Who is Authur Cayley invented matrix algebra Define: combinatorial includes problems of covering, packing and symmetry Define: Latin square of the n-th order a permutation of n symbols arranged in n rows and n columns - used in design of experiments that will be subjected to statistical analysis Define: fractal curve when the estimated length of a curve becomes arbitrarily large as the measuring stick becomes smaller and smaller Define: fractal dimension a measure of the degree of irregularity considered at all scales ratio between the logarithms of the number of copies and the size of the seed relative to each copy (how much smaller is the next level of detail to the previous) Who coined the term fractals and developed Benoit Mandelbrot much of the base of math Define: mode locking in dynamic systems the tendency to fall back into a behavior pattern called an attractor, even when external pertubations knock the system off the attractor momentarily. Define: Sierpinski's Triangle You begin with a triangle. Then divide the triangle into 4 equal pieces. Then divide the outer three pieces in the same way as the initial, continuing into infinity. Define: Riemannian geometry that are obtained by changing the way distance is measured in the plane one example would by hyperbolic geometry Who is Hippasus of Mepapontum Discovered irrational numbers and was reputadely drowned by Pythagoreans Who is Augustin Louis Cauchy French mathematician of 19th century who started a project of formulating and proving the theorems of calculus in a rigrous manner Cantor's definition of an infinite set a set that can be put into a one-to-one correspondence with a proper subset of itself Define: Bertrand Russel's barber paradox deals with classes. A class of books is not a book an so does not belong to itself, but a class of ideas is an idea and does belong to itself. A village barber advertises that he doesn't shave any people in the village who shave themselves, but he does shave all those who don't shave themselves. One does it occurred to him to ask whether he should shave himself. If he does shave himself then he can't shave himself because he only shaves people who don't shave themselves.

Define: ordered set and well ordered set ordered - means, as in the case of the whole numbers, that if a and b are any 2 members of the set, either a precedes b or b precedes a.well ordered - Further, if a precedes b and b precedes c, then a precedes c. Godel's incompleteness theorem It states that, if an formal theory T adequate to embrace the theory of whole numbers is consistent, the T is incomplete. The price of consistency is incompleteness. No system of mathematical and logical axiom that can be arithmetized in some manner such as Godel used is adequate to encompass all the truths of even that one system, to say nothing about all of mathematics, because any such axiom system is incomplete. Banach-Tarski paradox Given any 2 solid spheres, one the size of a baseball and the other the size of the earth, both the ball and the earth can be divided into a finite number of non-overlapping little solid pieces, so that each part of one is congruent to one and only one part of the other. One can divide the entire earthup into little pieces and merely by rearranging them make up a sphere the size of a ball. Define: Lowenheim-Skolem theory Suppose one sets up axioms, logical and mathematical, for a branch of mathematics. One intends that these axioms should completely characterize the branch. But, suprisingly, one discovers that one can find interpretations - models that are drastically different and yet satisfy the axioms. Axiom systems that are designed to characterize a unique class of mathematical objects do not do so. A set of axioms permits many more essential interpretations than the one intended. Who is Apollonius of Perga (ca. 262 BC-ca. 190 BC) was a Greek geometer and astronomer, of the Alexandrian school, noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes.

Who created the foundation for group theory Evariste Galois Who is al-Khwarizmi was a Persian mathematician, astronomer, astrologer and geographer. He was born around 780 in Khwarizm (now Khiva, Uzbekistan) and died around 850. He worked most of his life as a scholar in the House of Wisdom in Baghdad. His Algebra was the first book on the systematic solution of linear and quadratic equations. Consequently he is considered to be the father of algebra, Define: rational numbers any number that is the ratio of 2 integers 1 / 2, 4 / 2 Define: ring number system A number system in which addition, subtraction, and multiplication are always defined and the associative and distributive laws are valid Define: field number system A number system in which addition, subtraction, and multiplication are always defined and the associative and distributive laws are valid (i.e. - ring system) Division (except by 0) can also be carried out

When did the general acceptance and during the 18th century algebraic use of negative numbers begin What is the application of prime numbers to often advantageous to use gear wheels gear design whose number of teeth are prime numbers - or any other number whose only common factor is 1. The same gear teeth will then mesh only at long intervals, limiting uncontrolled localized wear of gear flanks by high spots and other imperfections, and reducing gear noise. Define: perfect numbers/ deficient number, perfect number - an integer number that is abundant number equal to the sum of all its possible divisors deficient number - sum of all its possible integers is less than the number abundant number - the sum of all its possible integers is more than the number

Define: amicable numbers 2 integer numbers are said to be amicable if each is the sum of all the possible divisors of the other smallest pair is 220 and 284

2 types of irrational numbers algebraic numbers - roots of algebraic equations with irrational numbers transcendental numbers - not the roots of any algebraic equation. existence proved in 1844, includes e (base of natural logarithms) and pie

When/who introduced imaginary and Raffaele Bombelli in 1572 in conjunction with complex numbers solving cubic equations Define: to square the circle to construct a square with the same area as a given circle What does Greek capital letter pi, _ denote a product, so you multiply each of the terms in formulas together

Define: antecedent and consequent of a ratio first term of ratio is antecedent - asecond a / b term of ratio is consequent - b Define: complex fraction have a fraction for the numerator or denominator or both Define: commutative systems the order of adding or multiplying terms is inconsequential Example of non-commutative system vectors and matrices Define: radical a nth root of some number N Define: surd a radical expressing an irrational number example: square root of 2

Difference between: ln x and lg x lg x - common logarithm of x (to base 10) ln x - natural logarithm of x (to base e)

What is the definition of natural logarithm e e = limit as n approaches infinity of ( 1 + 1/n) ^ n Define: indirect proof establish the truth of a statement by showing that the contradiction of it is false, and therefore the statement must be true Define: combinatorics branch of mathematics that is concerned with the selection of objects - called elements - from a given set of elements Formula for the number of permutations with n! / q! * r! * s! ..... identical objects: n objects, q of one type, r of another, s of another, etc. to a total of p objects When sampling with replacement: what is n ^ p the number of samples formed with n n raised to the p power objects, sampled p at a time

What rule did Euler uncover about traversing Reduce paths to network with vertices and a network without crossing a path twice paths to arcs. The valence of a is the number of arcs that originate or end at the vertex. A path traversing the network by crossing every segment just once is possible only if the network is connected and has at most 2 vertices with odd valences

Contrast: Euler path vs. Hamilton path Euler path - traverses every segment of a network once, with no restrictions on the # of times each vertex may be passed through Hamilton path - goes through every vertex only once, with no obligation to traverse all segments but ending at the starting point

Define: transfinite numbers cardinal or ordinal numbers that are larger than all the finite numbers Meaning of the symbol:? contains as a member Meaning of the symbol: ? Does not contain as a member Who initiated partnership between G. W. von Leibniz (1646-1716) mathematics and logic and when Who /when created Boolean algebra George Boole in 1847 Define: Epiminides' paradox Liar paradox This statement is not true. What is this symbol in symbolic logic:¬ negation or not What is this symbol mean:? universal quantifier - indicates "for every" or "for all"

What is the symbol: Rdouble struckRused for the collection of all real numbers What is the meaning of the symbol: ? existential quantifier - indicates "there exists" What is the meaning of symbol: U conjunction - indicates "and"

What is the meaning of symbol:? disjunction symbol - indicates "or" What is the meaning of the symbol: _ implication - indicates "implies"

Define: converse of p implies q q implies p

Define: contrapositive of p? q ¬ q? ¬p Meaning of the symbol:? equivalence - indicates "is equavalent to" Meaning of symbol:? is a member of

Meaning of symbol:? not a member of What is symbol:Ø null set, empty set What set does: Z represent set of integers

What set does: Q represent set of rational numbers What set does: C represent set of complex numbers Contrast: normal sets with non-normal sets normal sets - sets that do not contain outlined by Bertrand Russell themselves as elements (The set of all dogs) non-normal sets - sets that contain themselves as elements (The set of all sets that are not dogs)

What is the set symbol: ? subset of or equal to What is the set symbol: ? subset of and not equal to (proper subset) What is the set symbol: ? includes properly (not equal to) What is the set symbol P mean power set - the set of all subsets of a given set, containing the empty set and the original set Contrast proper set with improper set proper set - includes elements of the set to which it is matched, but is not equal to that setimproper set - is supposed include all the elements of the set to which it is matched What is the set notation symbol: ? complement - the difference between the universal set and the subset Define the Cartesian products of sets A = A X B = { (1,a), (1,b), (2,a), (2,b), (3,a), (3,b) {1,2,3} and B={a,b} } What are the idempotent laws for sets A U A = A A insersect with A = A

Define: de Morgan's laws for sets 1. ? ( A U B) = (?A) intersect ?B - the complement of the union of 2 sets is the intersection of the complements 2. ?( A intersect B) = (?A) U ?B - the complement of the intersection of 2 sets is the union of the complements Define: denumerably infinite sets sets that are equivalent to the set of all natural numbers Define: alternating series has alternating positive and negative terms Formula for sum of a finite arithmetic series = n / 2( first term + last term) equals half the consisting of n terms product of the number of terms and sum of the first and last terms Define: geometric series a1 + a1*r + a1*r^2 + a1*r^3 + ...... Formula for the sum of a finite geometric a1 * (1 - r^n) / (1 - r); series of n terms r not equal to 1 a1 is the first term r is the common ratio

When is a geometric series convergent when the common ratio is between -1 and 1 What is the sum of a convergent, infinite lim as n approaches infinity of Sn = a1 / 1 - r geometric series a1 is the first term r is the common ratio What is d'Alembert's ratio test to determine The series S An = a1 + a2 + .... + an series convergence (assume positive terms) is convergent if limit as n approached infinity of (An+1) / An 1 and divergent if limit as n approached infinity of (An+1) / An 1 Define: triangular numbers the natural numbers which can be drawn as dots and arranged in a triangular shape * ** * * * 1,3,6,10,15, ...... Define: gnomons the geometric representations of odd numbers as dots on equally long legs of . The name refers to the angle's likeness to the Bablyonian sundial, the gnomon 3 basics schools of thought at end of 1.) logicism - all math could be reduced to 19th/start of 20th century on the basis of purely logical laws and definitions (failed) mathematical truth 2.) formalism - math is a kind of logic game, in which definitions are invented and then mathematicians explore the logical consequences of these terms in various combinations according to specified rules. Not connected to reality. 3.) intuitionist - believe that math is an example of the mind imposing order on experience. There is no connection between math and reality. We intuit the kinds of mathematical relationships that will be useful and interesting and then explore them logically.

Define: fundamental theorem of algebra every polynomial of degree n= 1, with real or complex coefficients, has n real or complex roots Define: Diophantine Equations polynomial equation that allows the variables to be integers only Define: even function function is said to be even if f(-x) = f(x) for every value of x in the domain of definition symmetrical about the ordinate axis Define: odd function a function is odd if f(-x) = - f(x) for every value of x in the domain of definition Define: interval of monotonicity in function interval over which a function either increases or decreases What does the following function notation The composite, or composition of the 2 mean: f ° g functions f and g - function of a function Define: composite function function of a function- f ° g is created by taking g(x) and then feeding the answer into function f

Who developed hyperbolic geometry and Nicolai Lobachevski - 1829 when Janos Bolyai in 1832

Who developed elliptic geometry and when Bernhard Riemann in 1871 Define: homeomorphic objects in topology topological equivalent - if one object can be transformed into another by topological transformations, that is, by bending, stretching, or twisting, but not by overlapping, tearing, or curring Define: topological invariant a property held in common by all homeomorphic objects Define: tensor analysis a generalization of vector analysis - a study of components, motions, and the like in n-dimensional space Define: topology branch of geometry that deals with are not altered by continuous deformations Define: centroid center of mass of objects Define: Fourier series expansion The principal idea is to represent a function f of period 2p as an infinite series of trigonometric functions a0/2 + a1cos x + a2 cos 2x + ..... + b1 sin x + b2 sin 2x + ...... If f(x) is an even function defined over -p x p cosine series then its Fourier expansion reduces to _____ If f(x) is an odd function defined over -p x p a sine series then its Fourier expansion reduces to _____ Meaning of symbol:? proportional to Math: What are 2 co-prime integers integers that have no positive divisor other than 1 Math: relatively prime numbers 2 integers are relatively prime (or co-prime) if their largest common denominator is 1 Math: meaning of number theory notation - a and b are co-prime or relatively prime a b Category: Default - (5 questions) Define: catenoid surface generated when a catenary is rotated about it's directrix Define hyperbolic sine in terms of e -natural sinh x = (e^x - e^-x) / 2 log Define hyperbolic cosine in terms of e - cosh x = (e^x + e^-x) / 2 natural log Formula for catenary curve f(x) = a cosh(x/a) where a is a positive constant equal to the y-intercept Define: arsinh x inverse hyperbolic sine of x Category: Default - (13 questions) Define: mixed partial derivatives ?^2f /?y ?x second or higher-order partial derivatives with respect to several independent variables What is the derivative of a quotient: f(x) / g(x) [ f ' (x) * g(x) - f(x) * g ' (x) ] / { g(x)^2 } What is the derivative of a composite df / dg * dg / dx - chain rule of differentiation function f composed of g

What is the derivative of sin x cos x What is the derivative of cos x - sin x What is the derivative of tan x 1 / cos^2 x = sec^2 x = 1 + tan^2 x What is the derivative of ln x 1 / x Define: implicit differentiation process of determining the derivative of one of two variables with respect to the other. Use implicit differentiation to solve x^2 + y^2 use the chain rule on = 1 with y assumed as the dependent y^2 2x + 2y(dy/dx) = 0 variable dy/dx = - x/y

Define: partial differentiation a function of more than one independent variable, y = f(x,y,z,t ....) is differentiated with respect to one of these variables, the other variables being held constant Meaning of syntax:?f /?x partial derivative of the function f (of several variables x,y,....) with respect to x Define: Rolle's Theorem Suppose f is a continous function that crosses the x-axis at 2 points a and b and is differential at all points between a and b - that is, it has a tangent at all points on the curve between a and b. Then there is at least one point between a and b where the derivative is 0, and the tangent is parallel to the x-axis Define: Lagrange's Mean-Value Theorem A function f(x) that is continous in the closed interval [a,b] and differentiable in the open interval ]a,b[ has in this interval at least one value c such that f ' (c) = ( f(b) - f(a) ) / (b - a) Category: Default - (3 questions) Define: integrand the function to be integrated Define: indefinite and definite integral indefinite - integral with no restrictions imposed on its independent variable definite - defined by limit values a and b of the independent variable Define: improper integral Where an integrand that is infinite whithin the range of integrange, or an integral on an infinite range of integration Category: Default - (3 questions) Define: point attractor is a set to which the system evolves after a long enough time. For the set to be an attractor, trajectories that get close enough to the attractor must remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with fractal structures known as a strange attractor. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory. Define: self-organized criticality the theory that many interacting systems naturally and inevitably evolve to a critical state existing at the between choas and stability. Popular example is a pile of sand forming on a round tabletop. When a grain of sand is added it is unknown how many grains will fall off, but the shape of pile is maintained. Define: flicker noise versus white noise flicker noise - behavior that indicates a correlation between current system activities and past system activities white noise - indicates no correlation between current system behavior and past activity

Category: Default - (6 questions) Define: indegree of a node the number of distinct edges ending at the notde Define: edges pairs of adjacent connected nodes in the graph Define: outdegree of a node the number of distinct edges beginning of distinct edges beginning at node Define: source node of a graph has a positive outdegree but an indegree of zero has edges leading from, but not to, the node Define: sink node of a graph has a positive indegree but an outdegree of zero has edges leading to, but not from, the node Define: internal node of a graph has an indegree greater than zero and an outdegree greater than zero both a source and a sink Category: Default - (6 questions) Define: modulus of complex number length of the vector from origin representing the complex number Define: argument of complex number the polar angle of vector representing the complex number Formula for product of 2 complex numbers convert to polar coordinates the product has a modulus that is equal to multiplying the modulus of 2 original numbers has an argument that is the addition of 2 original modulus Formula for quotient of 2 complex numbers convert to polar coordinates the product has a modulus that is equal to dividing the modulus of 2 original numbers has an argument that is the subtraction of 2 original modulus de Moivre's formula for finding powers of z^n = r^n(cos n*_ + i sin n*_) where r is the complex numbers modulus of complex number and _ is the argument

What is i ^ i - the imaginary number raised to = e ^(-p/2) imaginary power Category: Default - (4 questions) If f(x) is a function that has a second concave downward in ]a,b[ derivative f ''(x) < 0 for all x in the inverval ]a,b[, the graph of f(x) is _____ If f(x) is a function that has a second concave upward in the inverval ]a,b[ derivative f ''(x)> 0 for all x in the inverval ]a,b[, the graph of f(x) is _____

How do you find local min/max and inflection Where f '(x) = 0;solve for x; x = cif f(x) is points on f(x) twice differentiable, and f ''(c) 0, then c represents a relative maximum and f ''(c) 0, then c represents a relative minimum How do you find critcal points on where both their first-order parital derivatives differentiable functions of 2 independent are zero variables Category: x2, y2 - (5 questions) Formula for distance between 2 points (x1, d = square root of ( (x1 - x2)^2 + (y1 - y2)^2 ) y1) and (x2, y2) Formula for the slope of a line with 2 points m = (y1 - y2) / (x1 - x2) (x1, y1) and Sum of two vectorsa (x1, y1)b (x2, y2) [ (x1 + x2), (y1 + y2) ] Dot product of 2 vectorsa (x1, y1) and b (x2, x1*x2 + y1 * y2 y2) Cross product of two vectors a (x1, y1) and b | a x b | = |a| |b| sin O Category: trig - (12 questions) Define: angle in standard form has its initial side on the positive x axis and the terminal (ending) side is rotated counterclockwise from that initial side. Up to 90 degrees. Law of sines sinA / a = sinB / b = sinC / c Law of cosines a^2 = b^2 + c^2 - 2bc cosA b^2 = a^2 + c^2 - 2ac cosB c^2 = b^2 + a^2 - 2ba cosC

Define: sine in terms of sides of right triangle opposite side of angle / hypotenuse Define: cosine in terms of sides of right adjacent side of angle / hypotenuse triangle Define: tangent in terms of sides of right opposite side / adjacent side triangle Define: arcos function inverse trigonometric function How would you solve an arbitrary triangle law of sines : sin A / a = sin B / b = sin C / c when we know 2 angles and one opposite side How would you solve an arbitrary triangle law of sines : sin A / a = sin B / b = sin C / c when you know 2 sides and an opposite angle How would you solve an arbitrary triangle law of cosines - a^2 = b^2 + c^ - 2*b*c*cosA when you know 2 sides and the included angle

How would you solve an arbitrary triangle law of cosines - a^2 = b^2 + c^ - 2*b*c*cosA when you know 3 sides What is the area of a triangle when we know = a * b * sin(C) / 2 2 sides and the included angle Category: vector analysis - (6 questions) A vector that is the sum of a given set of resultant vectors is called _____ Define: position vector a vector represented by a line segment beginning at the origin of an orthoganal coordinate system Distinguish between scalar products and scalar product a • b - the sum of the products vector products of corresponding components of the vectors and useful for determing the angles between 2 vectors vector product a x b - produces a vector orthogonal to the plane that contains vectors a and b

What is the notations: a • b a dot b - the scalar product for a = x1 i + y1 j b = x2 i + y2 j a • b = x1*x2 + y1*y2

What is the scalar product useful for a • b determining the angle between 2 vectorscos angle = a * b/(|a| * |b| ) What is the geometric interpretation of vector vector orthoganal to the plan that contains product axb vectors a and b the magnitude of the vector product is the area of the determined by a and b Category: matrices - (13 questions) Define: dimension or order of a matrix A matrix with m rows and n columns has order of m x n Define: leading element of matrix row first non-zero entry in a row Define: algebraic vector If either m or n is unity, the matrix is either a row vector or column vector Define: lower triangle matrix and upper lower triangle matrix - all entries above the triangle matrix main diagonal are zero upper triangle matrix - all entries below the main diagonal are zero

Define: trace of the matrix sum of the diagonal entries of a square matrix Meaning of notation: trA trace of the matrix Define: scalar matrix diagonal matrix with a11 = a22 = a33 = .... = k where k is a constant Define: determinant value representing sums and products of a square matrix The numerical value of an n-th order determinant is the algebraic sum of n! ters, each being the product of n different entries taken one each from every row and column of the determinant What technique is used to determine Laplace expansion into minor matrices determinants of order 3 or higher What is the geometric meaning of the area of a parallelogram spanned by vectors absolute value of a second order a1 and a2 whose coordinates are the determinant columns or rows of A What is the transpose of a matrix If a matrix A is reflected in its main diagonal, so that all rows become columns and all columns become rows without changing their relative order Define: orthogonal matrix a matrix A that is equal to the inverse of its transpose matrix What is the common algorithm used to Gaussian elimination reduce linear equations Category: power series - (7 questions) Define: power series ? a sub k * x^k of k=0 to ininity a + a1 * x^1 + a2 * x^2 ..... Define: power series centered at c f(x) = a0 + a1*(x - c) + a2*(x-c)^2 + a3*(x-c)^3 + ..... Define: Taylor's series f(x) = ? of k from 0 to infinity of kth derivitive of f(c) / k! * (x - c)^k Define: Maclaurin series a Taylor series centered at 0 (that is, c = 0) Expand sin x in a power series sin x = x - x^3 / 3! + x^5 / 5! - x^7 / 7! + ..... Expand cos x in a power series cosx =1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ....

Define: Euler's zeta function ?(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...... Category: topology - (1 questions) Define: genus of a surface the largest number of Jordan curves that can be drawn on the surface without cutting it into 2 unconnected parts Category: set operation - (2 questions) Define: partition a covering whose subsets do not intersect each other Define: covering collection of subsets, drawn from a set, whose union is the original set Category: analytic geometry - (20 questions) Description of hyperbola locus of points, in the plane, for which the difference of their distances from 2 fixed points (foci) is constant What is the general equation | x / a |^n + | when n = 2 it describes an ellipse, otherwise y/b |^n = 1 when n = 1 it describes straight lines between the a and b vertices, if n<1 the lines bends inward toward the origen and n>1 it starts bulging outward Define: translation of equation a parallel displacement of the original system along one or more of its axes Formula for rotating an equation z degrees x = x' cos z - y' sin z y = x' sin z + y' cos z

What determines the position of a point in a distance from center r direction in degrees of polar coordinate system angle from polar axis Define: spiral of Archimides with the polar equation r = a * angle where a is a proportionality constant, radius vector r increase with the polar angle Define: hyperbolic spiral r = a / polar angle spirals inward toward origin Define: logarithmic spiral log r = a * polar angle Define: lituus a spiral that is asymptotic to the polar axis and winds around and gets increasing closer to the pole but never reaches it r^2 = a / polar angle Define: lemniscate double loop around the origin with the 2 loops connecting at the origin r^2 = cos ( 2 * angle) Define: cylindrical coordinates combines the use of polar coordinates in the plane with the z-coordinate of 3 dimensional orthogonal coordinates Define parametric equations instead of representing a 2-dimensional graph by one equation in two variables, it may be represented by 2 equations, each of which gives the x and y in terms of a third variable, called a parameter and usually denoted t Equation for a parametric representation of a x = r * cos(t) circle y = r * sin(t)

Equation for parametric representation of an x = a cos(t) ellipse y = b sin(t)

Equation for parametric representation of x = a * sec(t) hyperbola y = b * tan(t)

Define: cycloid the curve generated by a point on the circumference of a circle which rolls on a straight line in its place Define: brachistochrone problem the curve along which a particle will slide - under the influence of gravity - in the shortest time between 2 points at different altitudes but no on the same vertical line the solution is along a cycloid Define: epicycloid curve generated by a point on the circumference of a circle (the epicycle) which rolls on the outside of a fixed circle (the deferent) Define: hypocycloid curve generated by a point on the circumference of a circle which rolls on the inside of a fixed circle Define: astroid a hypocycloid whose rolling circle has a diameter 1/4 of that of the fixed circle Category: calculus - (5 questions) What is the basic principle of finding the arc approximately equal to the sum of the length of f(x) lengths of the straight lines connecting subsequent points on the curve. integral over a to b of square root( 1 + [ f '(x) ]^2) dx ) Formula for the area between 2 curves Integral from a to b [f(x) - g(x)] dx

Formula for finding the area of a surface of S = 2 * pie * Integral over a to b of f(x) v(1 + [ revolution by revolving the curve f(x) about f '(x) ]^2 ) the x axis over inverval [a,b]

Formula for the volume of a solid of Integral over a to b of [f(x) ]^2 dx revolution generated by the region bounded by the graphs of y = f(x), y = 0, x = a and x = b, about the x axis

Formula for the volume of a solid of 2 * pie * Integral over a to b of xf(x) dx revolution generated by the region bounded by the graphs of y = f(x), y = 0, x = a and x = b, about they axis

Category: in radians - (1 questions) Formula for finding the length of an arc given s = degree (in radians) * radius the angle of the arc Category: geometry - (41 questions) Define Pythagorean triple a listing of three numbers that sastify the Pythagorean theorem ex: { 3, 4, 5 } Define: transversal when a line cuts through 2 parallel lines What is the area of a circle 2 * pie * radius squared Define: chord of a circle segment that is drawn from one point on a circle to another How many solids are there with flat sides of 5 - tetrahedron, cube, octahedron, equal area dodecahedron, icosahedron Define: complementary angles a pair of angles whose measurments add up to 90 degrees Define: lune spherical angle, the angle between any two semicircles passing through diametrically opposed points but not on the same circle Meaning of ~ symbol in geometry similarity - not differing in shape but only in size Define: parallelepiped a prism with 6 faces, all Define: prism polyhedron with 2 congruent and parallel faces (bases) whose remaining faces (lateral faces) are parallelograms Define: congruent figures when superimposed, figures that can be made to coincide exactly; have equal size and shape What does symbol mean: ~ similar to - same shape but different size Define: projective geometry the study of those properties of plane figures that are unchanged when a give set of points is projected into a second plane Define: topologically transformations bending, stretching, or the like, not cutting, overlapping, or tearing Definition of symbol:? corresponds to If 1 mm on a map corresponds to a distance of 1km: 1mm? 1km What is not allowed in a topological folding that brings once distant points into transformation direct contact/overlap or cutting unless followed by a regluing that reestablishes the preexisting relationships of continuity Define: Jordon curve simple closed curve - a continous curve that does not intersect itself and has no endpoints Define: hyperbolic geometry substitutes the Euclidean parallel postulate with the following: Through a given point outside a given straight line pass more than one line not intersecting the given line. Define: elliptic geometry substitutes the Euclidean parallel postulate with the following: Through a given point outside a given straight line there are no parallel lines, if if extended far enough, any 2 straight lines in a plane will meet Define: cusp double point on a curve where the curve has two coincident tangents (not smooth - two curves that have different limits when approached from either side) Define: crunode is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines. does a loop de loo Define: skew lines lines which lie in different planes and do not intersect each other Define: secant line on a curve straight line that intersects a curve Define: rhomboid a parallelogram whose adjacent sides are unequal Define: a rhomboid (parallelogram) with all sides equal Define: semi-regular polyhedra 3-D objects that are bounded by regular , but of more than one kind, generally 2 kinds, and they can all be inscribed in a sphere Define: frustum on a pyramid the part pyramid between the base and any transecting plane parallel to the base Define: nappes of cones 2 partes of a conical surface - one on either side of the vertex Define: pentadecagon 15 sided What is the golden ratio a line segment that is divided into 2 segmentsa greater a and a lesser b such thatthe length of a + b is to a as a is to b2 / square root(5) - 1 Define: Pythagoras's lute a progression of diminishing and , linking vertices together

Formula for the area of a rhomboid product of the length of one side and the corresponding height between the sides Formula for the area of a 1/2 * (a + b) * h half the product of the sum of the parallel sides times its height Formula for the area of an ellipse Area =p*radius to small side * radius to large side Formula for finding the area of a segment of 2/3 * width of segment * height of segment a parabola

Formula for the volume of a prism equal to the product of the base area B and the perpendicular height h V = B * h Formula for the volume of all pyramids or 1/3 Base * height cones Define: cardioid heart shaped r = a(1 + cos ?) Formula for surface area of a sphere with 4 p r^2 radius r Guldin's First Rule for calculating surface The surface area of a surface of revolution is area equal to the product of the arc length of the generating curve and the length of the path described by it's centroid Guldin's Second Rule for calculating volume The volume of a solid of revolution is equal to area the product of the generating area and the length of the path described by it's centroid Category: fractals - (7 questions) Define: Sierpinski carpet starts out as a square which is divided into 9 smaller , or which the central square is removed. Each of the remaining squares is divided into 9 smaller squares with the middle removed, and so forth Define: von Koch's curve described by Helge von Koch in 1904, start with equilater triangle of side length a, on the middle of every side construct a new of side length a/3, and so forth for each new triangle. Creates a snowflake pattern with infinite perimeter but finite area. Define: Cantor set generated by removing, in an iterative fashion, a midportion of a true line segment. left with an infinite number of points Define: Sierpinski sponge generated from a cube that is divided into 27 smaller cubes and the central cube, and the central cube and those at the center of each face of the cube are removed. This process is repeated for each remaining cube.

Define: Julia set formed by iterating a nonlinear function whose variables includes its own result set. a Julia set is a where f(z), f[f(z)] f{f[f(z)]}, ... where z is a complex number and it jumps around within a bounded region.ex: f(z) = z^2 + i

Define: Peano curve Divide a square into 4 sub-squares; join the centers of the sub-squares by a broken line. Then, divided every sub-square of a unit square int 4 more sub-squares, and repeat. Define: tent transformation x at n+1 = a * x at n if x <= 0.5 and a( 1 - x at n) if x>0.5

Category: stastics - (1 questions) What is the standard deviation of the square root ( (standard deviations of sampling distribution of 2 means population sample)^2 /size of sample 1 + (standard deviations of second population sample)^2 / size of second population )

Category: indeterminant limits - (1 questions) Define: L'Hospital's Rule If f and g are differentiable functions on an open interval that contains a, and limit as x approaches a on f(x) = 0 ; g(x) = 0 or f(x) = +/- infinity ; g(x) = +/- infinity then limit of f(x) / g(x) = f ' (x) / g ' (x) provided that the derivative of (x) is not equal to 0 at a Category: mathematics -set theory - (1 questions) Define: power set of a given set If you have any infinite set, then you can generate one that is infinitely bigger by considering the set that contains all its subsets.