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AP Calculus › Concept Labs AP Calculus AB/BC · Concept Labs · don't just watch it — manipulate it
Choose one investigation, predict before manipulating, and use the central model as mathematical evidence. Controls, observations, answers, and explanations stay attached to the same instrument.
Change visualization · Left and right limits Course scope BC unlocks Units 9–10
AP Calculus AB AP Calculus BC
Unit All units Unit 1 · Limits and Continuity Unit 2 · Differentiation Unit 3 · Composite and Implicit Functions Unit 4 · Contextual Applications Unit 5 · Analytical Applications of Differentiation Unit 6 · Integration and Accumulation of Change Unit 7 · Differential Equations Unit 8 · Applications of Integration Topic All topics 1.3 · One-Sided Limits 1.4 · Limits and Function Values 1.5 · Limits That Fail to Exist 1.10 · Infinite Limits and Vertical Asymptotes 1.13 · Defining Continuity at a Point 1.12 · Limits at Infinity and Horizontal Asymptotes 2.1 · Average and Instantaneous Rates of Change 2.2 · The Difference Quotient 2.3 · Estimating Derivatives 2.4 · The Derivative as a Function 3.1 · The Chain Rule 3.2 · Implicit Differentiation 3.3 · Differentiating Inverse Functions 4.2 · Straight-Line Motion 4.5 · Solving Related Rates 4.6 · Local Linearity and Linearization 5.1 · Using f, f′, and f″ 5.2 · Extreme Value and Critical Points 5.3 · Intervals of Increase and Decrease 5.6 · Concavity and Inflection Points 5.10 · Optimization Problems 6.2 · Approximating Areas with Riemann Sums 6.3 · Accumulation Functions 6.4 · The Fundamental Theorem of Calculus 7.3 · Sketching Slope Fields 7.4 · Reasoning Using Slope Fields 7.2 · Verifying Solutions 7.5 · Euler’s Method 8.4 · Area Between Curves 8.9 · Volume with Disk Method 8.10 · Volume with Washer Method 8.7 · Volumes with Cross Sections Lab type All types Approach Discontinuity Continuity Secant / tangent Derivative graph Composition Implicit curve Inverse relationship Motion Related rates Linearization Derivative analysis Optimization Riemann sum Accumulation / FTC Slope field Solution curve Euler method Area between curves Volume slice
Units 1–8 · AB catalog
36 shown U1 · 1.3 · Approach Left and right limits U1 · 1.4 · Discontinuity Holes and removable discontinuities U1 · 1.5 · Discontinuity Jump discontinuities U1 · 1.10 · Discontinuity Infinite discontinuities U1 · 1.13 · Continuity Continuity at a point U1 · 1.12 · Approach Limits at infinity U2 · 2.1 · Secant / tangent Secant to tangent U2 · 2.2 · Secant / tangent Changing h U2 · 2.3 · Secant / tangent Derivative at a point U2 · 2.4 · Derivative graph Derivative as a function U3 · 3.1 · Composition Chain-rule composition U3 · 3.2 · Implicit curve Implicit curves U3 · 3.3 · Inverse relationship Inverse-function relationships U4 · 4.2 · Motion Particle motion U4 · 4.2 · Motion Position / velocity / acceleration U4 · 4.5 · Related rates Related-rates geometry U4 · 4.6 · Linearization Local linearity U5 · 5.1 · Derivative analysis f / f′ / f″ U5 · 5.2 · Derivative analysis Extrema U5 · 5.3 · Derivative analysis Increasing / decreasing U5 · 5.6 · Derivative analysis Concavity U5 · 5.10 · Optimization Optimization U6 · 6.2 · Riemann sum Riemann sums U6 · 6.2 · Riemann sum Left / right / midpoint U6 · 6.2 · Riemann sum Changing n U6 · 6.3 · Accumulation / FTC Accumulation functions U6 · 6.4 · Accumulation / FTC Fundamental Theorem of Calculus U7 · 7.3 · Slope field Slope fields U7 · 7.4 · Slope field Draggable initial condition U7 · 7.2 · Solution curve Solution curves U7 · 7.5 · Euler method Euler approximation U8 · 8.4 · Area between curves Area between curves U8 · 8.4 · Area between curves Movable bounds U8 · 8.9 · Volume slice Disks U8 · 8.10 · Volume slice Washers U8 · 8.7 · Volume slice Cross sections
Course scope BC unlocks Units 9–10
AP Calculus AB AP Calculus BC
Unit All units Unit 1 · Limits and Continuity Unit 2 · Differentiation Unit 3 · Composite and Implicit Functions Unit 4 · Contextual Applications Unit 5 · Analytical Applications of Differentiation Unit 6 · Integration and Accumulation of Change Unit 7 · Differential Equations Unit 8 · Applications of Integration Topic All topics 1.3 · One-Sided Limits 1.4 · Limits and Function Values 1.5 · Limits That Fail to Exist 1.10 · Infinite Limits and Vertical Asymptotes 1.13 · Defining Continuity at a Point 1.12 · Limits at Infinity and Horizontal Asymptotes 2.1 · Average and Instantaneous Rates of Change 2.2 · The Difference Quotient 2.3 · Estimating Derivatives 2.4 · The Derivative as a Function 3.1 · The Chain Rule 3.2 · Implicit Differentiation 3.3 · Differentiating Inverse Functions 4.2 · Straight-Line Motion 4.5 · Solving Related Rates 4.6 · Local Linearity and Linearization 5.1 · Using f, f′, and f″ 5.2 · Extreme Value and Critical Points 5.3 · Intervals of Increase and Decrease 5.6 · Concavity and Inflection Points 5.10 · Optimization Problems 6.2 · Approximating Areas with Riemann Sums 6.3 · Accumulation Functions 6.4 · The Fundamental Theorem of Calculus 7.3 · Sketching Slope Fields 7.4 · Reasoning Using Slope Fields 7.2 · Verifying Solutions 7.5 · Euler’s Method 8.4 · Area Between Curves 8.9 · Volume with Disk Method 8.10 · Volume with Washer Method 8.7 · Volumes with Cross Sections Lab type All types Approach Discontinuity Continuity Secant / tangent Derivative graph Composition Implicit curve Inverse relationship Motion Related rates Linearization Derivative analysis Optimization Riemann sum Accumulation / FTC Slope field Solution curve Euler method Area between curves Volume slice
Units 1–8 · AB catalog
36 shown U1 · 1.3 · Approach Left and right limits U1 · 1.4 · Discontinuity Holes and removable discontinuities U1 · 1.5 · Discontinuity Jump discontinuities U1 · 1.10 · Discontinuity Infinite discontinuities U1 · 1.13 · Continuity Continuity at a point U1 · 1.12 · Approach Limits at infinity U2 · 2.1 · Secant / tangent Secant to tangent U2 · 2.2 · Secant / tangent Changing h U2 · 2.3 · Secant / tangent Derivative at a point U2 · 2.4 · Derivative graph Derivative as a function U3 · 3.1 · Composition Chain-rule composition U3 · 3.2 · Implicit curve Implicit curves U3 · 3.3 · Inverse relationship Inverse-function relationships U4 · 4.2 · Motion Particle motion U4 · 4.2 · Motion Position / velocity / acceleration U4 · 4.5 · Related rates Related-rates geometry U4 · 4.6 · Linearization Local linearity U5 · 5.1 · Derivative analysis f / f′ / f″ U5 · 5.2 · Derivative analysis Extrema U5 · 5.3 · Derivative analysis Increasing / decreasing U5 · 5.6 · Derivative analysis Concavity U5 · 5.10 · Optimization Optimization U6 · 6.2 · Riemann sum Riemann sums U6 · 6.2 · Riemann sum Left / right / midpoint U6 · 6.2 · Riemann sum Changing n U6 · 6.3 · Accumulation / FTC Accumulation functions U6 · 6.4 · Accumulation / FTC Fundamental Theorem of Calculus U7 · 7.3 · Slope field Slope fields U7 · 7.4 · Slope field Draggable initial condition U7 · 7.2 · Solution curve Solution curves U7 · 7.5 · Euler method Euler approximation U8 · 8.4 · Area between curves Area between curves U8 · 8.4 · Area between curves Movable bounds U8 · 8.9 · Volume slice Disks U8 · 8.10 · Volume slice Washers U8 · 8.7 · Volume slice Cross sections
Unit 1 · Topic 1.3 · Approach
Approach one input from both directions and decide when the two-sided limit exists.
01 Prediction 02 Interactive model 03 Controls 04 Observation 05 Question 06 Explanation 01 Prediction
Choose a claim you can test with the visualization.
For the piecewise graph, will the left- and right-hand limits at x=1 agree?
Yes No Only if f(1) is undefined A graph cannot show this
Open the model →
Curriculum-aligned learning support — not an official school, university, or ministry-approved course, and not a replacement for official credit.
