Review Sheets

These are review sheets for some of the courses I taught. For each course, Review Sheets 1, 2, and 3 cover material (in order) for the first, second, and third exams. The final review sheet reviews the whole course. Each review sheet consists of problems, followed by solutions to the problems.

These review sheets are intended to help guide your studying, but they are not comprehensive. They were written for my courses, and your instructor may cover different topics, and do things differently. Also, the review sheets don't explain topics from scratch; they assume you've seen the material, and want some problems to practice. If you want more background on the material, you can check my course notes.

College Algebra Review Sheets

These reviews cover linear equations, linear inequalities, solving equations for a variable, word problems involving linear equations, absolute value equations and inequalities, lines, systems of linear equations, word problems involving systems of linear equations, integer exponents, polynomials, factoring, solving by factoring, rational expressions, fraction equations, word problems involving fraction equations, graphing, variation, fractional exponents, radical equations, complex numbers, quadratics and the Quadratic Formula, circle equations, word problems involving quadratics, equations quadratic in form, quadratic inequalities, functions, inverse functions, exponential functions, and logarithmic functions.

Calculus 1 Review Sheets

These reviews cover limits, continuity and the Intermediate Value Theorem, rates of change and tangent lines, derivatives, differentiation rules (including trig, inverse trig, log, and exponential functions), the Mean Value Theorem, applications of differentiation (max-min word problems, related rates, absolute maxima and minima, linear approximations, Newton's methods, graphing), L'Hôpital's rule, sums and Riemann sums, the definite integral, the Fundamental Theorem, antiderivatives, substitution, area between curves, and growth and decay.

Calculus 2 Review Sheets

These reviews cover integration techniques, applications of integration (areas, volumes, arc length, surfaces of revolution), sequences, infinite series and convergence tests, power series, parametric equations, and polar coordinates.

Calculus 3 Review Sheets

These reviews cover lines and planes in space, vectors, curves (tangent, normal, binormal, curvature), parametrized surfaces, functions of several variables (continuity, partial derivatives, directional derivatives, max-min for two variables, Lagrange multipliers), double and triple integrals, applications of multiple integrals, conservative fields, vector integral calculus (Green's theorem, the Divergence Theorem, Stokes Theorem).

Math Proof Review Sheets

These reviews cover sentential logic, logic proofs, quantifiers, direct proof, conditional proofs, limits (epsilon-delta proofs), proof by contradiction, proof by cases, induction, counterexamples, sets, limits at infinity, infinite unions and intersections, equivalence relations, divisibility, modular arithmetic, functions, cardinality, and order relations.

Linear Algebra Review Sheets

Note: I taught linear algebra in a non-standard way (specifically, I used fields of nonzero characteristic for many examples). If you're taking a standard linear algebra course, many of the problems in the reviews will probably not make sense to you. In particular, if you see a problem which refers to "Zn" where n is some number (usually 2, 3, 5, 7), you should probably ignore it. (But I still think it was a good way to do linear algebra!)

These reviews cover number systems (commutative rings with identity and fields), matrices, row reduction, inverses, determinants, vector spaces, subspaces, independence and spanning, bases, subspaces for matrices (row space, column space, null space), linear transformations, change of basis, eigenvalues and eigenvectors, the Cayley-Hamilton theorem, systems of linear constant coefficient differential equations, the exponential of a matrix, inner product spaces, coordinate transformations in the plane, unitary and Hermitian matrices, the Spectral Theorem, and basic Fourier series.

Abstract Algebra Review Sheets

These reviews cover groups, subgroups, group maps, divisibility, greatest common divisors, primes, modular arithmetic, cyclic groups, the group of units, permutations, direct product, the Structure Theorem for Finitely Generated Abelian Groups, cosets, Lagrange's theorem, normal subgroups, quotient groups, the First Isomorphism Theorem, rings, ring maps, subrings and ideals, integral domains and fields, polynomial rings, quotient fields, and quotient rings.

Number Theory Review Sheets

These reviews cover the integers, the greatest integer function, sums, products, binomial coefficients, induction, the Fibonacci number, divisibility and primes, greatest common divisors, the Fundamental Theorem of Arithmetic, divisibility tests, Fermat factorization, Fermat numbers, linear Diophantine equations, modular arithmetic, a day-of-the-week algorithm, nonlinear Diophantine equations (examples only), linear congruences, the Chinese Remainder theorem, systems of linear congruences, prime power congruences, Wilson's theorem and Fermat's theorem, Euler's theorem, arithmetic functions and Möbius inversion, the sum and number of divisors functions, perfect numbers and Mersenne primes, character and block ciphers, exponential ciphers, the RSA cryptosystem, quadratic residues and quadratic reciprocity, Jacobi symbols, fractions and rational number, number bases, continued fractions. (finite, infinite, periodic), and the Fermat-Pell equation.

Contact information

Bruce Ikenaga's Home Page