Mathematics for AI/ML

Determine the domain and range of a real-valued function from its formula or graph.

Evaluate composite functions and decompose a composition into its component functions.

Determine whether a function is one-to-one and compute its inverse when it exists.

Sketch graph transformations produced by translations, reflections, and scaling.

Interpret slope, intercepts, and rates of change in linear models.

Model repeated growth or decay with an exponential function.

Use logarithm laws to simplify expressions and invert exponential relationships.

Solve elementary exponential and logarithmic equations.

Convert between degrees and radians and locate standard angles on the unit circle.

Calculate sine, cosine, and tangent from unit-circle or right-triangle definitions.

Use fundamental trigonometric identities to simplify elementary expressions.

Represent finite sequences and sums using indexed and sigma notation.

Compute arithmetic and geometric series, including convergent infinite geometric series.

Estimate limits numerically and graphically, including one-sided limits.

Evaluate elementary limits using limit laws and algebraic simplification.

Determine whether a function is continuous at a specified point.

Interpret a derivative as an instantaneous rate of change and tangent-line slope.

Compute the derivative of a polynomial directly from the limit definition.

Differentiate algebraic functions using the power, product, and quotient rules.

Differentiate composite functions using the chain rule.

Differentiate elementary exponential, logarithmic, and trigonometric functions.

Locate and classify local extrema using first-derivative information.

Analyze curvature and classify stationary points using the second derivative.

Construct first- and second-order Taylor approximations around a point.

Perform vector addition and scalar multiplication and calculate common vector norms.

Compute dot products, angles, orthogonality, and vector projections.

Perform matrix addition, multiplication, transposition, and dimension checks.

Represent a linear transformation with a matrix and determine its effect on vectors.

Solve a linear system using Gaussian elimination.

Use pivots and rank to determine whether a linear system has zero, one, or infinitely many solutions.

Compute a determinant and relate it to invertibility and geometric scaling.

Compute the inverse of a small matrix and relate inverse existence to unique solvability.

Determine whether vectors span a space or are linearly independent.

Find bases and dimensions for the column space and null space of a matrix.

Construct an orthonormal basis using the Gram–Schmidt process.

Solve a least-squares approximation by projecting onto a column space and verify that the residual is orthogonal to that space.

Compute eigenvalues and eigenvectors of a small square matrix.

Diagonalize a matrix when possible and use the diagonalization to compute matrix powers.

Classify a symmetric quadratic form using eigenvalues and definiteness.

Compute and interpret a small singular value decomposition and its low-rank approximation.

Interpret a definite integral as accumulated change and as a limit of Riemann sums.

Evaluate definite integrals using the Fundamental Theorem of Calculus.

Evaluate elementary integrals using substitution.

Compute first- and second-order partial derivatives of a multivariable function.

Compute a gradient and use it to find a directional derivative.

Construct the Jacobian matrix of a vector-valued function.

Differentiate a multivariable composition using the multivariable chain rule.

Use the gradient and Hessian to classify stationary points of a multivariable function.

Construct a sample space and represent events with set operations.

Count outcomes using the multiplication principle, permutations, and combinations.

Calculate event probabilities using the axioms of probability.

Compute conditional probabilities from joint probabilities.

Apply the law of total probability by partitioning a sample space.

Apply Bayes’ theorem to update probabilities from evidence.

Test whether two events are independent and distinguish independence from mutual exclusivity.

Model and solve a multistage probability problem using a tree or contingency table.

Construct the probability mass function and cumulative distribution function of a discrete random variable and compute its expectation and variance.

Select and parameterize Bernoulli, categorical, binomial, or Poisson distributions for binary, categorical, and count outcomes.

Calculate probabilities from a continuous probability density and cumulative distribution function.

Calculate probabilities, expectations, and variances for uniform and exponential random variables.

Standardize a normal random variable and calculate interval probabilities.

Derive marginal and conditional distributions from a joint distribution.

Compute covariance and correlation and distinguish them from independence.

Use the law of large numbers to explain convergence of empirical averages.

Calculate and interpret sample mean, variance, and covariance.

Use the central limit theorem to approximate a sampling distribution and standard error.

Evaluate an estimator in terms of bias, variance, and consistency.

Derive a maximum-likelihood estimator for a simple probabilistic model.

Derive a maximum-a-posteriori estimate and distinguish it from maximum likelihood.

Construct and interpret a confidence interval for a population mean.

Carry out and interpret a basic hypothesis test for a population mean.

Compute entropy, cross-entropy, and Kullback–Leibler divergence for finite discrete distributions.

Formulate an unconstrained or constrained optimization problem from a verbal objective.

Determine whether a set or differentiable objective function is convex using definitions or a Hessian test.

Derive the gradient-descent update from a first-order local approximation.

Analyze gradient-descent convergence and learning-rate stability on a quadratic objective.

Derive and apply Newton’s method using gradient and Hessian information.

Solve an equality-constrained optimization problem using Lagrange multipliers.

Check Karush–Kuhn–Tucker conditions for a simple inequality-constrained problem.

Construct an unbiased mini-batch gradient estimator and compute its variance in a simple case.

Express a supervised dataset and prediction model using vectors, matrices, and a scalar loss function.

Derive linear regression from a Gaussian likelihood, solve it using the normal equations, and state when the solution is unique.

Solve and compare simple L1- and L2-regularized objectives and explain their sparsity and shrinkage effects.

Derive the likelihood, cross-entropy loss, and gradient for binary logistic regression.

Derive softmax probabilities and the multiclass cross-entropy gradient.

Compute gradients through a small feedforward neural network using backpropagation.

Derive principal component directions from a covariance matrix and relate PCA to singular value decomposition.

Decompose expected prediction error into bias, variance, and irreducible noise in a simple setting.