# CalcLearn

> CalcLearn (https://www.calclearn.org) is a free interactive AP Calculus study platform covering AP Calculus AB and BC. Solvers compute answers; CalcLearn teaches the method: which integration technique to use, how to set up related rates and optimization word problems, which series convergence test applies, and what a derivative actually is, through draggable visualizers and step-teaching calculators. Everything is free with no account required. From the maker of EconLearn (econlearn.org).

## Who writes this, and how it is checked

Author: Jude Wallis, a high school student, who writes and builds every page. Author page: https://www.calclearn.org/jude-wallis

Verified recognition (each with its primary source):
- Economics Olympiad, 2nd internationally, awarded to Jude Wallis. Source: https://econolympiad.org
- "What Building My First Product in High School Taught Me About Customers", written by Jude Wallis for YFS Magazine, 2026-07-22. Source: https://yfsmagazine.com/2026/07/22/building-first-product-customers/
- EconLearn, the sibling site (not CalcLearn), is listed in the Colorado Department of Education's economics resource bank alongside Khan Academy and EconEdLink. Source: https://ed.cde.state.co.us/cosocialstudies/economicresources

Editorial process:
- Every derivative, integral, and limit in a worked example is re-derived independently before publication, without reference to the stated answer.
- Curriculum claims come from the official College Board Course and Exam Description, read directly. Unit weightings on this site are the current bands, which differ from older figures circulating elsewhere.
- The problem banks behind the calculators are cross-checked numerically by an automated test suite; a wrong answer fails the build.
- Every mathematical expression is rendered at build time with strict error checking, so broken notation fails the build rather than the page.

Citation: CalcLearn content may be quoted with attribution to CalcLearn (https://www.calclearn.org). Answer-first lines at the top of each page are written as plain text specifically so they can be quoted verbatim.

## What CalcLearn Is

CalcLearn teaches AP Calculus through interactive visualizers (drag a point and watch the derivative trace out, slide n and watch Riemann sums converge) and teaching calculators that show the method and the why, not just the final answer. Content follows the official College Board Course and Exam Description for AP Calculus AB and BC.

Full text of every page, in one file: https://www.calclearn.org/llms-full.txt

## Interactive Visualizers (5)
- Tangent Line Tracer: Watch the Derivative Appear: https://www.calclearn.org/interactives/tangent-line-tracer
- Riemann Sum Slider: Drag n, Watch the Error Vanish: https://www.calclearn.org/interactives/riemann-sum-slider
- Secant to Tangent: The Limit Definition, Made Physical: https://www.calclearn.org/interactives/secant-to-tangent
- Solid of Revolution Builder: Disk, Washer, and Shell: https://www.calclearn.org/interactives/solid-of-revolution-builder
- Related Rates Scene: Sliding Ladder and Cone Tank: https://www.calclearn.org/interactives/related-rates-scene

## Teaching Calculators (12)
- Riemann Sum Visualizer: Rectangles, Error, and Rules: https://www.calclearn.org/calculators/riemann-sum-visualizer
- Chain Rule Step Trainer: You Pick the Decomposition: https://www.calclearn.org/calculators/chain-rule-trainer
- U-Substitution Helper: Pick u, See Why It Works: https://www.calclearn.org/calculators/u-substitution-helper
- Series Convergence Test Chooser: Which Test and Why (BC): https://www.calclearn.org/calculators/convergence-test-chooser
- Limit Solver (Teaching Mode): Which Method and Why: https://www.calclearn.org/calculators/limit-method-chooser
- Definite Integral Step Tool: FTC, Shown Honestly: https://www.calclearn.org/calculators/ftc-step-tool
- Tangent and Normal Line Finder (Teaching Mode): https://www.calclearn.org/calculators/tangent-line-calculator
- Taylor Series Builder: Coefficients from Derivatives: https://www.calclearn.org/calculators/taylor-series-builder
- Derivative Practice Checker: Any Correct Form Passes: https://www.calclearn.org/calculators/derivative-practice-checker
- Related Rates Setup Builder: The Five-Step Method: https://www.calclearn.org/calculators/related-rates-setup-builder
- Optimization Setup Builder: Objective vs Constraint: https://www.calclearn.org/calculators/optimization-setup-builder
- Interval of Convergence Finder: Ratio Test and Endpoints: https://www.calclearn.org/calculators/interval-of-convergence-finder

## Study Guides (27)
- Chain Rule: Spot the Composition, Then Differentiate: https://www.calclearn.org/guides/chain-rule
- U-Substitution: How to Choose u and Integrate It: https://www.calclearn.org/guides/u-substitution
- Integration by Parts: LIATE, When to Use It, Examples: https://www.calclearn.org/guides/integration-by-parts
- U-Sub vs Integration by Parts vs Partial Fractions: https://www.calclearn.org/guides/which-integration-technique
- Related Rates: The 5-Step Method for Any Word Problem: https://www.calclearn.org/guides/related-rates
- Optimization Word Problems: Setup, Constraint, Solve: https://www.calclearn.org/guides/optimization-problems
- Which Series Convergence Test Should I Use?: https://www.calclearn.org/guides/which-convergence-test
- The Ratio Test: How It Works and When to Use It: https://www.calclearn.org/guides/ratio-test
- How to Find Limits: Every Method, in Order: https://www.calclearn.org/guides/how-to-find-limits
- L'Hopital's Rule: When You Can and Cannot Use It: https://www.calclearn.org/guides/lhopitals-rule
- Implicit Differentiation: Finding dy/dx Step by Step: https://www.calclearn.org/guides/implicit-differentiation
- The Fundamental Theorem of Calculus: Part 1 vs Part 2: https://www.calclearn.org/guides/fundamental-theorem-of-calculus
- Riemann Sums: Left, Right, Midpoint, and Trapezoidal: https://www.calclearn.org/guides/riemann-sums
- Disk vs Washer vs Shell: Choosing the Volume Method: https://www.calclearn.org/guides/disk-washer-shell
- Area Between Two Curves: Setup, dx vs dy, Examples: https://www.calclearn.org/guides/area-between-curves
- Product Rule vs Quotient Rule: How and When to Use Each: https://www.calclearn.org/guides/product-quotient-rule
- Derivatives of Trig Functions: Table and Memory Tricks: https://www.calclearn.org/guides/trig-derivatives
- Derivative of ln x, e^x, and a^x: Answers and Proofs: https://www.calclearn.org/guides/derivatives-of-exponentials-and-logs
- Taylor and Maclaurin Series: How to Build Them: https://www.calclearn.org/guides/taylor-maclaurin-series
- Separable Differential Equations: Step-by-Step Method: https://www.calclearn.org/guides/separable-differential-equations
- Slope Fields and Euler's Method Explained: https://www.calclearn.org/guides/slope-fields-eulers-method
- Parametric and Polar Calculus: Derivatives and Area: https://www.calclearn.org/guides/parametric-polar-calculus
- Continuity and the Three Types of Discontinuity: https://www.calclearn.org/guides/continuity-and-discontinuities
- How to Study for AP Calculus AB: Unit-by-Unit Plan: https://www.calclearn.org/guides/how-to-study-ap-calculus-ab
- AP Calculus AB vs BC: Difficulty, Units, Which to Take: https://www.calclearn.org/guides/ap-calculus-ab-vs-bc
- AP Calculus Exam Format: Sections, Timing, and Scoring: https://www.calclearn.org/guides/ap-calculus-exam-format
- AP Calculus Calculator Policy: What Is Allowed and When: https://www.calclearn.org/guides/ap-calculus-calculator-policy

## Course Units (10)
- AP Calculus AB: https://www.calclearn.org/ap-calculus-ab
- AP Calculus BC: https://www.calclearn.org/ap-calculus-bc
- Unit 1: https://www.calclearn.org/units/unit-1-limits-and-continuity
- Unit 2: https://www.calclearn.org/units/unit-2-differentiation-definition
- Unit 3: https://www.calclearn.org/units/unit-3-composite-implicit-inverse
- Unit 4: https://www.calclearn.org/units/unit-4-contextual-applications
- Unit 5: https://www.calclearn.org/units/unit-5-analytical-applications
- Unit 6: https://www.calclearn.org/units/unit-6-integration-and-accumulation
- Unit 7: https://www.calclearn.org/units/unit-7-differential-equations
- Unit 8: https://www.calclearn.org/units/unit-8-applications-of-integration
- Unit 9: https://www.calclearn.org/units/unit-9-parametric-polar-vector
- Unit 10: https://www.calclearn.org/units/unit-10-infinite-sequences-and-series

## Reference Tables (6)
- Derivative Rules: The Complete Table: https://www.calclearn.org/reference/derivative-rules
- Common Integrals: The Complete Table: https://www.calclearn.org/reference/common-integrals
- Limit Laws and Indeterminate Forms: https://www.calclearn.org/reference/limit-laws-and-indeterminate-forms
- Series Convergence Tests: The Complete Table (BC): https://www.calclearn.org/reference/convergence-tests
- Common Maclaurin Series Table (BC): https://www.calclearn.org/reference/common-maclaurin-series
- Trig Identities You Actually Need for Calculus: https://www.calclearn.org/reference/trig-identities-for-calculus

## Common Derivatives, Answer First (12)
Each page states the derivative in plain text up front, then proves it and lists the mistakes students actually make.
- Derivative of ln x: Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/ln-x
- Derivative of e^x: Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/e-x
- Derivative of sin x: Answer, Proof, Mistakes: https://www.calclearn.org/derivative-of/sin-x
- Derivative of cos x: Answer, Proof, Mistakes: https://www.calclearn.org/derivative-of/cos-x
- Derivative of tan x: Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/tan-x
- Derivative of sec x: Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/sec-x
- Derivative of sqrt(x): Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/sqrt-x
- Derivative of 1/x: Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/1-over-x
- Derivative of a^x: Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/a-x
- Derivative of x^x: Answer, Proof, and Mistakes: https://www.calclearn.org/derivative-of/x-x
- Derivative of arctan x: Answer, Proof, Mistakes: https://www.calclearn.org/derivative-of/arctan-x
- Derivative of arcsin x: Answer, Proof, Mistakes: https://www.calclearn.org/derivative-of/arcsin-x

## Glossary (110)
One page per term. Each states the definition in plain text up front, so it can be quoted directly.
- Absolute convergence: A series converges absolutely when the series formed by taking the absolute value of every term still converges. Absolute convergence is stronger than ordinary convergence and always implies it. https://www.calclearn.org/glossary/absolute-convergence
- Acceleration: Acceleration is the derivative of velocity with respect to time, and therefore the second derivative of position. Comparing its sign with the sign of velocity tells you whether the object is speeding up or slowing down. https://www.calclearn.org/glossary/acceleration
- Accumulation function: An accumulation function is a definite integral whose upper bound is the variable, so it defines a new function measuring how much has accumulated from a fixed starting point. Its derivative is the integrand evaluated at the upper bound. https://www.calclearn.org/glossary/accumulation-function
- Alternating series error bound: For an alternating series that satisfies the alternating series test, the error in using a partial sum is less than the absolute value of the first omitted term. It is the simplest error bound in the course. https://www.calclearn.org/glossary/alternating-series-error-bound
- Alternating series test: The alternating series test says that a series whose terms alternate in sign converges if the absolute values of its terms decrease and approach zero. Both conditions are needed, and together they are enough. https://www.calclearn.org/glossary/alternating-series-test
- Antiderivative: An antiderivative of a function is any function whose derivative equals it. Because the derivative of a constant is zero, a function has infinitely many antiderivatives, all differing from one another by a constant. https://www.calclearn.org/glossary/antiderivative
- Arc length: Arc length is the distance measured along a curve rather than straight across. It comes from adding up infinitesimal hypotenuses, which produces an integral of the square root of one plus the square of the derivative. https://www.calclearn.org/glossary/arc-length
- Average rate of change: The average rate of change of a function over an interval is the change in output divided by the change in input, which is the slope of the secant line joining the endpoints. It needs no calculus, unlike the instantaneous rate of change. https://www.calclearn.org/glossary/average-rate-of-change
- Average value of a function: The average value of a function on an interval is the definite integral over that interval divided by the length of the interval. It is the constant height a rectangle would need to have the same area as the region under the curve. https://www.calclearn.org/glossary/average-value-of-a-function
- Candidates test: The candidates test finds absolute extrema on a closed interval by evaluating the function at every critical point and at both endpoints, then comparing the resulting values. The largest is the absolute maximum and the smallest is the absolute minimum. https://www.calclearn.org/glossary/candidates-test
- Comparison test: The direct comparison test settles a series of positive terms by bounding it against a series whose behaviour is known. If it is term by term smaller than a convergent series it converges, and if it is larger than a divergent series it diverges. https://www.calclearn.org/glossary/comparison-test
- Composite function: A composite function is one function evaluated inside another, so the output of the inner function becomes the input of the outer one. Spotting a composition is what tells you to use the chain rule. https://www.calclearn.org/glossary/composite-function
- Concavity: Concavity describes which way a graph bends. A graph is concave up where the second derivative is positive and the slope is increasing, and concave down where the second derivative is negative and the slope is decreasing. https://www.calclearn.org/glossary/concavity
- Conditional convergence: A series converges conditionally when it converges as written but the series of its absolute values diverges. The cancellation between positive and negative terms is what makes it work, so the convergence depends on the order of the terms. https://www.calclearn.org/glossary/conditional-convergence
- Constant multiple rule: The constant multiple rule says a constant factor passes straight through a derivative, and the sum rule says the derivative of a sum is the sum of the derivatives. Together they make differentiation linear. https://www.calclearn.org/glossary/constant-multiple-rule
- Constant of integration: The constant of integration is the arbitrary constant added to an indefinite integral. It is required because differentiating any constant gives zero, so a function has infinitely many antiderivatives that differ only by a constant. https://www.calclearn.org/glossary/constant-of-integration
- Continuity: A function is continuous at a point when three things hold: the function is defined there, the limit exists there, and the limit equals the function value. Informally, you can draw the graph through that point without lifting your pencil. https://www.calclearn.org/glossary/continuity
- Convergence: A series converges when its sequence of partial sums approaches a finite limit, and diverges when it does not. Divergence covers both partial sums that grow without bound and ones that oscillate without settling. https://www.calclearn.org/glossary/convergence
- Corner: A corner is a point where a graph changes direction abruptly, with left and right derivatives that are different finite numbers. The function is continuous there but not differentiable, because no single tangent slope exists. https://www.calclearn.org/glossary/corner
- Critical point: A critical point of a function is an interior point of its domain where the derivative is either zero or undefined. Every local maximum and minimum occurs at a critical point, though not every critical point is an extremum. https://www.calclearn.org/glossary/critical-point
- Cross-section: A cross-section is the two-dimensional shape you get by slicing a solid perpendicular to an axis. If you know the area of each cross-section as a function of position, integrating that area function along the axis gives the volume. https://www.calclearn.org/glossary/cross-section
- Cusp: A cusp is a sharp point on a graph where the one-sided slopes grow without bound in opposite directions. The function stays continuous there but is not differentiable, because the tangent slope is not a finite number. https://www.calclearn.org/glossary/cusp
- Definite integral: A definite integral is a number defined as the limit of Riemann sums as the subintervals shrink to zero width. It represents the net signed accumulation of a quantity between two bounds, with area below the axis counting as negative. https://www.calclearn.org/glossary/definite-integral
- Derivative: The derivative of a function at a point is its instantaneous rate of change there, defined as the limit of the difference quotient as the gap shrinks to zero. Geometrically it is the slope of the tangent line at that point. https://www.calclearn.org/glossary/derivative
- Difference quotient: The difference quotient is the slope of the secant line joining two points on a curve, written as the change in output over the change in input. Taking its limit as the gap shrinks to zero produces the derivative. https://www.calclearn.org/glossary/difference-quotient
- Differentiability: A function is differentiable at a point when the limit defining the derivative exists there, which requires the graph to be locally smooth with a single well-defined tangent slope. Differentiability always implies continuity, but continuity does not imply differentiability. https://www.calclearn.org/glossary/differentiability
- Differential: The differential dy is the change in output along the tangent line corresponding to a small change dx in the input. It approximates the true change in the function and is the notation that makes substitution and separable equations work. https://www.calclearn.org/glossary/differential
- Differential equation: A differential equation is an equation that relates an unknown function to its own derivatives. Its solution is a function rather than a number, and there is usually a whole family of them until an initial condition picks one out. https://www.calclearn.org/glossary/differential-equation
- Direct substitution: Direct substitution evaluates a limit by putting the value straight into the function. It is valid exactly when the function is continuous at that point, which makes it the first thing to try on any limit. https://www.calclearn.org/glossary/direct-substitution
- Disk method: The disk method computes the volume of a solid of revolution by slicing it into circular disks perpendicular to the axis of rotation and integrating the area of each. It applies when the region being rotated touches the axis, leaving no hole. https://www.calclearn.org/glossary/disk-method
- Displacement: Displacement is the net change in position over a time interval, computed by integrating velocity. Total distance travelled integrates the absolute value of velocity instead, so it counts backtracking as additional distance. https://www.calclearn.org/glossary/displacement
- Domain: The domain of a function is the set of inputs for which it produces a value. Domain restrictions determine where a function can be continuous or differentiable, and they are what make some answers to calculus problems invalid. https://www.calclearn.org/glossary/domain
- Epsilon-delta definition: The epsilon-delta definition states that a limit equals L if, for every tolerance around L, there is a distance around the input point that keeps all outputs within that tolerance. It is the precise version of the informal idea of approaching a value. https://www.calclearn.org/glossary/epsilon-delta-definition
- Euler's method: Euler's method approximates the solution of a differential equation by starting at a known point and repeatedly stepping along the tangent line for a small increment. Each step uses the differential equation to compute the slope at the current point. https://www.calclearn.org/glossary/eulers-method
- Exponential growth: Exponential growth describes a quantity whose rate of change is proportional to its current amount. The differential equation says the derivative equals a constant times the quantity, and its solution is a constant multiple of e raised to that constant times time. https://www.calclearn.org/glossary/exponential-growth
- Extreme Value Theorem: The Extreme Value Theorem says that a function continuous on a closed, bounded interval must attain both an absolute maximum and an absolute minimum somewhere on that interval. It guarantees the extrema exist without saying where they are. https://www.calclearn.org/glossary/extreme-value-theorem
- Extremum: An extremum is a maximum or minimum value of a function. A local extremum is largest or smallest compared only to nearby points, while an absolute extremum is largest or smallest across the entire domain or interval under consideration. https://www.calclearn.org/glossary/extremum
- First derivative test: The first derivative test classifies a critical point by examining the sign of the derivative on both sides. A change from positive to negative gives a local maximum, negative to positive gives a local minimum, and no sign change means neither. https://www.calclearn.org/glossary/first-derivative-test
- Fundamental Theorem of Calculus Part 1: Part 1 of the Fundamental Theorem of Calculus says that differentiating an integral with a variable upper bound returns the integrand evaluated at that bound. It is the statement that differentiation and integration are inverse operations. https://www.calclearn.org/glossary/ftc-part-1
- Fundamental Theorem of Calculus Part 2: Part 2 of the Fundamental Theorem of Calculus says a definite integral equals any antiderivative evaluated at the upper bound minus the same antiderivative at the lower bound. It is what makes definite integrals computable without Riemann sums. https://www.calclearn.org/glossary/ftc-part-2
- General solution: The general solution of a differential equation is the entire family of functions that satisfy it, expressed with an arbitrary constant. Every specific solution is obtained by choosing a value for that constant. https://www.calclearn.org/glossary/general-solution
- Geometric series: A geometric series is one where each term is a fixed multiple of the previous term. It converges exactly when the absolute value of that common ratio is less than one, and then its sum is the first term divided by one minus the ratio. https://www.calclearn.org/glossary/geometric-series
- Harmonic series: The harmonic series is the sum of the reciprocals of the positive integers. Its terms shrink to zero, yet the series diverges, which makes it the standard proof that shrinking terms are not enough for convergence. https://www.calclearn.org/glossary/harmonic-series
- Higher-order derivative: A higher-order derivative is what you get by differentiating repeatedly. The second derivative is the derivative of the derivative and measures how the rate of change is itself changing, which is what determines concavity and acceleration. https://www.calclearn.org/glossary/higher-order-derivative
- Horizontal asymptote: A horizontal asymptote is a horizontal line that a graph approaches as the input grows without bound in either direction. A function has one exactly when its limit at infinity or at negative infinity is a finite number. https://www.calclearn.org/glossary/horizontal-asymptote
- Implicit function: An implicit function is one defined by an equation relating the variables without the output isolated, such as the equation of a circle. Its derivative is found by differentiating both sides and solving for the derivative rather than by rearranging first. https://www.calclearn.org/glossary/implicit-function
- Improper integral: An improper integral is one with an infinite bound of integration or an integrand that grows without bound somewhere on the interval. It is defined as a limit of ordinary definite integrals, and it converges if that limit is finite and diverges otherwise. https://www.calclearn.org/glossary/improper-integral
- Increasing function: A function is increasing on an interval where its derivative is positive and decreasing where its derivative is negative. Increasing behaviour is a property of an interval, not of a single point. https://www.calclearn.org/glossary/increasing-function
- Indefinite integral: An indefinite integral is the collection of all antiderivatives of a function, written with an integral sign and no limits. Its result is a family of functions plus a constant of integration, which is what distinguishes it from a definite integral. https://www.calclearn.org/glossary/indefinite-integral
- Indeterminate form: An indeterminate form is a limit expression whose value cannot be determined from its form alone, such as zero over zero or infinity minus infinity. It is not an answer but a signal that you need algebra or L'Hopital's rule to resolve the limit. https://www.calclearn.org/glossary/indeterminate-form
- Infinite discontinuity: An infinite discontinuity occurs where at least one one-sided limit is infinite, so the function grows without bound near the point. It always corresponds to a vertical asymptote and can never be removed by redefining a value. https://www.calclearn.org/glossary/infinite-discontinuity
- Infinite limit: An infinite limit means the function grows without bound as the input approaches a point, written as equal to positive or negative infinity. Because infinity is not a number, an infinite limit is a description of how the limit fails to exist, not a value. https://www.calclearn.org/glossary/infinite-limit
- Inflection point: An inflection point is a point on a graph where the concavity changes from up to down or from down to up. Finding one requires the second derivative to change sign there, which is a stronger condition than simply equalling zero. https://www.calclearn.org/glossary/inflection-point
- Integral test: The integral test says that for a positive, decreasing, continuous function, the series of its values and the improper integral of the function either both converge or both diverge. It is what proves the p-series rule. https://www.calclearn.org/glossary/integral-test
- Integrand: The integrand is the function being integrated, written between the integral sign and the differential. Identifying it clearly is what tells you which integration technique applies. https://www.calclearn.org/glossary/integrand
- Intermediate Value Theorem: The Intermediate Value Theorem says that if a function is continuous on a closed interval, it attains every value between the two endpoint values somewhere on that interval. It is the standard tool for proving a solution exists without finding it. https://www.calclearn.org/glossary/intermediate-value-theorem
- Interval of convergence: The interval of convergence is the complete set of inputs for which a power series converges. You find the radius with the ratio test, then test each endpoint separately, because the ratio test is inconclusive exactly there. https://www.calclearn.org/glossary/interval-of-convergence
- Inverse function derivative: The derivative of an inverse function at a point equals the reciprocal of the original function's derivative, evaluated at the matching point. You never need a formula for the inverse itself, only the point correspondence. https://www.calclearn.org/glossary/inverse-function-derivative
- Jump discontinuity: A jump discontinuity occurs at a point where both one-sided limits exist but disagree, so the graph steps abruptly from one value to another. Because the one-sided limits differ, the two-sided limit does not exist and no redefinition can repair the break. https://www.calclearn.org/glossary/jump-discontinuity
- Lagrange error bound: The Lagrange error bound limits how far a Taylor polynomial can be from the function it approximates. It uses the maximum size of the next derivative on the interval between the centre and the point of interest. https://www.calclearn.org/glossary/lagrange-error-bound
- Leibniz notation: Leibniz notation writes a derivative as dy over dx, which names both the changing quantity and the variable it changes with respect to. That explicitness makes it the standard notation for related rates, implicit differentiation, and any problem with several variables. https://www.calclearn.org/glossary/leibniz-notation
- Limit: A limit is the single value a function approaches as its input approaches a given number. The limit describes where the function is heading, not where it lands, so a limit can exist at a point where the function is undefined. https://www.calclearn.org/glossary/limit
- Limit at infinity: A limit at infinity is the value a function approaches as its input grows without bound in the positive or negative direction. It describes the end behavior of the graph, and a finite limit at infinity is exactly what produces a horizontal asymptote. https://www.calclearn.org/glossary/limit-at-infinity
- Limit comparison test: The limit comparison test compares two series of positive terms by taking the limit of their ratio. If that limit is finite and positive, both series converge or both diverge together. https://www.calclearn.org/glossary/limit-comparison-test
- Limit laws: The limit laws let you distribute a limit across sums, differences, products, quotients, and powers. Each law requires the individual limits to exist, and the quotient law additionally requires the denominator's limit to be nonzero. https://www.calclearn.org/glossary/limit-laws
- Linearization: Linearization approximates a function near a point by using its tangent line there. Because a smooth curve and its tangent line are nearly identical close to the point of tangency, the line gives a good estimate for inputs near that point. https://www.calclearn.org/glossary/linearization
- Logarithmic differentiation: Logarithmic differentiation is a technique where you take the natural logarithm of both sides of an equation before differentiating implicitly. It converts products into sums and exponents into coefficients, which is what makes a variable base raised to a variable power differentiable. https://www.calclearn.org/glossary/logarithmic-differentiation
- Logistic growth: Logistic growth models a quantity that grows nearly exponentially when small but levels off as it approaches a maximum called the carrying capacity. Its growth rate is fastest when the quantity is exactly half the carrying capacity. https://www.calclearn.org/glossary/logistic-growth
- Maclaurin series: A Maclaurin series is a Taylor series centred at zero. It is not a different object, just the most common special case, and the standard ones for the exponential, sine, and cosine functions are worth knowing by heart. https://www.calclearn.org/glossary/maclaurin-series
- Mean Value Theorem: The Mean Value Theorem says that if a function is continuous on a closed interval and differentiable on the open interval, then somewhere inside there is a point where the instantaneous rate of change equals the average rate of change across the whole interval. https://www.calclearn.org/glossary/mean-value-theorem
- Net change theorem: The net change theorem says that integrating a rate of change over an interval gives the net change in the underlying quantity. It is the interpretation of the Fundamental Theorem of Calculus that applied problems actually use. https://www.calclearn.org/glossary/net-change-theorem
- Normal line: The normal line at a point on a curve is the line through that point perpendicular to the tangent line. Its slope is the negative reciprocal of the derivative there, provided the derivative is not zero. https://www.calclearn.org/glossary/normal-line
- nth term test: The nth term test says that if the terms of a series do not approach zero, the series diverges. It can only ever prove divergence, because terms approaching zero says nothing about whether the series converges. https://www.calclearn.org/glossary/nth-term-test
- Objective function: The objective function in an optimization problem is the quantity being maximized or minimized. The constraint is a separate equation relating the variables, used to rewrite the objective in terms of a single variable before differentiating. https://www.calclearn.org/glossary/objective-function
- One-sided limit: A one-sided limit is the value a function approaches as the input comes in from one direction only. The left-hand limit uses inputs below the point and the right-hand limit uses inputs above it. The two-sided limit exists exactly when both agree. https://www.calclearn.org/glossary/one-sided-limit
- p-series: A p-series is the sum of the reciprocals of the positive integers raised to a fixed power p. It converges when p is greater than one and diverges when p is less than or equal to one. https://www.calclearn.org/glossary/p-series
- Parametric equations: Parametric equations define the coordinates of a point separately as functions of a third variable called the parameter, usually time. They describe not just the shape of a curve but the direction and speed with which it is traced. https://www.calclearn.org/glossary/parametric-equations
- Partial fractions: Partial fraction decomposition rewrites a rational function as a sum of simpler fractions with linear denominators. Each piece then integrates to a natural logarithm, which is what makes otherwise impossible rational integrals routine. https://www.calclearn.org/glossary/partial-fractions
- Partial sum: The nth partial sum of a series is the total of its first n terms. The series converges precisely when this sequence of partial sums approaches a finite limit, which is what makes an infinite sum meaningful. https://www.calclearn.org/glossary/partial-sum
- Particular solution: A particular solution is the one member of a differential equation's solution family that passes through a specified point. The initial condition is what determines the arbitrary constant and selects that single curve. https://www.calclearn.org/glossary/particular-solution
- Polar coordinates: Polar coordinates locate a point by its distance from the origin and the angle it makes with the positive horizontal axis. A polar curve expresses that distance as a function of the angle, which makes circles and roses far simpler to describe than in rectangular form. https://www.calclearn.org/glossary/polar-coordinates
- Power rule: The power rule says that to differentiate a power of the variable, multiply by the exponent and then reduce the exponent by one. It holds for every real exponent, including negative and fractional ones. https://www.calclearn.org/glossary/power-rule
- Power series: A power series is an infinite sum of constant multiples of powers of the variable minus a fixed centre. It converges for the centre always, and generally on an interval around it whose half-width is the radius of convergence. https://www.calclearn.org/glossary/power-series
- Product rule: The product rule differentiates a product of two functions: multiply the first by the derivative of the second, then add the second times the derivative of the first. The derivative of a product is never the product of the derivatives. https://www.calclearn.org/glossary/product-rule
- Properties of definite integrals: Definite integrals obey a small set of rules: swapping the bounds reverses the sign, equal bounds give zero, constants factor out, sums split apart, and an integral can be broken at any interior point. https://www.calclearn.org/glossary/definite-integral-properties
- Quotient rule: The quotient rule differentiates a fraction: the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the denominator squared. The order matters because of the subtraction. https://www.calclearn.org/glossary/quotient-rule
- Radius of convergence: The radius of convergence is the distance from the centre of a power series to the edge of the region where it converges. It is found by applying the ratio test and solving the resulting inequality for the variable. https://www.calclearn.org/glossary/radius-of-convergence
- Related rate: A related rate is the rate of change of one quantity expressed in terms of the rate of change of another that it depends on. These problems link the quantities with an equation, then differentiate both sides with respect to time. https://www.calclearn.org/glossary/related-rate
- Removable discontinuity: A removable discontinuity is a point where the limit exists but the function either is not defined there or is defined to be a different value. It appears as a hole in the graph, and redefining the single point would make the function continuous. https://www.calclearn.org/glossary/removable-discontinuity
- Riemann sum: A Riemann sum approximates a definite integral by slicing the interval into subintervals and adding up the area of a rectangle on each one. As the number of subintervals grows without bound, the sum approaches the exact value of the integral. https://www.calclearn.org/glossary/riemann-sum
- Rolle's Theorem: Rolle's Theorem says that if a function is continuous on a closed interval, differentiable inside it, and takes the same value at both endpoints, then there is an interior point where the derivative is zero. It is the Mean Value Theorem with equal endpoint values. https://www.calclearn.org/glossary/rolles-theorem
- Root test: The root test takes the limit of the nth root of the absolute value of the terms. A limit less than one means the series converges absolutely, greater than one means it diverges, and exactly one means the test tells you nothing. https://www.calclearn.org/glossary/root-test
- Secant line: A secant line is a straight line through two points on a curve. Its slope equals the average rate of change between those points, and as the two points slide together the secant line approaches the tangent line. https://www.calclearn.org/glossary/secant-line
- Second derivative test: The second derivative test classifies a critical point by evaluating the second derivative there. A negative value means a local maximum, a positive value means a local minimum, and a value of zero makes the test inconclusive. https://www.calclearn.org/glossary/second-derivative-test
- Sequence: A sequence is an ordered list of numbers indexed by the positive integers. A series is the sum of the terms of a sequence, so a sequence is a list while a series is a total. https://www.calclearn.org/glossary/sequence
- Series: A series is the sum of the terms of a sequence, often infinitely many. It converges when the sequence of its partial sums approaches a finite limit, and that limit is defined to be the sum of the series. https://www.calclearn.org/glossary/series
- Shell method: The shell method computes a volume of revolution by slicing the region parallel to the axis of rotation and treating each slice as a thin cylindrical shell. Its integrand is two pi times the radius times the height. https://www.calclearn.org/glossary/shell-method
- Slope field: A slope field is a grid of short line segments, each drawn with the slope that a differential equation assigns to that point. It shows the shape of every solution curve at once without solving the equation. https://www.calclearn.org/glossary/slope-field
- Solid of revolution: A solid of revolution is the three-dimensional shape swept out when a plane region is rotated about a line. Its volume is computed by integrating cross-sectional areas, using the disk method, the washer method, or the shell method. https://www.calclearn.org/glossary/solid-of-revolution
- Speed: Speed is the absolute value of velocity, so it is never negative and carries no direction information. Speed increases exactly when velocity and acceleration have the same sign. https://www.calclearn.org/glossary/speed
- Squeeze theorem: The squeeze theorem says that if a function is trapped between two others near a point, and those two share the same limit there, the trapped function must have that limit too. It is the standard way to evaluate limits that resist algebra, such as x squared times sine of one over x. https://www.calclearn.org/glossary/squeeze-theorem
- Tangent line: The tangent line to a curve at a point is the straight line through that point whose slope equals the derivative there. It is the best linear approximation of the curve near that point. https://www.calclearn.org/glossary/tangent-line
- Taylor polynomial: A Taylor polynomial is the finite piece of a Taylor series obtained by stopping after a chosen degree. It approximates the function near the centre, and higher degree generally means a better approximation over a wider range. https://www.calclearn.org/glossary/taylor-polynomial
- Taylor series: A Taylor series represents a function as an infinite polynomial whose coefficients come from the function's derivatives at a single centre point. Each coefficient is a derivative at the centre divided by the factorial of its order. https://www.calclearn.org/glossary/taylor-series
- Trapezoidal rule: The trapezoidal rule approximates a definite integral by joining consecutive points with straight segments and summing the resulting trapezoids. It equals the average of the left and right Riemann sums and is usually more accurate than either. https://www.calclearn.org/glossary/trapezoidal-rule
- Vector-valued function: A vector-valued function assigns a vector to each value of a parameter, most often describing the position of a particle over time. Differentiating each component separately gives the velocity vector, and differentiating again gives acceleration. https://www.calclearn.org/glossary/vector-valued-function
- Velocity: Velocity is the derivative of position with respect to time. It is a signed quantity, so its sign tells you the direction of motion, and its absolute value is the speed. https://www.calclearn.org/glossary/velocity
- Vertical asymptote: A vertical asymptote is a vertical line that a graph approaches without ever reaching, occurring wherever at least one one-sided limit is infinite. For a rational function, these sit at the zeros of the denominator that remain after all common factors cancel. https://www.calclearn.org/glossary/vertical-asymptote
- Vertical tangent: A vertical tangent occurs at a point where the derivative grows without bound with the same sign from both sides, so the tangent line is vertical. The function remains continuous there but is not differentiable, since infinite slope is not a number. https://www.calclearn.org/glossary/vertical-tangent
- Washer method: The washer method computes the volume of a solid of revolution with a hole through it by subtracting the inner circle's area from the outer circle's area on every slice. It is used whenever the rotated region does not touch the axis of rotation. https://www.calclearn.org/glossary/washer-method

## Which One Do I Use: Side-by-Side Comparisons (41)
The method-choice questions a solver structurally cannot answer. Each states the verdict in plain text up front.
- Continuity vs Differentiability: Differentiability is the stronger condition: every differentiable function is continuous, but plenty of continuous functions are not differentiable. Continuity means no break in the graph; differentiability additionally means no corner, cusp, or vertical tangent. https://www.calclearn.org/vs/continuity-vs-differentiability
- Removable vs Jump vs Infinite Discontinuity: A removable discontinuity is a hole where the limit exists but the value is missing or wrong. A jump has two different one-sided limits. An infinite discontinuity has at least one infinite one-sided limit. Only the removable kind can be repaired by redefining a single point. https://www.calclearn.org/vs/removable-vs-jump-discontinuity
- Average vs Instantaneous Rate of Change: Average rate of change measures over an interval and equals the slope of the secant line joining the endpoints. Instantaneous rate of change measures at a single point and equals the slope of the tangent line, which is the derivative. https://www.calclearn.org/vs/average-vs-instantaneous-rate-of-change
- Product Rule vs Chain Rule: Use the product rule when two functions are multiplied side by side, and the chain rule when one function sits inside another. The test is to ask what you would compute last if you evaluated by hand: a multiplication means product rule, an outer function means chain rule. https://www.calclearn.org/vs/product-rule-vs-chain-rule
- Explicit vs Implicit Differentiation: Differentiate explicitly when the equation is already solved for the output variable. Use implicit differentiation when the variables are mixed together and isolating one would be messy or impossible, as with a circle. https://www.calclearn.org/vs/explicit-vs-implicit-differentiation
- Related Rates vs Optimization: Related rates problems link two quantities changing over time and differentiate with respect to time. Optimization problems find a largest or smallest value and set a derivative equal to zero. Both start from a geometric relationship, but only optimization uses a constraint to eliminate a variable. https://www.calclearn.org/vs/related-rates-vs-optimization
- Local vs Absolute Extrema: A local extremum is the largest or smallest value compared only to nearby points. An absolute extremum is the largest or smallest across the entire interval. Local extrema occur only at critical points; absolute extrema occur at a critical point or at an endpoint. https://www.calclearn.org/vs/local-vs-absolute-extrema
- First vs Second Derivative Test: The first derivative test classifies a critical point by whether the derivative changes sign there, and it never fails. The second derivative test checks the sign of the second derivative at the point, which is faster but says nothing when that value is zero. https://www.calclearn.org/vs/first-vs-second-derivative-test
- Critical Point vs Inflection Point: A critical point is where the first derivative is zero or undefined, and it is where extrema can occur. An inflection point is where the second derivative changes sign, and it is where concavity flips. They answer different questions and often sit at different places. https://www.calclearn.org/vs/critical-point-vs-inflection-point
- Mean Value Theorem vs Intermediate Value Theorem: The Intermediate Value Theorem guarantees that a continuous function attains some output value, which is how you prove a root exists. The Mean Value Theorem guarantees that a differentiable function attains some slope, namely the average rate of change across the interval. https://www.calclearn.org/vs/mvt-vs-ivt
- Mean Value Theorem vs Rolle's Theorem: Rolle's Theorem is the special case of the Mean Value Theorem where the two endpoint values are equal. When they match, the average rate of change is zero, so the guaranteed slope is zero and the tangent line is horizontal. https://www.calclearn.org/vs/mvt-vs-rolles-theorem
- Speed vs Velocity: Velocity is signed, so it tells you both how fast and in which direction. Speed is the absolute value of velocity and is never negative. Speed increases exactly when velocity and acceleration have the same sign, which is not the same as acceleration being positive. https://www.calclearn.org/vs/speed-vs-velocity
- Displacement vs Total Distance Travelled: Displacement is the net change in position, found by integrating velocity, and it can be zero even after a long trip. Total distance travelled integrates the absolute value of velocity, so movement in both directions adds up. https://www.calclearn.org/vs/displacement-vs-distance-travelled
- Definite vs Indefinite Integral: A definite integral has bounds and evaluates to a number. An indefinite integral has no bounds and evaluates to a family of functions plus a constant of integration. One is a value, the other is a function, and that single difference explains when the constant is needed. https://www.calclearn.org/vs/definite-vs-indefinite-integral
- FTC Part 1 vs Part 2: Part 1 differentiates an accumulation function and returns the integrand, which is the statement that differentiation undoes integration. Part 2 evaluates a definite integral as an antiderivative at the top bound minus the same antiderivative at the bottom. https://www.calclearn.org/vs/ftc-part-1-vs-part-2
- U-Substitution vs Integration by Parts: Use substitution when the integrand contains a function and something close to its derivative, since substitution reverses the chain rule. Use integration by parts when the integrand is a product of two unlike types, such as a polynomial times an exponential, since parts reverses the product rule. https://www.calclearn.org/vs/u-substitution-vs-integration-by-parts
- Left vs Right Riemann Sum: Left and right Riemann sums differ only in whether each rectangle takes its height from the left or right endpoint of its subinterval. For an increasing function the left sum underestimates and the right sum overestimates, and for a decreasing function the roles swap. https://www.calclearn.org/vs/left-vs-right-riemann-sum
- Midpoint vs Trapezoidal Rule: The trapezoidal rule joins consecutive points with straight chords, so on a concave up curve it overestimates. The midpoint rule takes each rectangle's height from the centre of its subinterval and errs in the opposite direction, underestimating on a concave up curve. https://www.calclearn.org/vs/midpoint-vs-trapezoidal-rule
- Area vs Net Signed Area: A definite integral computes net signed area, so region below the horizontal axis counts as negative and can cancel region above it. True geometric area integrates the absolute value of the function, so every piece contributes positively. https://www.calclearn.org/vs/area-vs-net-signed-area
- Average Value vs Average Rate of Change: Average value integrates a function over an interval and divides by the interval length, answering what constant height would give the same area. Average rate of change divides the change in output by the change in input, answering how fast the function moved on average. https://www.calclearn.org/vs/average-value-vs-average-rate-of-change
- General vs Particular Solution: The general solution of a differential equation is the whole family of functions that satisfy it, written with an arbitrary constant. A particular solution is the single member of that family passing through a given initial condition. https://www.calclearn.org/vs/general-vs-particular-solution
- Exponential vs Logistic Growth: Exponential growth has a rate proportional to the current amount and increases without any ceiling. Logistic growth multiplies that by a braking factor, so growth slows as the quantity nears a carrying capacity and is fastest at exactly half of it. https://www.calclearn.org/vs/exponential-vs-logistic-growth
- Slope Field vs Euler's Method: A slope field is a picture: short segments showing the slope a differential equation assigns at each point, sketching every solution at once. Euler's method is a numerical procedure that starts from an initial condition and steps along tangent lines to produce actual approximate values. https://www.calclearn.org/vs/slope-field-vs-eulers-method
- Disk vs Washer Method: Use the disk method when the region being rotated touches the axis of rotation, so each slice is a solid circle. Use the washer method when a gap separates the region from the axis, because that gap becomes a hole and each slice becomes a ring. https://www.calclearn.org/vs/disk-vs-washer-method
- Washer vs Shell Method: Washers slice perpendicular to the axis of rotation and integrate in the axis variable. Shells slice parallel to it and integrate in the other variable. Choose whichever lets you avoid rewriting the equation in terms of a variable it is not already solved for. https://www.calclearn.org/vs/washer-vs-shell-method
- Known Cross-Sections vs Solids of Revolution: Both methods integrate cross-sectional area along an axis. A solid of revolution always has circular or ring-shaped slices because it is swept by rotation. A solid with known cross-sections can have squares, semicircles, or triangles instead, and the problem tells you which. https://www.calclearn.org/vs/cross-sections-vs-solids-of-revolution
- Parametric vs Polar Curves: Parametric equations define the coordinates separately as functions of a parameter, usually time. Polar equations define distance from the origin as a function of angle. Polar is really a special case of parametric, which is why the slope formulas match. https://www.calclearn.org/vs/parametric-vs-polar
- Sequence vs Series: A sequence is an ordered list of numbers; a series is the sum of that list. A sequence converges when its terms approach a limit, while a series converges when its partial sums approach a limit, which is a much stronger requirement. https://www.calclearn.org/vs/sequence-vs-series
- Ratio Test vs Root Test: The ratio test compares consecutive terms and is the right choice when factorials or products of consecutive integers appear. The root test takes the nth root and is the right choice when the whole term is raised to the nth power. Both are inconclusive when the limit equals one. https://www.calclearn.org/vs/ratio-test-vs-root-test
- Direct Comparison vs Limit Comparison Test: Direct comparison requires you to prove a term by term inequality against a known series. Limit comparison only requires the two series to grow at comparable rates, judged by the limit of their ratio, which makes it easier to apply when the inequality is awkward. https://www.calclearn.org/vs/comparison-vs-limit-comparison-test
- Absolute vs Conditional Convergence: A series converges absolutely when the series of absolute values also converges. It converges conditionally when it converges as written but the absolute values diverge, meaning the convergence depends entirely on cancellation between positive and negative terms. https://www.calclearn.org/vs/absolute-vs-conditional-convergence
- Geometric Series vs p-Series: A geometric series has a constant ratio between consecutive terms, with the index in the exponent, and converges when the absolute ratio is below one. A p-series has the index in the base raised to a fixed power and converges when that power exceeds one. https://www.calclearn.org/vs/geometric-vs-p-series
- Taylor vs Maclaurin Series: A Maclaurin series is a Taylor series centred at zero. They are not different objects: the Maclaurin case is simply the most common centre, which is why the standard series for the exponential, sine, and cosine functions are all Maclaurin series. https://www.calclearn.org/vs/taylor-vs-maclaurin-series
- Taylor Polynomial vs Taylor Series: A Taylor polynomial is a finite truncation, so it approximates the function and carries an error you can bound. A Taylor series continues forever and, inside its interval of convergence, equals the function exactly. https://www.calclearn.org/vs/taylor-polynomial-vs-taylor-series
- Lagrange vs Alternating Series Error Bound: The alternating series error bound is the absolute value of the first omitted term, and it applies only when the series alternates and passes the alternating series test. The Lagrange error bound works for any Taylor polynomial but requires bounding the next derivative on the interval. https://www.calclearn.org/vs/lagrange-vs-alternating-series-error-bound
- Radius vs Interval of Convergence: The radius of convergence is a single number, the distance from the centre to where convergence stops, found with the ratio test. The interval of convergence is the full set of inputs where the series converges, which requires testing each endpoint separately. https://www.calclearn.org/vs/radius-vs-interval-of-convergence
- Secant Line vs Tangent Line: A secant line passes through two points on a curve and its slope is the average rate of change. A tangent line touches at a single point and its slope is the derivative there. Shrinking the gap between the two points turns the secant into the tangent. https://www.calclearn.org/vs/secant-vs-tangent-line
- Tangent Line vs Normal Line: The tangent line at a point has slope equal to the derivative there. The normal line is perpendicular to it, so its slope is the negative reciprocal of the derivative. Both pass through the same point on the curve. https://www.calclearn.org/vs/tangent-vs-normal-line
- Vertical vs Horizontal Asymptote: A vertical asymptote occurs where a one-sided limit is infinite, and a graph can never cross one. A horizontal asymptote comes from a finite limit as the input grows without bound, describes long-run behaviour only, and can be crossed any number of times. https://www.calclearn.org/vs/vertical-vs-horizontal-asymptote
- Increasing vs Concave Up: Increasing means the first derivative is positive, so the function is rising. Concave up means the second derivative is positive, so the slope itself is rising. The two are completely independent: a function can be increasing while concave down. https://www.calclearn.org/vs/increasing-vs-concave-up
- Power Rule vs Exponential Rule: Use the power rule when the variable is the base and the exponent is a constant, as in x cubed. Use the exponential rule when the variable is the exponent and the base is a constant, as in two to the x. When the variable is in both places, neither rule works and you need logarithmic differentiation. https://www.calclearn.org/vs/power-rule-vs-exponential-rule

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