AP Calculus mastery tools
Beyond the lessons: adaptive Level 1–5 practice, misconception remediation, spaced review, MCQ + free response, and separate AP / Advanced / Elite readiness.
Advanced Placement (AP)MathematicsAdvanced
AP Calculus AB/BC
The full College Board AP Calculus AB & BC course — ten units from limits to infinite series, taught with deep theory, adaptive self-verifying practice, misconception remediation, spaced review, MCQ + free-response, evidence-based readiness scoring, and optional Advanced Reasoning and Elite Challenge problems.
10
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111
lessons
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1Unit 1: Limits and Continuity0/16 lessons · Current
- ✓1.1 Change at an instant
- ✓1.2 Defining limits and notation
- ✓1.3 Estimating limits from graphs
- ✓1.4 Estimating limits from tables
- ✓1.5 Limit laws (algebraic properties)
- ✓1.6 Limits by algebraic manipulation
- ✓1.7 Selecting a procedure for a limit
- ✓1.8 The Squeeze (Sandwich) Theorem
- ✓1.9 Connecting representations of a limit
- ✓1.10 Types of discontinuities
- ✓1.11 Continuity at a point
- ✓1.12 Continuity over an interval
- ✓1.13 Removing removable discontinuities
- ✓1.14 Infinite limits and vertical asymptotes
- ✓1.15 Limits at infinity and horizontal asymptotes
- ✓1.16 The Intermediate Value Theorem
2Unit 2: Differentiation: Definition & Fundamental Properties0/10 lessons · Not started
- ✓2.1 Average and instantaneous rate at a point
- ✓2.2 Defining the derivative (first principles) & notation
- ✓2.3 Estimating the derivative at a point
- ✓2.4 Differentiability and continuity
- ✓2.5 The power rule
- ✓2.6 Constant, sum, difference & constant-multiple rules
- ✓2.7 Derivatives of sin, cos, eˣ, ln x
- ✓2.8 The product rule
- ✓2.9 The quotient rule
- ✓2.10 Derivatives of tan, cot, sec, csc
3Unit 3: Differentiation: Composite, Implicit & Inverse Functions0/6 lessons · Not started
4Unit 4: Contextual Applications of Differentiation0/7 lessons · Not started
5Unit 5: Analytical Applications of Differentiation0/12 lessons · Not started
- ✓5.1 The Mean Value Theorem
- ✓5.2 Extreme Value Theorem & critical points
- ✓5.3 Increasing and decreasing intervals
- ✓5.4 The first derivative test
- ✓5.5 The candidates test (absolute extrema)
- ✓5.6 Concavity and inflection points
- ✓5.7 The second derivative test
- ✓5.8 Sketching f and its derivatives
- ✓5.9 Connecting f, f′, and f″
- ✓5.10 Setting up optimization problems
- ✓5.11 Solving optimization problems
- ✓5.12 Behaviours of implicit relations
6Unit 6: Integration and Accumulation of Change0/14 lessons · Not started
- ✓6.1 Accumulation of change
- ✓6.2 Approximating area with Riemann sums
- ✓6.3 Riemann sums and definite-integral notation
- ✓6.4 The FTC and accumulation functions
- ✓6.5 Behaviour of accumulation functions
- ✓6.6 Properties of definite integrals
- ✓6.7 Evaluating definite integrals with the FTC
- ✓6.8 Antiderivatives and indefinite integrals
- ✓6.9 Integration by u-substitution
- ✓6.10 Long division & completing the square (BC)
- ✓6.11 Integration by parts (BC)
- ✓6.12 Integration with linear partial fractions (BC)
- ✓6.13 Improper integrals (BC)
- ✓6.14 Selecting an antidifferentiation technique
7Unit 7: Differential Equations0/9 lessons · Not started
- ✓7.1 Modeling situations with differential equations
- ✓7.2 Verifying solutions for differential equations
- ✓7.3 Sketching slope fields
- ✓7.4 Reasoning using slope fields
- ✓7.5 Approximating solutions using Euler's method (BC)
- ✓7.6 Finding general solutions using separation of variables
- ✓7.7 Finding particular solutions using initial conditions
- ✓7.8 Exponential models with differential equations
- ✓7.9 Logistic models with differential equations (BC)
8Unit 8: Applications of Integration0/13 lessons · Not started
- ✓8.1 Average value of a function
- ✓8.2 Position, velocity, acceleration via integrals
- ✓8.3 Accumulation in applied contexts
- ✓8.4 Area between curves (in terms of x)
- ✓8.5 Area between curves (in terms of y)
- ✓8.6 Area with more than two intersections
- ✓8.7 Volumes by cross section: squares & rectangles
- ✓8.8 Volumes by cross section: triangles & semicircles
- ✓8.9 Volume by discs (around an axis)
- ✓8.10 Volume by discs (around other axes)
- ✓8.11 Volume by washers (around an axis)
- ✓8.12 Volume by washers (around other axes)
- ✓8.13 Arc length & distance traveled (BC)
9Unit 9: Parametric, Polar & Vector-Valued Functions (BC)0/9 lessons · Not started
- ✓9.1 Differentiating parametric equations (BC)
- ✓9.2 Second derivatives of parametric equations (BC)
- ✓9.3 Arc length of a parametric curve (BC)
- ✓9.4 Differentiating vector-valued functions (BC)
- ✓9.5 Integrating vector-valued functions (BC)
- ✓9.6 Motion with parametric/vector functions (BC)
- ✓9.7 Polar coordinates and differentiation (BC)
- ✓9.8 Area inside a single polar curve (BC)
- ✓9.9 Area between two polar curves (BC)
10Unit 10: Infinite Sequences and Series (BC)0/15 lessons · Not started
- ✓10.1 Convergent and divergent series (BC)
- ✓10.2 Geometric series (BC)
- ✓10.3 The nth-term test for divergence (BC)
- ✓10.4 The integral test (BC)
- ✓10.5 Harmonic series and p-series (BC)
- ✓10.6 Comparison and limit-comparison tests (BC)
- ✓10.7 The alternating series test (BC)
- ✓10.8 The ratio test (BC)
- ✓10.9 Absolute vs conditional convergence (BC)
- ✓10.10 The alternating series error bound (BC)
- ✓10.11 Taylor polynomial approximations (BC)
- ✓10.12 The Lagrange error bound (BC)
- ✓10.13 Radius and interval of convergence (BC)
- ✓10.14 Taylor and Maclaurin series (BC)
- ✓10.15 Representing functions as power series (BC)
What is AP Calculus AB/BC?
Short answer
AP Calculus is a College Board college-level calculus course and exam offered in two versions. AP Calculus AB covers limits, differentiation, and integration (roughly a first semester of university calculus). AP Calculus BC includes all of AB plus additional topics — parametric, polar, and vector functions, and infinite sequences and series. Both build on strong pre-calculus.
Course overview
AP Calculus develops two central ideas — the derivative (instantaneous rate of change) and the integral (accumulation) — and the Fundamental Theorem that links them. AB focuses on limits, differentiation, and integration and their applications; BC adds series and parametric/polar/vector calculus. Both reward students who can represent a problem numerically, graphically, analytically, and verbally, and justify their reasoning.
What you will learn
- Evaluate limits and reason about continuity
- Differentiate using all standard rules and apply derivatives to motion, rates, and optimization
- Integrate and apply the Fundamental Theorem of Calculus
- Model with and solve basic differential equations
- Apply integration to area, volume, and accumulation
- BC only: work with parametric, polar, and vector functions, and infinite sequences and series
Who this course is for
- • Students taking AP Calculus AB or BC who want deep understanding and exam readiness
- • Strong Pre-Calculus 12 / Calculus 12 students ready for college-level rigour
- • Students who want adaptive practice targeted at their specific weak skills
Commonly difficult topics
- • Limits — understanding what a limit describes rather than just substituting
- • The chain rule and implicit differentiation — identifying inner and outer functions
- • Choosing an integration approach and setting up applications
- • BC series — selecting and justifying the right convergence test
How Dr. Shreyank Educare teaches this course
01 · Diagnose
Find the exact skill causing difficulty — not the chapter, the skill.
02 · Understand
Learn it through explanations, worked examples and visual reasoning.
03 · Practice
Work problems that adapt to how you are actually doing.
04 · Fix
A wrong answer becomes targeted remediation for the misconception behind it.
05 · Prove
Solve new, unseen problems independently before mastery is awarded.
Course curriculum
1Unit 1: Limits and Continuity16 topics
- — 1.1 Change at an instant
- — 1.2 Defining limits and notation
- — 1.3 Estimating limits from graphs
- — 1.4 Estimating limits from tables
- — 1.5 Limit laws (algebraic properties)
- — 1.6 Limits by algebraic manipulation
- — 1.7 Selecting a procedure for a limit
- — 1.8 The Squeeze (Sandwich) Theorem
- — 1.9 Connecting representations of a limit
- — 1.10 Types of discontinuities
- — 1.11 Continuity at a point
- — 1.12 Continuity over an interval
- — 1.13 Removing removable discontinuities
- — 1.14 Infinite limits and vertical asymptotes
- — 1.15 Limits at infinity and horizontal asymptotes
- — 1.16 The Intermediate Value Theorem
2Unit 2: Differentiation: Definition & Fundamental Properties10 topics
- — 2.1 Average and instantaneous rate at a point
- — 2.2 Defining the derivative (first principles) & notation
- — 2.3 Estimating the derivative at a point
- — 2.4 Differentiability and continuity
- — 2.5 The power rule
- — 2.6 Constant, sum, difference & constant-multiple rules
- — 2.7 Derivatives of sin, cos, eˣ, ln x
- — 2.8 The product rule
- — 2.9 The quotient rule
- — 2.10 Derivatives of tan, cot, sec, csc
3Unit 3: Differentiation: Composite, Implicit & Inverse Functions6 topics
- — 3.1 The chain rule
- — 3.2 Implicit differentiation
- — 3.3 Differentiating inverse functions
- — 3.4 Derivatives of inverse trigonometric functions
- — 3.5 Selecting procedures for derivatives
- — 3.6 Higher-order derivatives
4Unit 4: Contextual Applications of Differentiation7 topics
- — 4.1 Interpreting the derivative in context
- — 4.2 Straight-line motion: position, velocity, acceleration
- — 4.3 Rates of change in applied contexts
- — 4.4 Introduction to related rates
- — 4.5 Solving related-rates problems
- — 4.6 Local linearity and linearization
- — 4.7 L'Hôpital's Rule for indeterminate forms
5Unit 5: Analytical Applications of Differentiation12 topics
- — 5.1 The Mean Value Theorem
- — 5.2 Extreme Value Theorem & critical points
- — 5.3 Increasing and decreasing intervals
- — 5.4 The first derivative test
- — 5.5 The candidates test (absolute extrema)
- — 5.6 Concavity and inflection points
- — 5.7 The second derivative test
- — 5.8 Sketching f and its derivatives
- — 5.9 Connecting f, f′, and f″
- — 5.10 Setting up optimization problems
- — 5.11 Solving optimization problems
- — 5.12 Behaviours of implicit relations
6Unit 6: Integration and Accumulation of Change14 topics
- — 6.1 Accumulation of change
- — 6.2 Approximating area with Riemann sums
- — 6.3 Riemann sums and definite-integral notation
- — 6.4 The FTC and accumulation functions
- — 6.5 Behaviour of accumulation functions
- — 6.6 Properties of definite integrals
- — 6.7 Evaluating definite integrals with the FTC
- — 6.8 Antiderivatives and indefinite integrals
- — 6.9 Integration by u-substitution
- — 6.10 Long division & completing the square (BC)
- — 6.11 Integration by parts (BC)
- — 6.12 Integration with linear partial fractions (BC)
- — 6.13 Improper integrals (BC)
- — 6.14 Selecting an antidifferentiation technique
7Unit 7: Differential Equations9 topics
- — 7.1 Modeling situations with differential equations
- — 7.2 Verifying solutions for differential equations
- — 7.3 Sketching slope fields
- — 7.4 Reasoning using slope fields
- — 7.5 Approximating solutions using Euler's method (BC)
- — 7.6 Finding general solutions using separation of variables
- — 7.7 Finding particular solutions using initial conditions
- — 7.8 Exponential models with differential equations
- — 7.9 Logistic models with differential equations (BC)
8Unit 8: Applications of Integration13 topics
- — 8.1 Average value of a function
- — 8.2 Position, velocity, acceleration via integrals
- — 8.3 Accumulation in applied contexts
- — 8.4 Area between curves (in terms of x)
- — 8.5 Area between curves (in terms of y)
- — 8.6 Area with more than two intersections
- — 8.7 Volumes by cross section: squares & rectangles
- — 8.8 Volumes by cross section: triangles & semicircles
- — 8.9 Volume by discs (around an axis)
- — 8.10 Volume by discs (around other axes)
- — 8.11 Volume by washers (around an axis)
- — 8.12 Volume by washers (around other axes)
- — 8.13 Arc length & distance traveled (BC)
9Unit 9: Parametric, Polar & Vector-Valued Functions (BC)9 topics
- — 9.1 Differentiating parametric equations (BC)
- — 9.2 Second derivatives of parametric equations (BC)
- — 9.3 Arc length of a parametric curve (BC)
- — 9.4 Differentiating vector-valued functions (BC)
- — 9.5 Integrating vector-valued functions (BC)
- — 9.6 Motion with parametric/vector functions (BC)
- — 9.7 Polar coordinates and differentiation (BC)
- — 9.8 Area inside a single polar curve (BC)
- — 9.9 Area between two polar curves (BC)
10Unit 10: Infinite Sequences and Series (BC)15 topics
- — 10.1 Convergent and divergent series (BC)
- — 10.2 Geometric series (BC)
- — 10.3 The nth-term test for divergence (BC)
- — 10.4 The integral test (BC)
- — 10.5 Harmonic series and p-series (BC)
- — 10.6 Comparison and limit-comparison tests (BC)
- — 10.7 The alternating series test (BC)
- — 10.8 The ratio test (BC)
- — 10.9 Absolute vs conditional convergence (BC)
- — 10.10 The alternating series error bound (BC)
- — 10.11 Taylor polynomial approximations (BC)
- — 10.12 The Lagrange error bound (BC)
- — 10.13 Radius and interval of convergence (BC)
- — 10.14 Taylor and Maclaurin series (BC)
- — 10.15 Representing functions as power series (BC)
A sample learning experience
Differentiate y = sin(x²) with respect to x. (Original example, not an exam question.)
- 1. Recognize a composite function: the outer function is sin(u), the inner is u = x².
- 2. Chain rule: dy/dx = (derivative of the outer at u) × (derivative of the inner).
- 3. Derivative of sin(u) is cos(u); derivative of x² is 2x.
- 4. So dy/dx = cos(x²) · 2x = 2x·cos(x²).
Frequently asked questions
- What is the difference between AP Calculus AB and BC?
- AB covers limits, differentiation, and integration — about a first semester of college calculus. BC includes everything in AB plus parametric, polar, and vector functions and infinite sequences and series. BC moves faster and covers more.
- How difficult is AP Calculus?
- It is rigorous but very learnable if your pre-calculus is solid. Most difficulty comes from weak algebra/trig foundations or from rushing setup; targeted practice on the specific skill you miss makes it manageable.
- Should I take AB or BC?
- AB is a strong, thorough single course. BC suits students who are confident with calculus concepts and want the additional series and parametric/polar topics (and often a BC subscore that reflects AB-level material).
- What prerequisites do I need?
- Fluency with functions, logarithms, and trigonometry — the content of Pre-Calculus 12. Calculus 12 is excellent additional preparation.
- How does Adaptive Mastery work?
- Practice adapts to demonstrated performance: a wrong answer never raises mastery, difficulty adjusts to what you are ready for, and you must solve new, unseen problems unaided before a skill is marked mastered.
- Does Dr. Shreyank Educare include FRQ and exam practice?
- Yes — free-response practice with feedback, multiple-choice practice, adaptive Level 1–5 drills, spaced review, and a timed exam simulator that estimates readiness.
- Are Advanced Reasoning and Elite Challenge required for AP readiness?
- No. AP readiness is measured separately from Advanced Reasoning and Elite Challenge, which are optional enrichment. You do not need them to be AP-ready.
Related courses
Curriculum & sources
Content source: original lessons, worked examples and diagrams created for Dr. Shreyank Educare.
Curriculum alignment: Advanced Placement (AP) Mathematics (College Board AP framework).
- Aligned with College Board — AP Calculus AB course framework
- Aligned with College Board — AP Calculus BC course framework
Curriculum-aligned learning support — not an official school, ministry, or College Board course, and not a replacement for official credit.
Last updated . Authored and maintained in-house by the Dr. Shreyank Educare teaching team — see our editorial standards, methodology and source policy.