Calculus library
Choose the topic.
Then choose the kind of help.
Move through calculus in a coherent order, or jump directly to the method, concept, decision, or exact answer you need.
30full resources across
six connected topics
six connected topics
01Direct answer
The result first, followed by the reasoning and a clean verification.
02Method guide
How to recognize the method, run it, and know when it is the wrong choice.
03Concept explainer
What the idea means, why its conditions matter, and where it connects.
04Decision guide
A practical comparison that turns a vague choice into a repeatable test.
Topic 1
Limits & Continuity
How functions behave near a point, when substitution is legal, and what continuity actually promises.
01Direct answerWhy is the limit of sin x over x equal to 1?A foundational limit whose real proof comes from geometry, not from plugging in zero and hoping.02Method guideHow to evaluate an indeterminate limitA decision process for 0/0 and ∞/∞ that starts with algebra before reaching for a theorem.03Concept explainerWhat continuity at a point really requiresThree conditions, one precise promise: nearby inputs produce nearby outputs without a break at the point.04Method guideHow and when to use the Squeeze TheoremTrap a difficult function between two easier functions that are forced to meet at the same limit.05Concept explainerInfinite limits and vertical asymptotesInfinity describes unbounded behavior, not a number a function eventually reaches.
Topic 2
Derivatives
Rules, structure, and the meaning of instantaneous change—from the chain rule to implicit curves.
01Direct answerWhat is the derivative of x to the x?The base and exponent both vary, so neither the ordinary power rule nor the simple exponential rule works alone.02Method guideThe Chain Rule as a structure-reading skillDifferentiate the outside function, keep the inside intact, then multiply by the inside rate.03Method guideImplicit differentiation without losing dy/dxDifferentiate an equation whose y-values are not isolated, treating y as a function of x every time it appears.04Method guideWhen logarithmic differentiation is the clean moveUse logarithms to untangle variable exponents and products or quotients with many factors.05Decision guideProduct Rule or Quotient Rule?Choose the derivative rule from the top-level operation—and simplify first when algebra can remove the choice entirely.
Topic 3
Applications of Derivatives
Use derivatives to model motion, optimize quantities, approximate values, and read the shape of a graph.
01Method guideRelated rates: translate the geometry before differentiatingConnect changing quantities with one equation, differentiate with respect to time, and substitute only after the rates appear.02Method guideA calculus optimization workflow that does not skip the modelTurn a word problem into one objective function, then use calculus and endpoint checks to justify the best feasible value.03Concept explainerLinear approximation: use the tangent line as a local calculatorNear a known input, a smooth function behaves like its tangent line—and the derivative measures the approximation’s sensitivity.04Concept explainerWhat the Mean Value Theorem actually guaranteesSome instantaneous rate must match the average rate—provided continuity and differentiability hold where required.05Method guideCurve sketching from first and second derivativesBuild a graph from domain, intercepts, asymptotes, monotonicity, extrema, concavity, and end behavior—in that order.
Topic 4
Integration Techniques
Choose an antiderivative strategy from the structure of the integrand instead of guessing from a list.
01Method guideu-substitution as the reverse Chain RuleReplace a repeated inner expression and its derivative with one variable so the antiderivative’s structure becomes visible.02Decision guideHow to choose an identity for a trigonometric integralThe parity of sine, cosine, secant, and tangent powers tells you what to save and what to convert.03Method guidePartial fractions: decompose before integratingTurn a proper rational function into simpler fractions whose antiderivatives are logarithmic or inverse-trigonometric.04Decision guideChoosing the right trigonometric substitutionMatch a quadratic radical to a Pythagorean identity so the square root simplifies instead of becoming worse.05Concept explainerImproper integrals are limits, not unusual notationReplace infinite bounds or unbounded integrands with limits before evaluating, then decide whether the result converges.
Topic 5
Applications of Integration
Translate geometry and physical accumulation into integrals with units, bounds, and meaning intact.
01Decision guideWasher method or shell method?Choose slices by geometry: perpendicular slices create washers, parallel slices create cylindrical shells.02Method guideArea between curves without guessing top and bottomFind intersections, choose a slice direction, and integrate the positive geometric difference across each interval.03Concept explainerWhere the arc-length formula comes fromApproximate a smooth curve by tiny line segments, then let the partition refine until the polygonal lengths converge.04Concept explainerAverage value of a function on an intervalTotal accumulated output divided by interval length—the continuous counterpart of an arithmetic mean.05Method guideSetting up work and fluid-force integralsSlice the changing force into small contributions, express every dimension in one variable, and integrate with units attached.
Topic 6
Sequences & Series
Decide convergence, work with power series, and understand what finite approximations can guarantee.
01Direct answerWhy does the harmonic series diverge?Its terms approach zero, but they do so too slowly for the accumulated sum to settle.02Concept explainerGeometric series: convergence, sum, and structureA constant ratio makes an infinite sum exactly solvable when repeated scaling shrinks toward zero.03Decision guideHow to choose a convergence testMatch the series’ structure to a test instead of cycling through tests in chapter order.04Method guideFinding a power series interval of convergenceFind the radius with the Ratio or Root Test, then test both endpoints separately because the general test goes silent there.05Concept explainerTaylor remainder: how accurate is the polynomial?A Taylor polynomial is useful only with a way to control what was left out.