FY12CE Mathematics study plan

Compute limits including one-sided limits, limits at infinity, and limits requiring L'Hôpital's rule or algebraic manipulation, and classify discontinuities.

State and apply the definitions of continuity and differentiability, and construct counterexamples showing continuity without differentiability.

Differentiate products, quotients, composites (chain rule), and implicit and logarithmic functions fluently without reference notes.

Set up and solve related-rates and linear-approximation problems from word descriptions alone.

Apply the Mean Value Theorem, Rolle's Theorem, and the First and Second Derivative Tests to sketch curves and justify monotonicity and concavity claims.

Solve optimization problems with closed and open constraint sets, and verify endpoints and boundary behavior.

Compute definite and indefinite integrals using substitution, integration by parts, and partial fractions.

Evaluate improper integrals and determine convergence or divergence with justification.

Apply the Fundamental Theorem of Calculus, including the derivative of an integral with variable limits.

Compute areas between curves, volumes of revolution (disk, washer, shell), and arc length, choosing the correct method per region.

Perform Gaussian elimination with row operations, determine rank, and classify solution sets of linear systems as consistent or inconsistent.

Compute determinants by cofactor expansion and row reduction, and use them to test invertibility.

Test sets of vectors for linear independence and spanning, and construct bases for column space, null space, and row space.

Compute eigenvalues and eigenvectors by hand and use diagonalization to evaluate matrix powers.

Solve vector geometry problems involving dot and cross products, projections, lines and planes in three dimensions, and distances between them.

Apply the Rank-Nullity Theorem and the invertible matrix theorem to connect properties of a linear map.

Classify first-order differential equations (separable, linear, exact) and solve each by its standard method, including integrating factors.

Solve second-order linear constant-coefficient equations, both homogeneous and with undetermined-coefficient or variation-of-parameters particular solutions.

Model and solve applied initial-value problems (exponential growth and decay, Newton cooling, simple harmonic motion) and interpret the constants.

Compute probabilities using conditional probability, Bayes' theorem, and independence, and derive means and variances of standard distributions.

Apply discrete and continuous distributions (binomial, geometric, Poisson, uniform, normal) to compute probabilities and to set up sampling-distribution and central-limit-theorem questions.

Construct confidence intervals and carry out one-sample hypothesis tests, stating assumptions and interpreting p-values correctly.

Translate English statements into propositional and predicate logic, build truth tables, and test validity with rules of inference.

Solve counting problems using permutations, combinations, and inclusion-exclusion, and apply basic proof techniques (induction, contradiction, contraposition).

Implement and compare root-finding methods (bisection, Newton, secant), stating convergence conditions and estimating error.

Apply numerical integration (trapezoid, Simpson) and compute truncation-error bounds for a given step size.

Perform interpolation (Lagrange, Newton divided differences) and a least-squares linear fit, and explain when each is appropriate.

Formulate a real-world scenario as a mathematical model, identify assumptions and parameters, and assess the sensitivity of the result to one assumption.

Review the error log from weeks 1-7 and rework every item flagged as a recurring error type until it is solved correctly without notes.

Complete two timed full-length practice exams under exam conditions and score them by domain to confirm weighting-appropriate performance.

Memorize and rehearse a compact formula and theorem sheet covering all seven domains, reciting derivations of the five most-used results.

Build a final-week schedule that allocates remaining study time in proportion to domain weight and personal weak spots.