#### From basic skills to complex equations, Brainfuse helps students calculate the answers.

**Brainfuse offers live Mathematics tutoring in the following areas:**

**BASIC MATH**

** Numbers and Operations:** number types, set operations, basic number properties, number systems with different bases, signed numbers, place value, operations with whole numbers, order of operations; fractional notation, comparing fractions, reciprocals and zero, operations with fractions, writing decimals, rounding decimals, comparing decimals, and operations with decimals; percentages, percent equations, and percent change; exponents, roots, and radicals; scientific notation; mean, median, mode, and weighted average; ratio, rate, proportion, and scale drawing;

**distance and motion, work rate, simple and compound interest;**

*Formulas*:**customary system, metric system, converting between systems;**

*Measurement*:**quadrilaterals, angles, triangles, polygons, similarity and dilations, perimeter, area, circumference, surface area and volume of geometric solids (prisms/cylinders/cones/pyramids/spheres);**

*Geometry*:**interpret and create graphs and Venn diagrams;**

*Data and Statistics*:**conditional statements, bi-conditional statements, truth tables, deductive reasoning, inductive reasoning, proofs;**

*Logic*:**Pascal’s Triangle, combinations, permutations; Application Problems**

*Combinatorics*:**ALGEBRA**

** Algebraic Expressions:** simplifying and evaluating;

**linear equations, applications, exponential equations, rules for exponents, solving radical equations, lines and regions, calculating slope, graphing relations/domain/range/linear equations/exponential functions, direct and inverse variation, linear inequalities, absolute value, set notation, quadratic or rational inequalities, solving linear systems;**

*Linear Equations and Inequalities:***coordinate plane, translation, shifting, reflecting, rotating, function notation, 1-to-1 functions, operations and evaluations of functions, inverse functions, determinant functions, slope-intercept form; solve equations with rational and radical expressions; conics—parabola, ellipse, hyperbola;**

*Functions and Graphs*:**substitution, elimination, graphing, Gaussian elimination;**

*Solving Systems of Equations*:**permutations, factorials, combinations, independent and dependent events;**

*Probability*:*products, quotients, negative and zero exponents, scientific notation;*

**Rules of Exponents:***distance and projectile motion, simple interest and compound interest, annuities, and amortization, growth and decay, work rate problems;*

**Algebraic Applications and Problem Solving:****factoring, expanding, and operations with polynomials;**

*Binomials and Polynomials*:**graphing, solving by factoring, grouping, square root property, completing the square, and use of the quadratic formula;**

*Quadratic Equations*:**simplifying radicals, solving radical equations;**

*Radicals*:**common and natural logs, logarithmic functions, operations and evaluations, composite functions, inverse relations and functions, graphing functions and logarithms;**

*Functions and Logarithms*:**lines and regions, asymptotes and graphing rational functions, graphing relations/domain/range/ exponential functions/radical functions/conic sections, finding zeros of polynomial functions, writing polynomial functions with specified zeros, finding roots by synthetic division;**

*Graphing Relations and Functions*:**simplifying rational expressions, operations with rational expressions, simplifying radical expressions, operations with radical expressions, simplifying complex fractions, complex and imaginary numbers (**

*Rational, Radical, and Complex Expressions:**i*);

**linear equations, special factoring patterns – differences of squares, sums and differences of cubes, square root property, completing the square, absolute value in equations, logarithmic properties and equations, radical equations, partial fractions, synthetic division; linear programming with the simplex method and geometric method;**

*Solving Equations*:**linear inequalities, absolute value, quadratic or rational inequalities; set notation; Sequences and Series;**

*Solving Inequalities*:**substitution, elimination, and graphing**

*Systems of Equations and Matrices:***LIBERAL ARTS MATH**

** Problem Solving:** Polya’s method, conditional statements, bi-conditional Statements; Sets and Groups;

**design, implementation, evaluation, interpretation and ways to display the findings;**

*Statistics and Data*:**number systems with different bases; number types;**

*Properties of Numbers*:**modular arithmetic and matrices;**

*Cryptography*:**golden rectangles; Hamilton/Euler Circuits, topology;**

*Geometry:***linear programming with the simplex method and geometric method, algorithms;**

*Computer Applications*:**percent change, simple and compound interest; History of Math;**

*Consumer Math*:**independent and dependent events;**

*Probability*:**Pascal’s Triangle, combination, permutation, and optimization techniques**

*Combinatorics*:**GEOMETRY**

** Describing Lines:** point, line, plane, collinear and coplanar lines, parallel and perpendicular lines, midpoint, intersections, segments, rays; theorems, postulates, and proofs;

**angle classification, segment and angle bisectors, angle pair relationships;**

*Angles:***perimeter and area of shapes; circumference of circles; surface area and volume of solids, Euler’s theorem;**

*Measurement:***proving statements about angles, parallel lines and transversals, proving lines are parallel, equations of lines in the coordinate plane, distance formula;**

*Lines and Angles:***classifying by sides and angles, finding angle measurements, proving congruence, SSS and SAS congruence postulates, ASA congruence postulate and AAS congruence theorem, properties, bisectors, medians, altitudes, mid-segment theorem, inequalities in one triangle, indirect proof, Pythagorean theorem;**

*Triangles*:**quadrilaterals, parallelograms, rhombuses, rectangles, squares, trapezoids, kites, special quadrilaterals;**

*Polygons*:**rigid motion, reflections, rotations, translations, vectors, frieze patterns, glide reflections and compositions;**

*Planes*:**ratio and proportion;**

*Scale Drawing:***tangents, arcs, chords, inscribed angles, segment lengths, equations, locus; Geometric Probability**

*Circles*:**TRIGONOMETRY**

Radian and Degree Measure; ** Trigonometric Functions:** the sine function, the cosine function, the tangent function, the secant function, the cosecant function, the cotangent function, law of sines, law of cosines, law of tangents, law of cotangents, Heron’s formula, right triangle trigonometry, and inverse trigonometric functions;

**verifying trigonometric identities,**

*Trigonometric Functions of Any Angle*:**graphs of sine, cosine, and tangent functions; graphs of the reciprocal functions; analytic trigonometry; solving trigonometric equations, trigonometric application problems, tau vs. pi, vector operations, complex numbers, and polar coordinates**

*Graph*s:**CALCULUS I**

** Defining Limits: **evaluating limits graphically and numerically, limit theorems, rates of change (average and instantaneous), L’Hôpital’s rule, continuity and one-sided limits, infinite limits and vertical asymptotes, intermediate value theorem, max/min;

**continuity and differentiability, basic differentiation rules, product and quotient rules, differentials, higher order derivatives, applications of derivatives, the chain rule; implicit differentiation, differentiation in business, related rates;**

*Differentiation*:**series convergence tests, Power series and Taylor series; parametric equations; extrema on an interval; Rolle’s theorem and the mean value theorem; increasing and decreasing functions; first derivative test; concavity; points of inflection; the second derivative test; limits at infinity and horizontal asymptotes; curve sketching (including monotonicity); graphing in polar coordinates analysis of graphs; optimization; local linear approximations;**

*Sequences and Series:***definition of integrals, antiderivatives and indefinite integration; differential equations; position, velocity, and acceleration problems; Riemann sums;**

*Introduction to Integral Calculus:***properties of definite integrals and the use of geometric formulas to solve them, displacement and definite integrals; trapezoidal rule; the fundamental theorem of calculus; average value of a function; second fundamental theorem of calculus;**

*Definite Integrals:***integrating by parts, integrating by**

*Integration Techniques*:*u*-substitution, integrating by rational functions; integrating exponentials and logs;

**the natural logarithmic function and differentiation; the natural logarithmic function and integration; inverse functions; exponential functions; bases other than**

*Transcendental Functions*:*e*and applications;

**growth and decay, separation of variables, and slope fields;**

*Differential Equations:***differentiation and integration;**

*Inverse Trigonometric Functions*:**Area of a region between two curves; volume (known cross-sections, disk method, washer method), surface area, and work**

*Applications of Integration*:**CALCULUS II**

** Functions and Function Notation:** definition of a function, vertical line test

*,*function notation

*,*finding input and output values

*,*domain and range

*;*

**piecewise-defined functions**

*Absolute Value and Piecewise Defined Functions:**,*the absolute value function, solving equations involving absolute values

*;*

**solving inequalities**

*Inequalities:**,*solving inequalities involving absolute values

*;*

**composition of functions**

*Composition and Combination of Functions:**,*inverse functions

*,*non-invertible functions

*;*

**the family of exponential functions, the number**

*Exponential and Logarithmic Functions:**e*

*,*logarithmic functions

*,*solving exponential and logarithmic functions

*,*changing the base of logarithmic functions;

**horizontal and vertical shifts**

*Transformation of Functions:**;*reflections and symmetry

*;*vertical stretches and compressions

*;*horizontal stretches and compressions

*;*

**radians and arc length, the sine, cosine, and tangent functions; inverse trigonometric functions**

*Trigonometric Functions:**;*trigonometric identities

*;*

**linear functions, quadratic functions, polynomials of higher degree, zeros of polynomials, rational functions, solving rational inequalities;**

*Polynomial and Rational Functions:***components of a vector**

*Vectors and Vector-Valued Functions:**;*addition, subtraction, and scalar multiplication

*;*the zero and unit vectors

*;*

**polar coordinates**

*Polar Coordinates and Graphs*:*,*coordinate conversion

*,*polar graphs

*,*and special polar graphs

*;*

**parametric graphing**

*Parametric Equations and Conic Sections:**,*Archimedean spiral, eliminating the parameter

*,*finding parametric equations

*;*

**using tables and graphs to find limit, using limits to find velocity**

*Intuitive Definition of a Limit:**;*

**calculating limits by using limit laws, direct substitution properties; indeterminate forms**

*Algebraic Techniques for Finding Limits:**;*

**definition, finding one-sided limits**

*One-Sided Limits:**,*existence of limits;

**definition of infinite limits**

*Infinite Limits:**,*vertical asymptotes

*;*

**end behavior of functions and horizontal asymptotes**

*Limits at Infinity:**;*limit laws for infinite limits; oblique asymptotes;

**simplification, substitution, special trigonometric limits**

*Limits of Trigonometric Functions*:*;*

**continuity at a point**

*Continuity**:**,*continuity on a closed interval

*,*continuity on an open interval, intermediate value theorem;

**the derivative as—the slope of a tangent, the rate of change, a function**

*Definition of the Derivative:**;*

**constant rule, constant multiple rule**

*Differentiation Rules:**,*power rule

*,*sum rule, difference rule

*,*product rule

*,*quotient rule

*;*

**the chain rule and its use with other rules**

*Trigonometric Rules:**;*higher-order derivatives

*;*

**exponential rule**

*Derivatives of Exponential Functions:**;*base-a exponentials rule

*;*

**natural logarithmic rule, general logarithmic rule**

*Derivative of Logarithmic Functions:**;*

**inverse trig rule, derivatives of inverses**

*Derivatives of Inverse Functions:**;*

**Differentiability implies continuity**

*Differentiability and Continuity:**,*non-differentiable functions

*;*

**derivatives of implicitly defined functions;**

*Implicit Differentiation:***derivatives of complicated expressions;**

*Logarithmic Differentiation:***parametric form of the derivative**

*Parametric Derivatives:**;*parametric form of the second derivative

*;*

**tangents to polar slopes**

*Differentiation with Polar Curves:**;*

**limits and continuity, derivatives of vector-valued functions**

*Differentiation of Vector-Valued Functions:**,*differentiation rules;

**tangent lines**

*Tangent and Normal Lines:**,*normal lines; Position, Velocity, and Acceleration (PVA); Related Rates;

**the first derivative test and critical numbers; increasing and decreasing functions and the first derivative test; concavity and the second derivative test**

*Relative Extrema:**;*

**extreme value theorem**

*Absolute Extrema and Optimization:**;*

**linear approximation, differentials; L’Hôpital’s rule; indeterminate forms**

*Differentials:***CALCULUS III**

The 3D Coordinate System; Equations of Lines; Equations of Planes; Quadric Surfaces; Functions of Several Variables;** Vectors: **lines, planes, dot product, cross product

*;*

**scalars or vectors, tangents, arc length, limits, derivatives, velocity and acceleration, curvature, cylindrical coordinates, spherical coordinates**

*Vector Functions**:**;*

**limits, interpretations of partial derivatives, higher order partial derivatives, differentials, chain rule, gradients, directional derivatives, Lagrange multipliers**

*Partial Derivatives:**;*

**tangent planes and linear approximations, gradient vector, tangent planes and normal lines, relative minimums and maximums, absolute minimums and maximums, Lagrange multipliers;**

*Applications of Partial Derivatives:***double integrals; iterated integrals; double Integrals over general regions; double integrals in polar coordinates; triple integrals; triple integrals in cylindrical coordinates; triple integrals in spherical coordinates; change of variables; moments about center of gravity; surface area; area and volume revisited;**

*Multiple Integrals*:**fundamental theorem for line integrals and conservative vector fields; Green`s theorem; curl and divergence; surface integrals: parametric surfaces; surface integrals of vector fields; Stokes’ theorem; divergence theorem**

*Line Integrals of Vector Fields**:***STATISTICS**

** Exploratory Analysis of Data: **univariate data displays—dotplot, stemplot, histogram, cumulative frequency plot;

*Measuring Center:*mean vs. median;

**standard deviation, quartiles and boxplots, clusters and gaps; outliers; describing shape; changing units, summarizing distributions of data;**

*Measuring Spread:***within group and between group variation; comparison of—center and spread, clusters and gaps, outliers and other features that stray from the expected pattern, shapes,**

*Comparing Distributions of Univariate Data:***Density curves, 68-95-99.7 rule, standard normal curve, nonstandard normal curve, assessing normality, normal distributions and calculations**

*Normal Distributions:**;*scatterplots, correlation, least-square regression, residuals, outliers, and influential observations;

**Explore Bivariate Data:***transform relationships for linearity; ladder of powers (power transformation), logarithmic transformations, power model, log-log transformations; frequency tables and bar charts; comparing distribution using bar charts; critical study of correlation and regression; causation, confounding, and common response, relations in categorical data, conditional distribution*

**Transformations:***;*methods of data collection—census, sample survey, experiment, observational study; qualities of a well-designed and well-conducted survey, selecting and defining the population, sources of bias and confounding, completely randomized design, randomized block design, generalizability of results, and valid conclusions that can be drawn; anticipating patterns and studying random variables

**Planning and Conducting an Experiment:***;*

**Probability****:**interpreting probability, “Law of Large Numbers” concept, addition rule, multiplication rule, conditional probability, and independence; discrete random variables and their probability distributions; simulation of random behavior and probability distributions; mean (expected value) and standard deviation of a random variable, linear transformation of a random variable; combining independent random variables; notion of independence versus dependence; mean and standard deviation for sums and differences of independent random variables.

**properties of the normal distribution, using tables of the normal distribution, the normal distribution as a model for measurements; sampling distributions of a sample proportion and of a sample mean; central limit theorem; sampling the distribution between two independent sample proportions; sampling distribution of a difference between the means of two independent samples; simulation; t-distribution, Chi-square distribution;**

*The Normal Distribution:***inference for means, inference for proportions, inference for two-way tables, Analysis of Variance (ANOVA), inference for regressions, regression model, predictions and conditions, and confidence intervals**

*Statistical Inference*:**ADVANCED STATISTICS**

** Domain of Multivariate Statistics:** numbers of independent and dependent variables (IVs and DVs); continuous, discrete, and dichotomous data, samples and populations; descriptive and inferential statistics; orthogonality; experimental and non-experimental research; computers and multivariate Statistics;

**data matrices, correlation matrices, variance-covariance matrices, sum-of-squares and cross-products matrices, residuals;**

*Appropriate Data:***nflated and deflated correlations; missing data; univariate and multivariate outliers; normality, linearity, and homoscedasticity, homogeneity of variance and homogeneity of variance-covariance matrices; common data transformations; multicollinearity and singularity**

*Screening Data:*i*;*Bivariate

**Research Questions:***r*, Multiple

*R*, Sequential

*R*, Canonical

*R*, multiway frequency analysis (log-linear analysis); multivariate analysis of variance and covariance; discriminant function analysis;

**one-way ANOVA and**

*Significance of Group Differences:**t*Test, one-way ANCOVA, factorial ANOVA, factorial ANCOVA; hotelling T

^{2}, one-way MANOVA (Wilk’s lamda criterion), one-way MANCOVA, factorial MANOVA, factorial MANCOVA;

*Structure:*principal components; factorial analysis, structural equation modeling; introduction to PRELIS for preparation of matrices; introduction to LISREL;

**generalized linear models, logistic regression, poisson regression, survival analysis, time series modeling, discriminant function analysis—linear and quadratic;**

*Predictive Models*:**one-way discrimination, sequential one-way discrimination, logistic regression, sequential logistic regression, factorial discriminant analysis, sequential factorial discriminant analysis, survival/failure analysis, and time-series analysis**

*Prediction of Group Membership*:**DIFFERENTIAL EQUATIONS**

** First-Order Differential Equations:** analytical technique–separation of variables, slope field; numerical technique–Euler’s Method; graphical technique–equilibria and the phase line; linear differential equations;

**bifurcations; modeling first-order systems; The Geometry of Systems;**

*First-Order Systems:***linear systems, linearity principle, eigenvalues and eigenvectors, phase planes for linear systems with real eigenvalues, repeated and zero eigenvalues; trace-determinant plane; Fourier series;**

*Linear Systems:***forced harmonic oscillators; periodic solutions, sinusoidal forcing; undamped forcing and resonance;**

*Forcing and Resonance:***equilibrium point analysis, qualitative analysis, Hamiltonian systems, complex numbers and exponentials, critical point analysis;**

*Nonlinear Systems:***discontinuous functions, second-order equations, delta functions, and impulse forcing**

*Laplace Transforms:***LINEAR ALGEBRA**

** Matrices:** matrix types, vectors, matrix equations, solution sets of linear systems, linear transformations, theorems

*;*

**Matrix Algebra****:**entries, operations, inverses, subspaces, dimension and rank, kernel and range, row-column rule, rank theorem, basis theorem, invertible matrix theorem; algebraic definitions; algebraic rules;

**cofactors, row operations, Cramer’s rule, vector spaces and subspaces, bases, coordinate systems, vector space dimensions; the basis theorem; vector-space axiom;**

*Determinants*:**characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues, dynamical systems, differential equations, diagonalization theorem; diagonalization representation;**

*Eigenvalues and Eigenvectors:***basics of vectors, orthogonal sets, orthonormal sets, decomposing vectors, Gram-Schmidt process, least-square problems; linear models, inner product space;**

*Orthogonality and Least Squares:***diagonalization and symmetric matrices;**

*Symmetric Matrices and Quadratic Forms:***geometric view of principal axes, spectral theorem for symmetric matrices, principal axes theorem, quadratic forms and eigenvalues**

*Quadratic Forms:***DISCRETE MATHEMATICS **

** Proofs:** compound Statements, truth tables, the algebra of propositions;

**sets, operations on sets, binary relations, equivalence relations, partial orders;**

*Sets and Relations*:*Functions:*domain, range, one-to-one, onto, inverses and composition, one-to-one correspondence and the cardinality of a set;

**isomorphism, connectivity, Euler and Hamiltonian paths, trees;**

*Graphical Representation:**Integers:*the division algorithm, divisibility and the Euclidean algorithm, prime numbers, congruence, applications of congruence;

**mathematical induction; recursively defined sequences, the characteristic polynomial, solving recurrence relations**

*Induction and Recursion:**,*generating functions;

**Baye’s theorem;**

*Discrete Probability:***: logic, functions, gates, circuits;**

*Boolean Algebra***the principle of inclusion-exclusion, the addition and multiplication rules, and the pigeonhole principle;**

*Principles of Counting*:**permutations, combinations, and repetitions**

*Combinatorics*:**SPANISH/ENGLISH BILINGUAL TUTORING**

*Tutoria bilingüe en Español e Inglés también esta disponible en matemáticas, lectura y escritura.* (Spanish/English bilingual tutoring is also available in math, reading, and writing.)