Calculus
Every lesson explains the concept, works through examples, and gives you practice problems with full solutions. Work through them in order, or jump to the topic you need.
52 lessons
- What Is a Limit? One-Sided Limits and Limits from GraphsThe 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.
- Limit Laws and Evaluating Limits AlgebraicallyEvaluate 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.
- Limits at Infinity, Infinite Limits and AsymptotesFind limits as x grows without bound and limits that blow up: horizontal asymptotes from leading terms, vertical asymptotes from infinite limits, and growth rates.
- Continuity and the Intermediate Value TheoremWhat 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.
- The Derivative: Slope of the Tangent Line and Rate of ChangeThe 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.
- Basic Derivative Rules: Power, Sum, Sine, Cosine and e^xDifferentiate 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.
- The Product and Quotient RulesDifferentiate 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.
- The Chain Rule: Derivatives of Composite FunctionsDifferentiate 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.
- Implicit DifferentiationFind 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.
- Derivatives of Inverse Functions, Logarithms and Inverse TrigThe 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.
- Rates of Change in Context: Motion, Velocity and AccelerationRead 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.
- Related RatesFind 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.
- Linear Approximation and DifferentialsEstimate 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.
- L'Hôpital's Rule and Indeterminate FormsEvaluate 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.
- The Mean Value Theorem and the Extreme Value TheoremRolle's theorem, the Mean Value Theorem and the Extreme Value Theorem: what each guarantees, why the hypotheses matter, and what follows from them.
- Increasing, Decreasing and the First Derivative TestUse 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.
- Absolute Maximum and Minimum ValuesFind 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.
- Concavity, Inflection Points and the Second Derivative TestUse 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.
- Curve Sketching: Connecting f, f′ and f″Sketch a graph from its derivatives: intercepts, asymptotes, increasing and decreasing, extrema, concavity and inflection points — and read f from a graph of f′.
- Optimization: Maximum and Minimum ProblemsSolve 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.
- Riemann Sums and the Accumulation of ChangeApproximate accumulated change and area: left, right and midpoint Riemann sums, trapezoidal sums, sums from tables, sigma notation, and over- and underestimates.
- The Definite Integral: Limits of Riemann Sums and Signed AreaThe 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 Fundamental Theorem of CalculusThe link between derivatives and integrals: accumulation functions, the derivative of an integral (Part 1), and evaluating integrals with antiderivatives (Part 2).
- Antiderivatives and Indefinite IntegralsFind antiderivatives: the family F(x) + C, the basic integration rules as derivative rules run backward, ln|x|, and initial value problems.
- Integration by Substitution (u-Substitution)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.
- Integrating with Long Division and Completing the SquareRewrite an integrand before integrating: divide when a fraction's numerator is too big, and complete the square to reach the arctangent.
- Differential Equations and Slope FieldsWhat a differential equation is, how to check a solution, general and particular solutions, and how a slope field shows every solution at once.
- Separable Differential Equations and Exponential ModelsSolve 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.
- Average Value of a FunctionFind 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.
- Motion and Net Change: Integrals in ContextIntegrate rates in context: position from velocity, displacement versus total distance, and amounts that change with a rate in and a rate out.
- Area Between CurvesFind the area between two curves: top minus bottom with vertical slices, right minus left with horizontal slices, and splitting where the curves cross.
- Volumes with Known Cross SectionsFind volumes by slicing: integrate the area of a cross section, for solids whose slices are squares, triangles, rectangles or semicircles on a base region.
- Volumes of Revolution: Disks and WashersFind 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.
- Volumes by Cylindrical ShellsFind volumes of revolution with the shell method, V = 2π∫(radius)(height) dx, and choose between shells and washers for a region and an axis.
- Arc LengthFind 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.
- Integration by PartsIntegrate products with ∫u dv = uv − ∫v du, the product rule run backward: choosing u and dv, using parts twice, and definite integrals.
- Integration by Partial FractionsIntegrate rational functions by splitting them into simpler fractions: distinct linear factors, finding the constants, repeated factors, and dividing first.
- Improper IntegralsIntegrals over infinite intervals and of unbounded functions, defined as limits: when they converge, when they diverge, and the p-integrals.
- Euler's MethodApproximate 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.
- Logistic GrowthThe 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.
- Calculus with Parametric EquationsSlopes, 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.
- Vector-Valued Functions and Motion in the PlaneDescribe motion in the plane with a position vector: velocity, speed and acceleration as derivatives, and displacement and distance as integrals.
- Calculus with Polar Curves: Slopes and TangentsFind 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.
- Area in Polar CoordinatesFind areas bounded by polar curves with A = ½∫r² dθ: one curve, the region between two curves, and finding the limits where curves meet.
- Sequences, Series and Geometric SeriesWhat it means for an infinite sum to converge: sequences and their limits, partial sums, geometric series, and the nth-term test for divergence.
- The Integral Test and p-SeriesDecide whether a series converges by comparing it with an integral: the integral test, the harmonic series, p-series, and bounding the remainder.
- The Comparison Tests for SeriesJudge a series by comparing it with one you know: the direct comparison test, the limit comparison test, and how to choose a benchmark series.
- Alternating Series and Absolute ConvergenceThe alternating series test, the error bound for stopping early, and the difference between absolute and conditional convergence.
- The Ratio and Root TestsTest 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.
- Taylor Polynomials and the Lagrange Error BoundApproximate a function near a point by a polynomial that matches its derivatives there, and bound the error with the Lagrange error bound.
- Power Series: Radius and Interval of ConvergenceSeries 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.
- Taylor and Maclaurin SeriesRepresent functions as power series: the Taylor series formula, the standard Maclaurin series, and new series by substitution, differentiation and integration.