University math

Find the absolute maximum and minimum of a function: the closed interval method with critical points and endpoints, and what to do on open or unbounded intervals.

The alternating series test, the error bound for stopping early, and the difference between absolute and conditional convergence.

Find antiderivatives: the family F(x) + C, the basic integration rules as derivative rules run backward, ln|x|, and initial value problems.

Find the length of a curve with an integral, L = ∫√(1 + f′(x)²) dx, built from the Pythagorean theorem on tiny pieces of the curve.

Find the area between two curves: top minus bottom with vertical slices, right minus left with horizontal slices, and splitting where the curves cross.

Find areas bounded by polar curves with A = ½∫r² dθ: one curve, the region between two curves, and finding the limits where curves meet.

Find the average value of a function with an integral, see it as the height of an equal-area rectangle, and use the Mean Value Theorem for integrals.

Differentiate without limits: the constant, power, constant-multiple and sum rules, the derivatives of sin x, cos x, e^x and ln x, and why each rule holds.

Judge a series by comparing it with one you know: the direct comparison test, the limit comparison test, and how to choose a benchmark series.

Use f″ to see how a graph bends: concave up and down, inflection points where the bend changes, and the second derivative test for maxima and minima.

What it means for a function to be continuous, the kinds of discontinuity, which functions are continuous, and how the Intermediate Value Theorem proves roots exist.

Sketch a graph from its derivatives: intercepts, asymptotes, increasing and decreasing, extrema, concavity and inflection points — and read f from a graph of f′.

Find volumes of revolution with the shell method, V = 2π∫(radius)(height) dx, and choose between shells and washers for a region and an axis.

The derivative of an inverse function is the reciprocal of a slope: use it to find derivatives of ln x, logarithms in any base, arcsin, arccos and arctan.

What a differential equation is, how to check a solution, general and particular solutions, and how a slope field shows every solution at once.

Find the volume of a solid of revolution with disks and washers: V = π∫R² dx or π∫(R² − r²) dx, around the axes and around other lines.

Approximate the solution of a differential equation step by step: follow the slope for a short step, find the new slope, repeat, and judge the error.

The link between derivatives and integrals: accumulation functions, the derivative of an integral (Part 1), and evaluating integrals with antiderivatives (Part 2).

Find dy/dx when y is not solved for: differentiate both sides, attach dy/dx to each y term by the chain rule, solve, and find tangent lines and second derivatives.

Integrals over infinite intervals and of unbounded functions, defined as limits: when they converge, when they diverge, and the p-integrals.

Use the sign of f′ to find where a function rises and falls, locate critical points, and classify local maxima and minima with the first derivative test.

Decide whether a series converges by comparing it with an integral: the integral test, the harmonic series, p-series, and bounding the remainder.

Integrate products with ∫u dv = uv − ∫v du, the product rule run backward: choosing u and dv, using parts twice, and definite integrals.

Evaluate limits of the forms 0/0 and ∞/∞ by differentiating top and bottom, and rewrite the forms 0·∞, ∞ − ∞, 1^∞, 0⁰ and ∞⁰ so the rule applies.

Evaluate limits exactly: the limit laws, direct substitution, factoring and conjugates for 0/0 forms, the squeeze theorem, and why sin x over x approaches 1.

Find limits as x grows without bound and limits that blow up: horizontal asymptotes from leading terms, vertical asymptotes from infinite limits, and growth rates.

Estimate values with the tangent line: local linearity, the linearization L(x), whether an estimate is too high or too low, and differentials for small changes.

The logistic equation dP/dt = kP(1 − P/L): growth that levels off at a carrying capacity, its S-shaped solution, and where growth is fastest.

Rewrite an integrand before integrating: divide when a fraction's numerator is too big, and complete the square to reach the arctangent.

Rolle's theorem, the Mean Value Theorem and the Extreme Value Theorem: what each guarantees, why the hypotheses matter, and what follows from them.

Integrate rates in context: position from velocity, displacement versus total distance, and amounts that change with a rate in and a rate out.

Read derivatives as rates with units, and use them for motion: velocity and acceleration from position, when an object speeds up or slows down, and total distance.

Solve optimization problems: model the quantity to maximize or minimize, use the constraint to reduce it to one variable, find critical points, and justify the answer.

Slopes, tangent lines, second derivatives and arc length for curves given by x = f(t) and y = g(t), where dy/dx is dy/dt divided by dx/dt.

Integrate rational functions by splitting them into simpler fractions: distinct linear factors, finding the constants, repeated factors, and dividing first.

Find slopes of polar curves r = f(θ) by treating them as parametric curves: dy/dx in polar form, tangent lines, and what dr/dθ tells you.

Series with a variable, the sum of cₙ(x − a)ⁿ: where they converge, finding the radius with the ratio test, and testing the two endpoints.

Differentiate products and quotients: the product rule, the quotient rule, why a product's derivative has two terms, and the derivatives of tan, cot, sec and csc.

Test series with factorials and powers by comparing each term with the one before: the ratio test, the root test, and what an inconclusive result means.

Find how fast one quantity changes from how fast another does: link them with an equation, differentiate with respect to time, and substitute values only at the end.

Approximate accumulated change and area: left, right and midpoint Riemann sums, trapezoidal sums, sums from tables, sigma notation, and over- and underestimates.

Solve dy/dx = g(x)h(y) by separating variables, find particular solutions, and model exponential growth, decay, half-life and Newton's law of cooling.

What it means for an infinite sum to converge: sequences and their limits, partial sums, geometric series, and the nth-term test for divergence.

Represent functions as power series: the Taylor series formula, the standard Maclaurin series, and new series by substitution, differentiation and integration.

Approximate a function near a point by a polynomial that matches its derivatives there, and bound the error with the Lagrange error bound.

Differentiate a function inside a function: the chain rule, why rates of change multiply, common patterns like powers and exponentials of an inner function, and a^x.

The definite integral as the limit of Riemann sums: signed area, the properties of integrals, evaluating integrals with geometry, and the difference from total area.

The derivative as a limit of difference quotients: from average to instantaneous rate of change, tangent lines, the derivative as a function, and where it fails.

Undo the chain rule: choose u, rewrite the integral in u and du, integrate, and substitute back, with constant adjustments and new limits for definite integrals.

Describe motion in the plane with a position vector: velocity, speed and acceleration as derivatives, and displacement and distance as integrals.

Find volumes by slicing: integrate the area of a cross section, for solids whose slices are squares, triangles, rectangles or semicircles on a base region.

The idea of a limit: what value f(x) approaches as x approaches a, estimated from tables and read from graphs, with one-sided limits and the ways a limit fails to exist.

A basis spans a subspace with no redundancy, and every basis has the same size: the dimension. The rank theorem splits a matrix's columns between Col A and Nul A.

Meet subspaces, the sets closed under sums and scalar multiples, and the two every matrix carries: its column space in ℝᵐ and its null space in ℝⁿ, with a basis for each.

Compute determinants by cofactor expansion or by row reducing to triangular form, see how row operations change them, and read det A as an area or volume factor.

Write A = PDP⁻¹ with eigenvectors in P and eigenvalues in D, use it to compute Aᵏ in one step, and see where a Markov chain settles in the long run.

Find the directions a matrix only stretches: solve det(A − λI) = 0 for the eigenvalues, then the null space of A − λI for each eigenvector.

Turn any basis into an orthogonal one by subtracting projections one vector at a time, normalize to get an orthonormal basis, and package the result as A = QR.

When Ax = b has no solution, find the x that comes closest: solve the normal equations AᵀAx = Aᵀb, and use them to fit a line to data.

Build new vectors by scaling and adding old ones, and describe everything reachable that way: the span of a set, a line, a plane or all of ℝⁿ through the origin.

A set of vectors is independent when the only combination giving zero is the trivial one. Test it by row reduction, and read independence from the pivot columns.

A function on vectors is linear when it respects sums and scalar multiples. Every such function is multiplication by a matrix, built column by column from e₁ to eₙ.

Read Ax as a linear combination of the columns of A, multiply matrices of any compatible size, and see the product AB as doing B first and then A.

With an orthogonal basis, coordinates come from dot products alone. Project a vector onto a line or subspace, and see why the projection is the closest point.

Solve any linear system by row reducing its augmented matrix: three row operations, echelon form, pivots, and the reduced form that reads off the answer.

Read the shape of a solution set from the pivots: none, exactly one, or infinitely many, written in parametric vector form as a particular solution plus directions.

Find A⁻¹ by row reducing [A | I], see why that works, use the inverse to solve Ax = b, and meet the Invertible Matrix Theorem, which ties the course together.

Extend vectors from the plane to lists of n numbers: componentwise addition and scaling, length and the dot product in ℝⁿ, and the standard basis vectors.

Find the rate of change of f in any direction as a dot product with the gradient, see why the gradient points uphill, and use it as a normal to level curves and surfaces.

Integrate over disks, rings and wedges with dA = r dr dθ, see where the extra factor r comes from, and use it to evaluate the Gaussian integral.

Integrate over regions bounded by curves: set up type I and type II limits, compute areas and volumes, and reverse the order of integration to unlock hard integrals.

Find the volume under a surface z = f(x, y) with Riemann sums of boxes, then compute double integrals exactly as iterated integrals using Fubini's theorem.

Functions with two or three inputs: their domains, their graphs as surfaces, and level curves, which draw a surface on flat paper the way a topographic map does.

Green's theorem turns a line integral around a closed curve into a double integral over the region inside. Use it to compute circulation and area.

Optimize f(x, y) along a constraint curve g(x, y) = k by solving ∇f = λ∇g, and see why the best point is where a level curve of f touches the constraint.

Integrate along a curve: find the mass of a wire with ∫f ds, and the work a force field does along a path with ∫F · dr, using a parametrization of the curve.

Describe a line in space with a point and a direction, a plane with a point and a normal vector, and find intersections, angles and distances between them.

Find the critical points of f(x, y), classify them with the second derivative test, and find absolute extremes on a closed region by checking its boundary too.

Differentiate a function of several variables in one variable at a time, holding the rest fixed. Read partials as slopes of slices, and compute second partials.

Build the plane that touches a surface z = f(x, y) at a point from the two partial derivatives, use it to approximate f nearby, and estimate errors with differentials.

Multiply two vectors in space to get a third, perpendicular to both, whose length is the area of the parallelogram they span. Includes the triple product and torque.

For a gradient field, a line integral is the change in the potential between the endpoints. Test whether a field is conservative and find its potential.

Differentiate a function of several variables along a path, or through a change of variables, by adding one term per route, and differentiate implicit curves.

Locate points in space with three coordinates, find distances and midpoints, write the equation of a sphere, and read simple equations in x, y and z as surfaces.

Integrate a function over a solid region in space: set up limits from the inside out, find volumes and masses, and locate the center of mass of a solid.

Describe cylinders, cones and balls with cylindrical and spherical coordinates, and integrate over them using dV = r dz dr dθ and dV = ρ² sin φ dρ dφ dθ.

A vector field attaches an arrow to every point: wind, flowing water, gravity. Sketch fields, compute gradient fields and follow flow lines.