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Every one of the 1001 pages on CalcLearn, grouped by section, with the date each was last updated. Machine-readable versions: /sitemap.xml for crawlers, /llms.txt and /llms-full.txt for AI engines.
Start here 10
Course units 10
- Unit 1: Limits and Continuity
- Unit 2: Differentiation: Definition and Fundamental Properties
- Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
- Unit 4: Contextual Applications of Differentiation
- Unit 5: Analytical Applications of Differentiation
- Unit 6: Integration and Accumulation of Change
- Unit 7: Differential Equations
- Unit 8: Applications of Integration
- Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- Unit 10: Infinite Sequences and Series
Study guides 37
- Chain Rule: Spot the Composition, Then Differentiate
- U-Substitution: How to Choose u and Integrate It
- Integration by Parts: LIATE, When to Use It, Examples
- U-Sub vs Integration by Parts vs Partial Fractions
- Related Rates: The 5-Step Method for Any Word Problem
- Optimization Word Problems: Setup, Constraint, Solve
- Which Series Convergence Test Should I Use?
- The Ratio Test: How It Works and When to Use It
- How to Find Limits: Every Method, in Order
- L'Hopital's Rule: When You Can and Cannot Use It
- Implicit Differentiation: Finding dy/dx Step by Step
- The Fundamental Theorem of Calculus: Part 1 vs Part 2
- Riemann Sums: Left, Right, Midpoint, and Trapezoidal
- Disk vs Washer vs Shell: Choosing the Volume Method
- Area Between Two Curves: Setup, dx vs dy, Examples
- Product Rule vs Quotient Rule: How and When to Use Each
- Derivatives of Trig Functions: Table and Memory Tricks
- Derivative of ln x, e^x, and a^x: Answers and Proofs
- Taylor and Maclaurin Series: How to Build Them
- Separable Differential Equations: Step-by-Step Method
- Slope Fields and Euler's Method Explained
- Parametric and Polar Calculus: Derivatives and Area
- Continuity and the Three Types of Discontinuity
- How to Study for AP Calculus AB: Unit-by-Unit Plan
- AP Calculus AB vs BC: Difficulty, Units, Which to Take
- AP Calculus Exam Format: Sections, Timing, and Scoring
- AP Calculus Calculator Policy: What Is Allowed and When
- 30 AP Calculus Bell Ringers by Unit
- First Day of AP Calculus: A Complete Plan
- AP Calculus Discussion Questions by Topic
- Riemann Sum Slider: A 40-Minute Classroom Lesson
- AP Calculus Sub Plans With Full Answer Keys
- AP Calculus Review Games That Are Not Just Kahoot
- AP Calculus Project Ideas for High School
- 30 AP Calculus Exit Tickets by Unit
- A 32-Week AP Calculus Pacing Guide
- Tangent Line Tracer: A 40-Minute Classroom Lesson
Teaching calculators 12
- Riemann Sum Visualizer: Rectangles, Error, and Rules
- Chain Rule Step Trainer: You Pick the Decomposition
- U-Substitution Helper: Pick u, See Why It Works
- Series Convergence Test Chooser: Which Test and Why (BC)
- Limit Solver (Teaching Mode): Which Method and Why
- Definite Integral Step Tool: FTC, Shown Honestly
- Tangent and Normal Line Finder (Teaching Mode)
- Taylor Series Builder: Coefficients from Derivatives
- Derivative Practice Checker: Any Correct Form Passes
- Related Rates Setup Builder: The Five-Step Method
- Optimization Setup Builder: Objective vs Constraint
- Interval of Convergence Finder: Ratio Test and Endpoints
Interactive visualizers 5
Widgets to embed 30
- Put a visualizer on a class page
- For teachers
- For assistants
- Embed graphs
- Embed in Google Sites
- Embed in Canvas
- Embed in Schoology
- Embed in Moodle
- Embed in WordPress
- Embed in Notion
- First derivatives pack
- Integrals pack
- Series pack
- Embed Tangent Line Tracer: Watch the Derivative Appear
- Embed Riemann Sum Slider: Drag n, Watch the Error Vanish
- Embed Secant to Tangent: The Limit Definition, Made Physical
- Embed Solid of Revolution Builder: Disk, Washer, and Shell
- Embed Related Rates Scene: Sliding Ladder and Cone Tank
- Embed Riemann Sum Visualizer: Rectangles, Error, and Rules
- Embed Chain Rule Step Trainer: You Pick the Decomposition
- Embed U-Substitution Helper: Pick u, See Why It Works
- Embed Series Convergence Test Chooser: Which Test and Why (BC)
- Embed Limit Solver (Teaching Mode): Which Method and Why
- Embed Definite Integral Step Tool: FTC, Shown Honestly
- Embed Tangent and Normal Line Finder (Teaching Mode)
- Embed Taylor Series Builder: Coefficients from Derivatives
- Embed Derivative Practice Checker: Any Correct Form Passes
- Embed Related Rates Setup Builder: The Five-Step Method
- Embed Optimization Setup Builder: Objective vs Constraint
- Embed Interval of Convergence Finder: Ratio Test and Endpoints
Cram sheets and walkthroughs 7
Derivative lookups 129
- Derivative of ln x: Answer, Proof, and Mistakes
- Derivative of e^x: Answer, Proof, and Mistakes
- Derivative of sin x: Answer, Proof, Mistakes
- Derivative of cos x: Answer, Proof, Mistakes
- Derivative of tan x: Answer, Proof, and Mistakes
- Derivative of sec x: Answer, Proof, and Mistakes
- Derivative of sqrt(x): Answer, Proof, and Mistakes
- Derivative of 1/x: Answer, Proof, and Mistakes
- Derivative of a^x: Answer, Proof, and Mistakes
- Derivative of x^x: Answer, Proof, and Mistakes
- Derivative of arctan x: Answer, Proof, Mistakes
- Derivative of arcsin x: Answer, Proof, Mistakes
- Derivative of x^2: Answer, Proof, and Mistakes
- Derivative of cot x: Answer, Proof, and Mistakes
- Derivative of csc x: Answer, Proof, and Mistakes
- Derivative of sin 2x: Answer, Chain Rule, Mistakes
- Derivative of cos 2x: Answer, Chain Rule, Mistakes
- Derivative of sin^2 x: Answer, Chain Rule, Mistakes
- Derivative of cos^2 x: Answer, Chain Rule, Mistakes
- Derivative of tan^2 x: Answer, Chain Rule, Mistakes
- Derivative of e^2x: Answer, Chain Rule, Mistakes
- Derivative of ln 2x: Answer, Why the 2 Vanishes
- Derivative of log x: Answer, Base Change, Mistakes
- Derivative of arccos x: Answer, Proof, and Mistakes
- Derivative of x sin x: Answer, Product Rule, Mistakes
- Derivative of x ln x: Answer, Product Rule, Mistakes
- Derivative of e^(x^2): Answer, Chain Rule, Mistakes
- Derivative of 1/x^2: Answer, Power Rule, Mistakes
- Derivative of x^3: Answer, Proof, and Mistakes
- Derivative of 1/sqrt(x): Answer, Proof, Mistakes
- Derivative of e^3x: Answer, Chain Rule, Mistakes
- Derivative of sin 3x: Answer, Chain Rule, Mistakes
- Derivative of cos 3x: Answer, Chain Rule, Mistakes
- Derivative of x cos x: Answer, Product Rule, Mistakes
- Derivative of x e^x: Answer, Product Rule, Mistakes
- Derivative of sin x cos x: Answer, Proof, Mistakes
- Derivative of ln(x^2): Answer, Proof, Mistakes
- Derivative of e^(sin x): Answer, Proof, Mistakes
- Derivative of ln(cos x): Answer, Proof, Mistakes
- Derivative of sqrt(x^2+1): Answer, Proof, Mistakes
- Derivative of x/(x+1): Answer, Proof, Mistakes
- Derivative of arctan(2x): Answer, Proof, Mistakes
- Derivative of x tan x: Answer, Proof, Mistakes
- Derivative of 1/x^3: Answer, Proof, and Mistakes
- Derivative of x^4: Answer, Proof, and Mistakes
- Derivative of e^(-x): Answer, Chain Rule, Mistakes
- Derivative of sin(x^2): Answer, Chain Rule, Mistakes
- Derivative of cos(x^2): Answer, Chain Rule, Mistakes
- Derivative of x^2 e^x: Answer, Product Rule, Mistakes
- Derivative of x^2 sin x: Answer, Product Rule, Mistakes
- Derivative of e^x cos x: Answer, Product Rule, Mistakes
- Derivative of e^x sin x: Answer, Product Rule, Mistakes
- Derivative of tan 2x: Answer, Proof, and Mistakes
- Derivative of sec 2x: Answer, Proof, and Mistakes
- Derivative of ln 3x: Answer, Proof, and Mistakes
- Derivative of the Cube Root of x: Answer, Proof, Mistakes
- Derivative of (x^2+1)^3: Answer, Proof, Mistakes
- Derivative of sin^3 x: Answer, Proof, and Mistakes
- Derivative of arcsin 2x: Answer, Proof, Mistakes
- Derivative of cos^3 x: Answer, Proof, and Mistakes
- Derivative of x/ln x: Answer, Proof, Mistakes
- Derivative of sin x/x: Answer, Proof, Mistakes
- Derivative of e^x/x: Answer, Proof, Mistakes
- Derivative of ln(sin x): Answer, Proof, Mistakes
- Derivative of sqrt(sin x): Answer, Proof, Mistakes
- Derivative of arctan(x^2): Answer, Proof, Mistakes
- Derivative of x arctan x: Answer, Proof, Mistakes
- Derivative of x arcsin x: Answer, Proof, Mistakes
- Derivative of e^2x sin x: Answer, Proof, Mistakes
- Derivative of tan(x^2): Answer, Proof, and Mistakes
- Derivative of (ln x)/x: Answer, Proof, Mistakes
- Derivative of x^2 ln x: Answer, Proof, and Mistakes
- Derivative of sec^2 x: Answer, Proof, and Mistakes
- Derivative of 3^x: Answer, Proof, and Mistakes
- Derivative of sqrt(1-x^2): Answer, Proof, Mistakes
- Derivative of e^(cos x): Answer, Proof, Mistakes
- Derivative of x/(x^2+1): Quotient Rule
- Derivative of x^2/(x+1): Quotient Rule
- Derivative of sqrt(x+1): Chain Rule
- Derivative of ln(x+1): Answer and Chain Rule
- Derivative of cot 2x: Chain Rule and Sign
- Derivative of csc 2x: Chain Rule and Sign
- Derivative of sin 4x: Answer and Chain Rule
- Derivative of tan 3x: Answer and Chain Rule
- Derivative of x^3 e^x: Product Rule
- Derivative of x/e^x: Answer and Maximum
- Derivative of ln(ln x): Nested Chain Rule
- Derivative of e^(tan x): Chain Rule
- Derivative of sin(e^x): Chain Rule
- Derivative of sqrt(tan x): Chain Rule
- Derivative of 1/ln x: Chain Rule
- Derivative of x^5: The Power Rule
- Derivative of 2^x: Answer and Why ln 2 Appears
- Derivative of cos 4x: Answer and Chain Rule
- Derivative of x^2 cos x: Product Rule
- Derivative of ln(x^3): The Log Rule Shortcut
- Derivative of sec x tan x: Proof and Common Mistakes
- Derivative of x/sin x: Quotient Rule Proof, Mistakes
- Derivative of ln(tan x): Answer, Proof, Simplification
- Derivative of arcsin(x^2): Chain Rule Proof, Mistakes
- Derivative of arccos(2x): Answer, Proof, and Domain
- Derivative of arctan(1/x): Proof and the Minus Sign
- Derivative of x^(sin x) by Log Differentiation
- Derivative of x^(ln x): Answer, Proof, Mistakes
- Derivative of sqrt(ln x): Chain Rule Proof
- Derivative of ln(sqrt x): Answer and Shortcut
- Derivative of sin(sqrt x): Answer, Proof, Mistakes
- Derivative of e^sqrt(x): Answer, Proof, Mistakes
- Derivative of cos(1/x): Answer, Proof, Mistakes
- Derivative of ln(x + sqrt(x^2+1)): Answer and Proof
- Derivative of 1/(1+e^x): Answer, Proof, Mistakes
- Derivative of e^x/(1+e^x): Answer, Proof, Mistakes
- Derivative of x/sqrt(x^2+1): Proof and Graph Shape
- Derivative of sec^3 x: Answer, Proof, and Mistakes
- Derivative of (x^2-1)/(x^2+1): Quotient Rule Proof
- Derivative of sin x/(1+cos x): Answer and Proof
- Derivative of arctan(e^x): Answer, Proof, Mistakes
- Derivative of x^6: The Power Rule
- Derivative of csc^2 x: Chain Rule
- Derivative of arccot x: Answer and Sign
- Derivative of x^3 sin x: Product Rule
- Derivative of x arccos x: Product Rule
- Derivative of sin x / x^2: Quotient Rule
- Derivative of cos x / x: Quotient Rule
- Derivative of tan x / x: Quotient Rule
- Derivative of x^2 arctan x: Product Rule
- Derivative of ln x / x^2: Quotient Rule
- Derivative of arctan(sqrt x): Chain Rule
- Derivative of sec(x^2): Chain Rule
Integral lookups 125
- Integral of 1/x: Answer, Proof, and Mistakes
- Integral of ln x: Answer, Proof, and Mistakes
- Integral of e^x: Answer, Proof, and Mistakes
- Integral of sin x: Answer, Proof, and Mistakes
- Integral of cos x: Answer, Proof, and Mistakes
- Integral of tan x: Answer, Derivation, and Mistakes
- Integral of sec x: Proof, Formula, and Mistakes
- Integral of sec^2 x: Answer, Proof, and Mistakes
- Integral of csc x: Answer, Derivation, Mistakes
- Integral of cot x: Answer, u-Substitution, Mistakes
- Integral of 1/(1+x^2): Answer, Proof, and Mistakes
- Integral of 1/sqrt(1-x^2): Answer, Proof, Mistakes
- Integral of sqrt x: Answer, Steps, and Mistakes
- Integral of x^n: The Power Rule for Antiderivatives
- Integral of a^x: Answer, Proof, and Mistakes
- Integral of sin^2 x
- Integral of tan^2 x: Answer, Steps, and Domain
- Integral of x e^x: Answer, Parts Setup, Mistakes
- Integral of e^(2x): Answer, Steps, and Mistakes
- Integral of arcsin x: Answer, Parts Setup, Mistakes
- Integral of cos^2 x: Answer and Power Reduction
- Integral of x ln x: Answer, Parts Setup, Mistakes
- Integral of 1/(x ln x): Answer and Substitution
- Integral of arctan x: Answer, Parts Setup, Mistakes
- Integral of 1/x^2: Answer, Power Rule, Mistakes
- Integral of x^2: Answer, Proof, and Steps
- Integral of x^3: Answer, Proof, and Steps
- Integral of e^(3x): Answer, Steps, and Mistakes
- Integral of sin(2x): Answer, Proof, and Steps
- Integral of cos(2x): Answer, Proof, and Steps
- Integral of sin(3x): Answer, Proof, and Steps
- Integral of cos(3x): Answer, Proof, and Steps
- Integral of 1/(x+1): Answer, Proof, and Mistakes
- Integral of csc^2 x: Answer, Proof, and Steps
- Integral of sec x tan x: Answer and Proof
- Integral of csc x cot x: Answer and Proof
- Integral of x sin x: Answer, Parts, Mistakes
- Integral of x cos x: Answer, Parts, Mistakes
- Integral of 1/x^3: Answer, Power Rule, Mistakes
- Integral of sin x cos x: Answer and Proof
- Integral of e^x sin x: Answer and Proof
- Integral of x^4: Answer, Proof, and Steps
- Integral of e^(-x): Answer, Proof, and Steps
- Integral of the Cube Root of x: Answer and Steps
- Integral of 1/(9+x^2): Answer, Proof, and Steps
- Integral of 1/sqrt(4-x^2): Answer and Steps
- Integral of e^x cos(x): Answer, Proof, and Steps
- Integral of x e^(x^2): Answer, Proof, and Steps
- Integral of x/(x^2+1): Answer, Proof, and Steps
- Integral of sin^3 x: Answer, Proof, and Steps
- Integral of cos^3 x: Answer, Proof, and Steps
- Integral of x^2 e^x: Answer, Proof, and Steps
- Integral of x^2 sin x: Answer, Proof, and Steps
- Integral of x^2 cos x: Answer, Proof, and Steps
- Integral of sin 2x cos 2x: Proof and Steps
- Integral of 1/(2x): Answer, Proof, and Steps
- Integral of e^(4x): Answer, Proof, and Steps
- Integral of ln(x)/x: Answer, Proof, and Steps
- Integral of x sqrt(x^2+1): Answer and Proof
- Integral of sin(x)cos^2(x): Answer and Proof
- Integral of e^(sin x)cos(x): Answer and Proof
- Integral of 1/(x^2+4): Answer, Proof, and Steps
- Integral of 1/sqrt(9-x^2): Answer and Proof
- Integral of x/sqrt(x^2+1): Answer and Proof
- Integral of sin(4x): Answer, Proof, and Steps
- Integral of cos 4x: Answer, Proof, and Steps
- Integral of x cos 2x: Answer, Proof, and Steps
- Integral of x sin 2x: Answer, Proof, and Steps
- Integral of x^2/(x^3+1): Answer and Steps
- Integral of e^2x sin x: Answer and Steps
- Integral of ln 2x: Answer, Proof, and Steps
- Integral of x^(3/2): Answer, Proof, and Steps
- Integral of 1/(x+1)^2: Answer and Steps
- Integral of 1/(x-1): Answer and Bars
- Integral of x/(x+1): Divide First
- Integral of tan 2x: Answer and Substitution
- Integral of cot 2x: Answer and Substitution
- Integral of sec 2x: Answer and the Trick
- Integral of e^x/(1+e^x): Substitution
- Integral of 1/(1-x): Watch the Sign
- Integral of x^2 e^(x^3): Substitution
- Integral of sin x/(1+cos x)
- Integral of x/sqrt(1-x^2)
- Integral of arctan x/(1+x^2)
- Integral of (ln x)^2/x: Substitution
- Integral of x arctan x: Parts Then Division
- Integral of 1/(x^2-1): Partial Fractions
- Integral of sin^2 x cos x: Substitution
- Integral of x^2 sqrt(x): Rewrite as One Power
- Integral of x/(x^2+4): Log, Not Arctan
- Integral of sin 5x: Answer and the Divided Factor
- Integral of tan x sec^2 x: Two Right Answers
- Integral of sqrt(1-x^2): The Circle Answer
- Integral of x^2 ln x: Why ln x Should Be u
- Integral of (ln x)^2: Integration by Parts Twice
- Integral of x/(x^2-1): Substitution, Not Partials
- Integral of 1/(x^2+2x+5): Complete the Square
- Integral of 1/sqrt(1-4x^2): Arcsine With a Half
- Integral of x sec^2 x: Parts Then tan x
- Integral of sin x e^cos x: The Minus Sign
- Integral of cos^2(2x): Double Angle Twice
- Integral of x^3 e^(x^2): Substitute, Then Parts
- Integral of sqrt(x) ln x: Parts With a Fraction
- Integral of 1/(x^2+x): Partial Fractions to One Log
- Integral of 1/(x^2-4): A Log, Not an Arctangent
- Integral of (2x+3)/(x^2+3x): Spot the Derivative
- Integral of cos x / sin^2 x: The Answer Is -csc x
- Integral of sin x / cos^3 x: Two Minus Signs Cancel
- Integral of x e^-x: Parts and Three Minus Signs
- Integral of arcsin x / sqrt(1-x^2)
- Integral of ln x / x^2: Parts, Not Substitution
- Integral of sqrt(x+1): The Free Substitution
- Integral of sin 2x / sin x: Simplify First
- Integral of x^3 e^x: Parts Three Times
- Integral of e^(3x) cos x: The Loop Trick
- Integral of sec^3 x: The Classic Loop
- Integral of cos^2 x sin x: Substitution
- Integral of 1/sqrt(x+1): Answer and Rule
- Integral of x^3/(x^2+1): Divide First
- Integral of ln(x^3): Simplify First
- Integral of sin^3 x cos x: Substitution
- Integral of tan^3 x: Split the Power
- Integral of 1/(x (ln x)^2): Substitution
- Integral of x sqrt(x+1): Substitution
- Integral of 1/(1+e^x): Answer and Trick
Limit lookups 152
- Limit of sin x / x as x Approaches 0 Is 1
- Limit of (1 - cos x) / x as x Approaches 0 Is 0
- Limit of (1 - cos x) / x^2 as x Approaches 0 Is 1/2
- Limit of tan x / x as x Approaches 0 Is 1
- Limit of sin 2x / x as x Approaches 0 Is 2
- Limit of sin 3x / sin 5x as x Approaches 0 Is 3/5
- Limit of tan x / sin x as x Approaches 0 Is 1
- Limit of (e^x-1)/x as x Approaches 0 Is 1
- Limit of (x^2-1)/(x-1) as x Approaches 1 Is 2
- Limit of (x^3-8)/(x-2) as x Approaches 2 Is 12
- Limit of (x^2-4)/(x^2-x-2) at x = 2 Is 4/3
- Limit of (sqrt x - 2)/(x-4) at x = 4 Is 1/4
- Limit of (1+1/x)^x as x Approaches Infinity Is e
- Limit of (1+x)^(1/x) as x Approaches 0 Is e
- Limit of x^x as x Approaches 0 from the Right
- Limit of x ln x as x Approaches 0
- Limit of ln x as x Approaches 0
- Limit of 1/x^2 as x Approaches 0
- Limit of 1/x as x Approaches Infinity
- Limit of (2x^2+3)/(x^2-1) at Infinity
- Limit of (3x+1)/(x^2+2) at Infinity
- Limit of x^2/(x+1) as x Approaches Infinity
- Limit of e^x as x Approaches Negative Infinity
- Limit of ln(x)/x as x Approaches Infinity
- Limit of sin(x)/x as x Approaches Infinity
- Limit of arctan(x) as x Approaches Infinity
- Limit of x/sqrt(x^2+1) at Infinity
- Limit of sqrt(x^2+x) - x at Infinity
- Limit of x sin(1/x) as x Approaches 0 Is 0
- Limit of x^2 sin(1/x) as x Approaches 0 Is 0
- Limit of (x - sin x)/x^3 as x Approaches 0 Is 1/6
- Limit of (tan x - x)/x^3 as x Approaches 0 Is 1/3
- Limit of (1 - cos 2x)/x^2 as x Approaches 0 Is 2
- Limit of (sqrt(x+1) - 1)/x as x Approaches 0 Is 1/2
- Limit of sin x/(x + x^2) as x Approaches 0 Is 1
- Limit of ln(x+1)/x as x Approaches 0 Is 1
- Limit of (2^x-1)/x as x Approaches 0 Is ln 2
- Limit of (e^(2x)-1)/x as x Approaches 0 Is 2
- Limit of (x-3)/(x^2-9) as x Approaches 3 Is 1/6
- Limit of (1/x-1/2)/(x-2) at x = 2 Is -1/4
- Limit of x/e^x as x Approaches Infinity Is 0
- Limit of x^2e^(-x) as x Approaches Infinity Is 0
- Limit of (ln x)^2/x at Infinity Is 0
- Limit of (5x^3-2x)/(3x^3+7) at Infinity Is 5/3
- Limit of (x^2+1)/(2x^2-3) at Infinity Is 1/2
- Limit of (2x+3)/sqrt(x^2+1) at Infinity Is 2
- Limit of 3^x/4^x at Infinity Is 0
- Limit of x^(1/x) at Infinity Is 1
- Limit of sqrt(x^2+1)/x at Infinity Is 1
- Limit of 1/x as x Approaches 0 from the Right
- Limit of 1/x as x Approaches 0 from the Left
- Limit of |x|/x as x Approaches 0 from the Right
- Limit of |x|/x as x Approaches 0 from the Left
- Limit of ln x as x Approaches Infinity
- Limit of arctan(x) at Negative Infinity
- Limit of x^3 as x Approaches Negative Infinity
- Limit of cos x / x at Infinity Is 0
- Limit of x / sin x as x Approaches 0 Is 1
- Limit of sin 5x / 3x as x Approaches 0
- Limit of arcsin x / x as x Approaches 0 Is 1
- Limit of arctan x / x as x Approaches 0 Is 1
- Limit of tan 3x / x as x Approaches 0 Is 3
- Limit of sin(x^2) / x as x Approaches 0 Is 0
- Limit of sin x / x^3 at 0 Is Infinity
- Limit of (sec x - 1) / x at 0 Is 0
- Limit of ln x / (x - 1) as x Approaches 1
- Limit of (e^x - e^-x) / x at 0 Is 2
- Limit of (cos x - 1) / sin x at 0 Is 0
- Limit of ln(1 + 2x) / x as x Approaches 0 Is 2
- Limit of (x - 1) / (sqrt x - 1) at 1 Is 2
- Limit of (sqrt(x+4) - 2) / x at 0 Is 1/4
- Limit of (x - 2) / (sqrt(x+2) - 2) at 2
- Limit of x / (x + 1) at Infinity Is 1
- Limit of (3x^2 + 2x) / (x^2 - 5) at Infinity
- Limit of (x + sin x) / x at Infinity Is 1
- Limit of sqrt(x^2 - x) - x at Infinity
- Limit of cot x as x Approaches 0 from the Right
- Limit of e^(-x) as x Approaches Infinity Is 0
- Limit of 5^x as x Approaches Negative Infinity
- Limit of sqrt x as x Approaches Infinity Is Infinity
- Limit of (cos x - 1)/x^2 at 0 Is -1/2
- Limit of (e^x - 1 - x)/x^2 at 0 Is 1/2
- Limit of (sin x - x)/x as x Approaches 0 Is 0
- Limit of x^2/e^x at Infinity Is 0
- Limit of x^3/e^x at Infinity Is 0
- Limit of sqrt(x+1) - sqrt(x) at Infinity Is 0
- Limit of x ln(1 + 1/x) at Infinity Is 1
- Limit of (1 - 1/x)^x at Infinity Is 1/e
- Limit of (1 + 2/x)^x at Infinity Is e^2
- Limit of tan x / x^3 at 0 Is Infinity
- Limit of arctan x / sqrt x at Infinity Is 0
- Limit of ln(x+1)/ln(x) at Infinity Is 1
- Limit of sqrt(x^2+4) - x at Infinity Is 0
- Limit of x / ln x at Infinity Is Infinity
- Limit of sin 3x / tan 2x at 0 Is 3/2
- Limit of (1-cos x)/(x sin x) at 0 Is 1/2
- Limit of sin x / tan x as x Approaches 0 Is 1
- Limit of (sqrt x - 3)/(x - 9) at 9 Is 1/6
- Limit of (e^x - e)/(x - 1) at 1 Is e
- Limit of ln(sin x)/ln(x) at 0 from the Right
- Limit of (x^2 - 9)/(x - 3) at 3 Is 6
- Limit of (x^3 + 1)/(x + 1) at -1 Is 3
- Limit of (x^4 - 16)/(x - 2) at 2 Is 32
- Limit of (sqrt(x + 9) - 3)/x at 0 Is 1/6
- Limit of (x - 4)/(sqrt x - 2) at 4 Is 4
- Limit of sin(pi x)/(x - 1) at 1 Is -pi
- Limit of (e^x - 1)/sin x at 0 Is 1
- Limit of ln(cos x)/x^2 at 0 Is -1/2
- Limit of sin(1/x) at Infinity Is 0
- Limit of x^(sin x) at 0 from the Right Is 1
- Limit of sqrt(4x^2+1)/(x+3) at Infinity Is 2
- Limit of (3x-2)/sqrt(2x^2+1) at Infinity Is 3/sqrt2
- Limit of 1/(x - 1) at 1 From the Right Is Infinity
- Limit of (x^2 - x - 6)/(x - 3) at 3 Is 5
- Limit of sqrt(x^2 + 3x) - x at Infinity Is 3/2
- Limit of sin x / sqrt x at 0 From the Right Is 0
- Limit of tan 2x / tan 3x at 0 Is 2/3
- Limit of arcsin 2x / x at 0 Is 2
- Limit of arctan 3x / 5x at 0 Is 3/5
- Limit of (e^3x - 1)/x at 0 Is 3
- Limit of ln(1 + x^2)/x^2 at 0 Is 1
- Limit of x^2/(1 - cos x) at 0 Is 2
- Limit of x^2 sin(1/x) at Infinity Is Infinite
- Limit of sqrt(x+1)/sqrt x at Infinity Is 1
- Limit of tan x as x Approaches pi/2 from the Left
- Limit of sec x as x Approaches pi/2 from the Left
- Limit of csc x as x Approaches 0 from the Right
- Limit of x/(x-1) as x Approaches 1 from the Right
- Limit of 1/(x^2-4) at 2 from the Right
- Limit of e^(1/x) as x Approaches 0 from the Left
- Limit of arctan(1/x) at 0 from the Right
- Limit of (1-cos 3x)/x^2 as x Approaches 0
- Limit of sin(4x)/sin(2x) as x Approaches 0
- Limit of tan(5x)/x as x Approaches 0
- Limit of arcsin(3x)/x as x Approaches 0
- Limit of sin x/(x + sin x) as x Approaches 0
- Limit of x/tan x as x Approaches 0
- Limit of sin^2(x)/x as x Approaches 0
- Limit of sin^2(x)/x^2 as x Approaches 0
- Limit of (3^x - 1)/x as x Approaches 0
- Limit of ln(1+3x)/x as x Approaches 0
- Limit of e^(-x^2) as x Approaches Infinity
- Limit of x^(1/ln x) as x Approaches Infinity
- Limit of x^3/(x^2+1) as x Approaches Infinity
- Limit of (x^2-16)/(x-4) as x Approaches 4
- Limit of (x^3-27)/(x-3) as x Approaches 3
- Limit of (sqrt(x+3)-2)/(x-1) as x Approaches 1
- Limit of (4x^2-1)/(2x-1) as x Approaches 1/2
- Limit of sqrt(9x^2+1)/x as x Approaches Infinity
- Limit of x cos(1/x) as x Approaches 0
- Limit of sin(x)/x^2 as x Approaches Infinity
- Limit of tan(x)/(x-pi) as x Approaches pi
Series convergence 93
- Does the Sum of 1/n Converge? No
- Does the Sum of 1/n^2 Converge? Yes
- Does the Sum of 1/sqrt(n) Converge? No
- Does the Sum of 1/n^3 Converge? Yes
- Does the Sum of 1/n^(3/2) Converge? Yes
- Does the Sum of 1/2^n Converge? Yes, to 1
- Does the Sum of 1/3^n Converge? Yes, to 1/2
- Does the Sum of 3^n/4^n Converge? Yes, to 3
- Does the Sum of (-1)^n/2^n Converge?
- Does the Sum of n/(n+1) Converge? No
- Does the Sum of n^2/(2n^2+1) Converge? No
- Does the Sum of 2^n/n^2 Converge? No
- Does the Sum of n/2^n Converge? Yes, to 2
- Does the Alternating Harmonic Series Converge?
- Does the Sum of (-1)^n/n^2 Converge?
- Does the Sum of (-1)^n/sqrt(n) Converge?
- Does the Sum of 1/(n ln n) Converge? No
- Does the Sum of 1/(n(n+1)) Converge? Yes, to 1
- Does the Sum of sin(n)/n^2 Converge? Yes
- Does the Sum of 1/(n^2+1) Converge? Yes
- Does the Sum of 1/ln(n) Converge? No
- Does the Sum of 1/n! Converge? Yes
- Does the Sum of 2^n/n! Converge? Yes
- Does the Sum of 3^n/n! Converge? Yes
- Does the Sum of n^2/n! Converge? Yes
- Does the Sum of n!/n^n Converge? Yes
- Does the Sum of n^n/n! Converge? No
- Does the Sum of (n!)^2/(2n)! Converge?
- Does the Sum of 1/n^n Converge? Yes
- Does the Sum of n/3^n Converge? Yes, to 3/4
- Does the Sum of 2/3^n Converge? Yes, to 1
- Does the Sum of 2^n/5^n Converge? Yes, to 2/3
- Does the Sum of 1/(n^2-1) Converge? Yes
- Does the Sum of n/(n^2+1) Converge? No
- Does the Sum of 1/(2n+1) Converge? No
- Does the Sum of sqrt(n)/(n^2+1) Converge?
- Does the Sum of n/(n^3+1) Converge? Yes
- Does the Sum of 1/(n + sqrt n) Converge? No
- Does the Sum of arctan(n)/n^2 Converge? Yes
- Does the Sum of (-1)^n/(n+1) Converge?
- Does the Sum of (-1)^n/(2n+1) Converge?
- Does the Sum of 1/n^4 Converge? Yes
- Does the Sum of 1/n^(2/3) Converge? No
- Does the Sum of (-1)^n/3^n Converge? Yes
- Does the Sum of 5^n/4^n Converge? No
- Does the Sum of 1/e^n Converge? Yes
- Does the Sum of 2^(n+1)/3^n Converge? Yes
- Does the Sum of sqrt(n)/(sqrt(n)+1) Converge? No
- Does the Sum of 3n^2/(n^2+1) Converge? No
- Does the Sum of n^2/2^n Converge? Yes
- Does the Sum of n^3/3^n Converge? Yes
- Does the Sum of 3^n/n Converge? No
- Does the Sum of 1/(ln n)^n Converge? Yes
- Does the Sum of (-1)^n ln(n)/n Converge? Yes
- Does the Sum of (-1)^n/n! Converge? Yes, to 1/e - 1
- Does the Sum of (-1)^n/n^3 Converge? Yes, Absolutely
- Does the Sum of ln(n)/n Converge? No
- Does the Sum of ln(n)/n^2 Converge? Yes
- Does the Sum of 1/(n(ln n)^2) Converge? Yes
- Does the Sum of 1/(n(n+2)) Converge? Yes, to 3/4
- Does the Sum of 1/(4n^2-1) Converge? Yes, to 1/2
- Does the Sum of 1/(sqrt(n)+1) Converge? No
- Does the Sum of sin^2(n)/n^2 Converge? Yes
- Does the Sum of arctan(n)/n Converge? No
- Does the Sum of 1/(2^n+1) Converge? Yes
- Does the Sum of 1/5^n Converge? Yes, to 1/4
- Does the Sum of 1/10^n Converge? Yes, to 1/9
- Does the Sum of 4^n/5^n Converge? Yes, to 4
- Does the Sum of (-1)^n/5^n Converge? Yes, to -1/6
- Does the Sum of 1/n^5 Converge? Yes
- Does the Sum of 1/n^(5/2) Converge? Yes
- Does the Sum of 1/n^(1/3) Converge? No
- Does the Sum of n^4/2^n Converge? Yes
- Does the Sum of 1/(n·2^n) Converge? Yes, to ln 2
- Does the Sum of (2n)!/(n!)^2 Converge? No
- Does the Sum of n/4^n Converge? Yes, to 4/9
- Does the Sum of 2^n·n!/n^n Converge? Yes
- Does the Sum of 2^n/n^n Converge? Yes
- Does the Sum of (arctan n)^n Converge? No
- Does the Sum of (-1)^n/n^4 Converge? Yes, Absolutely
- Does the Sum of (-1)^n·n/2^n Converge? Yes, to -2/9
- Does the Sum of (-1)^n·n/(n^2+1) Converge? Conditionally
- Does the Sum of (-1)^n/sqrt(n+1) Converge? Conditionally
- Does the Sum of n^2/(n^2+1) Converge? No
- Does the Sum of n/sqrt(n^2+1) Converge? No
- Does the Sum of cos(1/n) Converge? No
- Does the Sum of n·sin(1/n) Converge? No
- Does the Sum of ln(n)/n^3 Converge? Yes
- Does the Sum of cos(n)/n^2 Converge? Yes
- Does the Sum of 1/(3n+2) Converge? No
- Does the Sum of sqrt(n+1)/n^2 Converge? Yes
- Does the Sum of (n^2+1)/(n^3+1) Converge? No
- Does the Sum of 1/(n(n+3)) Converge? Yes, to 11/18
Theorems: hypotheses and counterexamples 22
- When the Mean Value Theorem Fails: Counterexamples
- When Rolle's Theorem Fails: Three Counterexamples
- When the Intermediate Value Theorem Fails
- When the Extreme Value Theorem Fails
- When the Squeeze Theorem Fails to Apply
- Why Continuity Does Not Imply Differentiability
- When FTC Part 1 Fails: The Jump Counterexample
- When FTC Part 2 Fails: Integrating Across a Pole
- When L'Hopital's Rule Fails: Two Counterexamples
- When the MVT for Integrals Fails
- When the Inverse Derivative Formula Fails
- Critical Points That Are Not Extrema
- When the Second Derivative Test Is Inconclusive
- Positive Derivative but Not Increasing: The Catch
- When the Limit Laws Cannot Be Applied
- When the Chain Rule Does Not Apply
- Why the nth Term Test Cannot Prove Convergence
- When a Geometric Series Diverges
- Why p Must Exceed 1 for a p-Series
- When the Alternating Series Test Does Not Apply
- Convergent but Not Absolutely Convergent
- When a Taylor Series Does Not Equal Its Function
Reference tables 6
Comparisons: which one do I use 93
- Continuity vs Differentiability
- Removable vs Jump vs Infinite Discontinuity
- Average vs Instantaneous Rate of Change
- Product Rule vs Chain Rule
- Explicit vs Implicit Differentiation
- Related Rates vs Optimization
- Local vs Absolute Extrema
- First vs Second Derivative Test
- Critical Point vs Inflection Point
- Mean Value Theorem vs Intermediate Value Theorem
- Mean Value Theorem vs Rolle's Theorem
- Speed vs Velocity
- Displacement vs Total Distance Travelled
- Definite vs Indefinite Integral
- FTC Part 1 vs Part 2
- U-Substitution vs Integration by Parts
- Left vs Right Riemann Sum
- Midpoint vs Trapezoidal Rule
- Area vs Net Signed Area
- Average Value vs Average Rate of Change
- General vs Particular Solution
- Exponential vs Logistic Growth
- Slope Field vs Euler's Method
- Disk vs Washer Method
- Washer vs Shell Method
- Known Cross-Sections vs Solids of Revolution
- Parametric vs Polar Curves
- Sequence vs Series
- Ratio Test vs Root Test
- Direct Comparison vs Limit Comparison Test
- Absolute vs Conditional Convergence
- Geometric Series vs p-Series
- Taylor vs Maclaurin Series
- Taylor Polynomial vs Taylor Series
- Lagrange vs Alternating Series Error Bound
- Radius vs Interval of Convergence
- Secant Line vs Tangent Line
- Tangent Line vs Normal Line
- Vertical vs Horizontal Asymptote
- Increasing vs Concave Up
- Power Rule vs Exponential Rule
- Riemann Sum vs Definite Integral
- Average Value vs MVT for Integrals
- Definite vs Improper Integral
- U-Substitution vs Partial Fractions
- Disk Method vs Shell Method
- Arc Length vs Distance Travelled
- Continuity vs Limit Existence
- Horizontal Asymptote vs Limit at Infinity
- Removable vs Infinite Discontinuity
- Derivative vs Differential
- Derivative vs Antiderivative
- Critical Point vs Extremum
- Velocity vs Acceleration
- Position vs Displacement
- Euler's Method vs Exact Solution
- Slope Field vs Solution Curve
- Chain Rule vs Implicit Differentiation
- Polar Area vs Cartesian Area
- Taylor Series vs Power Series
- Alternating Series Test vs Absolute Convergence
- Integral Test vs p-Series Test
- nth Term Test vs Ratio Test
- Convergence vs Divergence
- Maclaurin Series vs Binomial Series
- Chain Rule vs u-Substitution
- Product Rule vs Integration by Parts
- Quotient Rule vs Product Rule
- One-Sided vs Two-Sided Limit
- Direct Substitution vs Indeterminate Form
- Squeeze Theorem vs L'Hopital's Rule
- Extreme Value Theorem vs MVT
- Candidates Test vs First Derivative Test
- Logarithmic vs Implicit Differentiation
- Corner vs Cusp
- Vertical Tangent vs Vertical Asymptote
- Implicit Differentiation vs Related Rates
- Objective Function vs Constraint
- Partial Fractions vs Long Division
- Infinite Limit vs Infinite Integrand
- Tangent Line vs Taylor Polynomial
- Accumulation Function vs Definite Integral
- Improper Integral vs Infinite Series
- Parametric vs Cartesian Second Derivative
- Overestimate vs Underestimate
- Acceleration vs Speeding Up
- Linearization vs Differential
- Antiderivative vs Indefinite Integral
- Average Velocity vs Average Speed
- Radians vs Degrees in Calculus
- Error Bound vs Actual Error
- Limit Does Not Exist vs Undefined
- Riemann Sum vs Trapezoidal Rule
Glossary 270
- Absolute convergence
- Absolute extremum
- Absolute maximum
- Absolute minimum
- Absolute value function
- Acceleration
- Accumulated change
- Accumulation function
- Alternate form of the derivative
- Alternating harmonic series
- Alternating series error bound
- Alternating series test
- Antiderivative
- Arc length
- Area under a curve
- Average acceleration
- Average rate of change
- Average value of a function
- Average velocity
- Axis of revolution
- Binomial series
- Bounded Function
- Bounded sequence
- Candidates test
- Carrying capacity
- Center of a power series
- Changing limits of integration
- Common ratio
- Comparison test
- Completing the square
- Composite function
- Concave down
- Concave up
- Concavity
- Concavity test
- Conditional convergence
- Conjugate method
- Constant multiple rule
- Constant of integration
- Constraint equation
- Continuity
- Continuity on an interval
- Continuous function
- Convergence
- Convergent sequence
- Convergent Series
- Corner
- Critical number
- Critical point
- Cross-section
- Curve Sketching
- Cusp
- Decreasing function
- Definite integral
- Derivative
- Derivative at a point
- Derivative of a constant
- Derivative of a Piecewise Function
- Difference quotient
- Differentiability
- Differentiability implies continuity
- Differentiable function
- Differential
- Differential equation
- Direct substitution
- Discontinuity
- Disk method
- Displacement
- Divergence
- Divergent Sequence
- Divergent series
- Domain
- Double-Angle Identity
- Doubling time
- Eliminating the parameter
- End behavior
- Endpoint analysis
- Endpoint of an interval of convergence
- Epsilon-delta definition
- Equilibrium solution
- Estimating a Derivative From a Table
- Euler's method
- Euler's number
- Even function
- Explicit Function
- Exponential decay
- Exponential function
- Exponential growth
- Exponential model
- Extreme Value Theorem
- Extremum
- Factorial
- First derivative test
- Fundamental Theorem of Calculus Part 1
- Fundamental Theorem of Calculus Part 2
- General solution
- Geometric Sequence
- Geometric series
- Graph of the derivative
- Greatest integer function
- Half-life
- Harmonic series
- Higher-order derivative
- Horizontal asymptote
- Horizontal Tangent
- Implicit function
- Implicit Second Derivative
- Improper integral
- Increasing function
- Increasing/decreasing test
- Indefinite integral
- Indeterminate form
- Index of summation
- Infinite discontinuity
- Infinite limit
- Inflection point
- Initial condition
- Initial value problem
- Initial velocity
- Inner function
- Instantaneous rate of change
- Integral test
- Integrand
- Intermediate Value Theorem
- Interval notation
- Interval of convergence
- Inverse function
- Inverse function derivative
- Inverse trigonometric derivatives
- Inverse trigonometric function
- Inverse Trigonometric Integrals
- Jump discontinuity
- Justification
- Lagrange error bound
- Left Riemann sum
- Left-hand limit
- Leibniz notation
- Limit
- Limit at infinity
- Limit comparison test
- Limit definition of the derivative
- Limit does not exist
- Limit laws
- Limit of a composite function
- Limit of a sequence
- Limits of integration
- Linear approximation
- Linearization
- Local linearity
- Logarithm properties
- Logarithmic differentiation
- Logistic differential equation
- Logistic growth
- Maclaurin series
- Mean Value Theorem
- Mean Value Theorem for Integrals
- Midpoint Riemann sum
- Monotonic
- Motion in the plane
- Natural exponential function
- Natural logarithm
- Net change theorem
- Net signed area
- Newton's Law of Cooling
- Nonremovable Discontinuity
- Normal line
- nth term test
- Numerical integration
- Objective function
- Odd function
- One-sided limit
- One-to-one function
- Optimization
- Order of a differential equation
- Orientation of a parametric curve
- Oscillating discontinuity
- Outer function
- Overestimate and underestimate
- p-series
- Parameter
- Parametric arc length
- Parametric derivative
- Parametric equations
- Partial fractions
- Partial sum
- Particular solution
- Periodic Function
- Piecewise function
- Polar area
- Polar coordinates
- Polar curve
- Polar symmetry
- Polar to Cartesian Conversion
- Polynomial function
- Position function
- Power function
- Power rule
- Power series
- Power series representation
- Prime notation
- Product rule
- Properties of definite integrals
- Pythagorean Identity
- Quotient rule
- Radian measure
- Radius Function
- Radius of convergence
- Range
- Rate In Minus Rate Out
- Rational function
- Rectilinear motion
- Recursive Sequence
- Region bounded by curves
- Related rate
- Relative extremum
- Relative maximum
- Relative minimum
- Relative rate of change
- Removable discontinuity
- Representative rectangle
- Reverse power rule
- Riemann sum
- Right Riemann sum
- Right-hand limit
- Rolle's Theorem
- Root of a function
- Root test
- Sample point
- Secant line
- Second derivative
- Second derivative of a parametric curve
- Second derivative test
- Separation of variables
- Sequence
- Series
- Shell method
- Sigma notation
- Sign chart
- Slant asymptote
- Slope field
- Slope of a polar curve
- Smooth curve
- Solid of revolution
- Special trigonometric limits
- Speed
- Speeding up and slowing down
- Squeeze theorem
- Sum and difference rule
- Sum of a geometric series
- Tangent line
- Taylor polynomial
- Taylor remainder
- Taylor series
- Telescoping series
- Term-by-term differentiation
- Term-by-term integration
- Total area
- Total distance
- Trapezoidal rule
- Turning point
- Unit Circle
- Units of a Derivative
- Variable of integration
- Vector magnitude
- Vector-valued function
- Velocity
- Vertical asymptote
- Vertical tangent
- Volume by cross sections
- Washer method