Higher Ed
Higher Ed Mathematics

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

Brainfuse offers live Mathematics tutoring in the following areas:


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; Formulas: distance and motion, work rate, simple and compound interest; Measurement: customary system, metric system, converting between systems; Geometry: quadrilaterals, angles, triangles, polygons, similarity and dilations, perimeter, area, circumference, surface area and volume of geometric solids (prisms/cylinders/cones/pyramids/spheres); Data and Statistics: interpret and create graphs and Venn diagrams; Logic: conditional statements, bi-conditional statements, truth tables, deductive reasoning, inductive reasoning, proofs; Combinatorics: Pascal’s Triangle, combinations, permutations; Application Problems


Algebraic Expressions: simplifying and evaluating; Linear Equations and Inequalities: 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; Functions and Graphs: 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; Solving Systems of Equations: substitution, elimination, graphing, Gaussian elimination; Probability: permutations, factorials, combinations, independent and dependent events; Rules of Exponents: products, quotients, negative and zero exponents, scientific notation; Algebraic Applications and Problem Solving: distance and projectile motion, simple interest and compound interest, annuities, and amortization, growth and decay, work rate problems; Binomials and Polynomials: factoring, expanding, and operations with polynomials; Quadratic Equations: graphing, solving by factoring, grouping, square root property, completing the square, and use of the quadratic formula; Radicals: simplifying radicals, solving radical equations; Functions and Logarithms: common and natural logs, logarithmic functions, operations and evaluations, composite functions, inverse relations and functions, graphing functions and logarithms; Graphing Relations and Functions: 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; Rational, Radical, and Complex Expressions: simplifying rational expressions, operations with rational expressions, simplifying radical expressions, operations with radical expressions, simplifying complex fractions, complex and imaginary numbers (i); Solving Equations: 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 Inequalities: linear inequalities, absolute value, quadratic or rational inequalities; set notation; Sequences and Series; Systems of Equations and Matrices: substitution, elimination, and graphing


Problem Solving: Polya’s method, conditional statements, bi-conditional Statements; Sets and Groups; Statistics and Data: design, implementation, evaluation, interpretation and ways to display the findings; Properties of Numbers: number systems with different bases; number types; Cryptography: modular arithmetic and matrices; Geometry: golden rectangles; Hamilton/Euler Circuits, topology; Computer Applications: linear programming with the simplex method and geometric method, algorithms; Consumer Math: percent change, simple and compound interest; History of Math; Probability: independent and dependent events; Combinatorics: Pascal’s Triangle, combination, permutation, and optimization techniques


Describing Lines: point, line, plane, collinear and coplanar lines, parallel and perpendicular lines, midpoint, intersections, segments, rays; theorems, postulates, and proofs;  Angles: angle classification, segment and angle bisectors, angle pair relationships; Measurement: perimeter and area of shapes; circumference of circles; surface area and volume of solids, Euler’s theorem;  Lines and Angles: proving statements about angles, parallel lines and transversals, proving lines are parallel, equations of lines in the coordinate plane, distance formula; Triangles: 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; Polygons: quadrilaterals, parallelograms, rhombuses, rectangles, squares, trapezoids, kites, special quadrilaterals; Planes: rigid motion, reflections, rotations, translations, vectors, frieze patterns, glide reflections and compositions; Scale Drawing: ratio and proportion; Circles: tangents, arcs, chords, inscribed angles, segment lengths, equations, locus; Geometric Probability


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; Trigonometric Functions of Any Angle: verifying trigonometric identities, Graphs: 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


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; Differentiation: 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; Sequences and Series: 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; Introduction to Integral Calculus: definition of integrals, antiderivatives and indefinite integration; differential equations; position, velocity, and acceleration problems; Riemann sums; Definite Integrals: 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; Integration Techniques: integrating by parts, integrating by u-substitution, integrating by rational functions; integrating exponentials and logs; Transcendental Functions: the natural logarithmic function and differentiation; the natural logarithmic function and integration; inverse functions; exponential functions; bases other than e and applications; Differential Equations: growth and decay, separation of variables, and slope fields; Inverse Trigonometric Functions: differentiation and integration; Applications of Integration: Area of a region between two curves; volume (known cross-sections, disk method, washer method), surface area, and work


Functions and Function Notation: definition of a function, vertical line test, function notation, finding input and output values, domain and range; Absolute Value and Piecewise Defined Functions: piecewise-defined functions, the absolute value function, solving equations involving absolute values; Inequalities: solving inequalities, solving inequalities involving absolute values; Composition and Combination of Functions: composition of functions, inverse functions, non-invertible functions; Exponential and Logarithmic Functions: the family of exponential functions, the number e, logarithmic functions, solving exponential and logarithmic functions, changing the base of logarithmic functions; Transformation of Functions: horizontal and vertical shifts; reflections and symmetry; vertical stretches and compressions; horizontal stretches and compressions; Trigonometric Functions: radians and arc length, the sine, cosine, and tangent functions; inverse trigonometric functions; trigonometric identities; Polynomial and Rational Functions: linear functions, quadratic functions, polynomials of higher degree, zeros of polynomials, rational functions, solving rational inequalities; Vectors and Vector-Valued Functions: components of a vector; addition, subtraction, and scalar multiplication; the zero and unit vectors; Polar Coordinates and Graphs: polar coordinates, coordinate conversion, polar graphs, and special polar graphs; Parametric Equations and Conic Sections: parametric graphing, Archimedean spiral, eliminating the parameter, finding parametric equations; Intuitive Definition of a Limit: using tables and graphs to find limit, using limits to find velocity; Algebraic Techniques for Finding Limits: calculating limits by using limit laws, direct substitution properties; indeterminate forms; One-Sided Limits: definition, finding one-sided limits, existence of limits; Infinite Limits: definition of infinite limits, vertical asymptotes; Limits at Infinity: end behavior of functions and horizontal asymptotes; limit laws for infinite limits; oblique asymptotes; Limits of Trigonometric Functions: simplification, substitution, special trigonometric limits; Continuity: continuity at a point, continuity on a closed interval, continuity on an open interval, intermediate value theorem; Definition of the Derivative: the derivative as—the slope of a tangent, the rate of change, a function; Differentiation Rules: constant rule, constant multiple rule, power rule, sum rule, difference rule, product rule, quotient rule; Trigonometric Rules: the chain rule and its use with other rules; higher-order derivatives; Derivatives of Exponential Functions: exponential rule; base-a exponentials rule; Derivative of Logarithmic Functions: natural logarithmic rule, general logarithmic rule; Derivatives of Inverse Functions: inverse trig rule, derivatives of inverses; Differentiability and Continuity: Differentiability implies continuity, non-differentiable functions; Implicit Differentiation: derivatives of implicitly defined functions; Logarithmic Differentiation: derivatives of complicated expressions; Parametric Derivatives: parametric form of the derivative; parametric form of the second derivative; Differentiation with Polar Curves: tangents to polar slopes; Differentiation of Vector-Valued Functions: limits and continuity, derivatives of vector-valued functions, differentiation rules; Tangent and Normal Lines: tangent lines, normal lines; Position, Velocity, and Acceleration (PVA); Related Rates; Relative Extrema: the first derivative test and critical numbers; increasing and decreasing functions and the first derivative test; concavity and the second derivative test; Absolute Extrema and Optimization: extreme value theorem; Differentials: linear approximation,  differentials; L’Hôpital’s rule; indeterminate forms


The 3D Coordinate System; Equations of Lines; Equations of Planes; Quadric Surfaces; Functions of Several Variables; Vectors: lines, planes, dot product, cross product; Vector Functions: scalars or vectors, tangents, arc length, limits, derivatives, velocity and acceleration, curvature, cylindrical coordinates, spherical coordinates; Partial Derivatives: limits, interpretations of partial derivatives, higher order partial derivatives, differentials, chain rule, gradients, directional derivatives, Lagrange multipliers; Applications of Partial Derivatives: tangent planes and linear approximations, gradient vector, tangent planes and normal lines, relative minimums and maximums, absolute minimums and maximums, Lagrange multipliers; Multiple Integrals: 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; Line Integrals of Vector Fields: 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


Exploratory Analysis of Data: univariate data displays—dotplot, stemplot, histogram, cumulative frequency plot; Measuring Center: mean vs. median; Measuring Spread: standard deviation, quartiles and boxplots, clusters and gaps; outliers; describing shape; changing units, summarizing distributions of data; Comparing Distributions of Univariate Data: 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, Normal Distributions: Density curves, 68-95-99.7 rule, standard normal curve, nonstandard normal curve, assessing normality, normal distributions and calculations; Explore Bivariate Data: scatterplots, correlation, least-square regression, residuals, outliers, and influential observations; Transformations: 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; Planning and Conducting an Experiment: 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; 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.  The Normal Distribution: 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; Statistical Inference: 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


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; Appropriate Data: data matrices, correlation matrices, variance-covariance matrices, sum-of-squares and cross-products matrices, residuals; Screening Data: inflated 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; Research Questions: Bivariate r, Multiple R, Sequential R, Canonical R, multiway frequency analysis (log-linear analysis); multivariate analysis of variance and covariance; discriminant function analysis; Significance of Group Differences: one-way ANOVA and t Test, one-way ANCOVA, factorial ANOVA, factorial ANCOVA; hotelling T2, 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; Predictive Models: generalized linear models, logistic regression, poisson regression, survival analysis, time series modeling, discriminant function analysis—linear and quadratic; Prediction of Group Membership: 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


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; First-Order Systems: bifurcations; modeling first-order systems; The Geometry of Systems; Linear 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; Forcing and Resonance: forced harmonic oscillators; periodic solutions, sinusoidal forcing; undamped forcing and resonance; Nonlinear Systems: equilibrium point analysis, qualitative analysis, Hamiltonian systems, complex numbers and exponentials, critical point analysis; Laplace Transforms: discontinuous functions, second-order equations, delta functions, and impulse forcing


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; Determinants: cofactors, row operations, Cramer’s rule, vector spaces and subspaces, bases, coordinate systems, vector space dimensions; the basis theorem; vector-space axiom; Eigenvalues and Eigenvectors: characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues, dynamical systems, differential equations, diagonalization theorem; diagonalization representation; Orthogonality and Least Squares: basics of vectors, orthogonal sets,  orthonormal sets, decomposing vectors, Gram-Schmidt process, least-square problems; linear models,  inner product space; Symmetric Matrices and Quadratic Forms: diagonalization and symmetric matrices; Quadratic Forms: geometric view of principal axes, spectral theorem for symmetric matrices, principal axes theorem, quadratic forms and eigenvalues


Proofs: compound Statements, truth tables, the algebra of propositions; Sets and Relations: sets, operations on sets, binary relations, equivalence relations, partial orders; Functions: domain, range, one-to-one, onto, inverses and composition, one-to-one correspondence and the cardinality of a set; Graphical Representation: isomorphism, connectivity, Euler and Hamiltonian paths, trees; Integers: the division algorithm, divisibility and the Euclidean algorithm, prime numbers, congruence, applications of congruence; Induction and Recursion: mathematical induction; recursively defined sequences, the characteristic polynomial, solving recurrence relations, generating functions; Discrete Probability: Baye’s theorem; Boolean Algebra: logic, functions, gates, circuits; Principles of Counting: the principle of inclusion-exclusion, the addition and multiplication rules, and the pigeonhole principle; Combinatorics: permutations, combinations, and repetitions


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.)