Grade 11 AP Calculus AB: One-Semester Course
An 18-week, 72-session self-learning course in AP Calculus AB for accelerated Grade 11 students. The course follows the eight units of the College Board AP Calculus AB Course and Exam Description: limits and continuity, differentiation, contextual and analytical applications of derivatives, integration and accumulation of change, differential equations, and applications of integration. Each session combines teacher-style explanations, worked examples, self-checks, and AP-style practice, supported by unit mastery assessments, a midterm, and a full-length AP-style final exam.
📚 Course Curriculum
Session Focus. Welcome to AP Calculus AB. In this first session you will see the central question of calculus: how can something change at a …
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Session Focus. In Session 1 you watched secant slopes settle toward a single number. Today you give that idea a name and a notation: the …
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Session Focus. Graphs show limits visually, but you will not always have a graph. Today you estimate limits from tables of values, compute one-sided limits …
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Session Focus. Graphs and tables give estimates. Today you begin finding limits exactly. You will learn the algebraic properties of limits, use direct substitution correctly, …
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Session Focus. Session 4 ended with the indeterminate form \(\frac{0}{0}\). Today you learn the standard algebraic tools that resolve it: factoring, rationalizing with a conjugate, …
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Session Focus. Some limits cannot be found by algebra alone. When a function wiggles too much to factor or simplify, you can trap it between …
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Session Focus. A limit tells you where a function is heading. Continuity asks whether the function actually arrives there. Today you define continuity at a …
Open trial session →Session Focus. Session 7 tested continuity one point at a time. Today you scale up: you decide where a function is continuous on whole intervals, …
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Session Focus. Some functions do not settle toward a number; they grow without bound. Today you describe that behavior with infinite limits, find vertical asymptotes, …
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Session Focus. So far every limit has been about \(x\) approaching a finite number. Today \(x\) grows without bound. Limits at infinity describe the end …
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Session Focus. Continuity is not just a definition to check. It has consequences. Today you meet the first major theorem of the course, the Intermediate …
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Session Focus. This session closes Unit 1. You will review the unit's key ideas, complete one AP-style free-response question with a scoring guide, and then …
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Session Focus. Unit 2 begins with the question that opened the course: how fast is something changing at a single instant? In Unit 1 you …
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Session Focus. Session 13 found rates of change one point at a time. Today you find them at every point at once. The result is …
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Session Focus. Not every derivative comes from a formula. On the AP exam you will estimate derivatives from tables and graphs, and you must know …
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Session Focus. Computing every derivative from the limit definition is slow. Today you learn the first differentiation rules, which turn that work into quick pattern …
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Session Focus. Today you add four essential functions to your derivative toolkit: sine, cosine, the natural exponential function, and the natural logarithm. You will see …
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Session Focus. The derivative of a sum is the sum of the derivatives. It is tempting to assume the same pattern holds for products, but …
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Session Focus. Today you complete the basic differentiation rules. You will learn the quotient rule and use it to find the derivatives of the remaining …
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Session Focus. This session closes Unit 2. You will review the definition of the derivative and the basic rules, complete an AP-style free-response question with …
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Session Focus. Most functions in calculus are built by putting one function inside another: \(\sin(x^2)\), \(e^{3x}\), \((3x^2 + 1)^5\). None of the Unit 2 rules …
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Session Focus. Real derivative problems rarely use one rule at a time. Today you apply the chain rule to functions with several layers, and combine …
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Session Focus. Not every curve is the graph of a function \(y = f(x)\). A circle such as \(x^2 + y^2 = 25\) fails the …
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Session Focus. If you know the derivative of a function, you can find the derivative of its inverse without ever finding a formula for the …
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Session Focus. The inverse trigonometric functions answer the question "which angle has this sine, cosine, or tangent?" Today you derive their derivatives with implicit differentiation …
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Session Focus. You now know every differentiation rule in AP Calculus AB. Today you practice choosing the right rule quickly and then differentiate again: second …
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Session Focus. This session closes Unit 3. You will review the chain rule, implicit differentiation, inverse function derivatives, and higher-order derivatives; complete an AP-style free-response …
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Session Focus. Unit 4 is about meaning. On the AP exam, a large share of free-response points come from explaining what a derivative tells you …
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Session Focus. A particle moving along a line is the AP exam's favorite context for derivatives. Today you connect position, velocity, and acceleration, decide when …
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Session Focus. When two quantities are linked by an equation and both change over time, their rates of change are linked too. Today you learn …
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Session Focus. Session 30 introduced the core move of related rates. Today you solve the classic multi-step problems: a sliding ladder, a filling cone, a …
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Session Focus. Zoom in far enough on a differentiable curve and it looks like its tangent line. Today you use that fact twice: to approximate …
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Session Focus. This session closes Unit 4. You will review how derivatives describe change in context, complete an AP-style free-response question with a scoring guide, …
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Session Focus. Unit 5 uses derivatives to analyze the shape of graphs. It begins with a theorem that connects average rate of change to instantaneous …
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Session Focus. Where does a function reach its highest and lowest values? Today you learn when such values are guaranteed to exist (the Extreme Value …
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Session Focus. The sign of the derivative tells you whether a function is rising or falling. Today you use sign charts of \(f'\) to find …
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Session Focus. Session 35 told you where absolute extrema can occur. Today you turn that into a complete procedure, the Candidates Test (also called the …
Open trial session →Session Focus. The first derivative tells you whether a graph rises or falls. The second derivative tells you how it bends. Today you study concavity, …
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Session Focus. This session pulls Unit 5 together. You will move fluently between the graphs of \(f\), \(f'\), and \(f''\), read the behavior of \(f\) …
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Session Focus. Optimization is where calculus pays off in practical decisions: the largest area for a fixed amount of fencing, the box with the greatest …
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Session Focus. Today you extend optimization to problems in three dimensions and in business, then apply the tools of Unit 5 to curves defined implicitly: …
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Session Focus. This session closes Unit 5, the most heavily weighted differentiation unit on the AP Calculus AB exam. You will review the analytical tools …
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Session Focus. You have completed Units 1 through 5, which together account for roughly 54 to 70 percent of the AP Calculus AB exam, based …
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Session Focus. This is the midterm examination for Units 1 through 5. It follows the structure of the AP Calculus AB exam in shortened form. …
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Session Focus. Unit 6 turns the main question of calculus around. Derivatives start with an amount and find its rate of change. Integrals start with …
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Session Focus. When the area under a curve is not made of simple shapes, you can approximate it with rectangles or trapezoids. Today you compute …
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Session Focus. Approximations improve as rectangles get thinner. Today you take that idea to its limit. You will use summation notation, write Riemann sums in …
Open trial session →Session Focus. What happens if the upper limit of a definite integral is a variable? The integral becomes a function, called an accumulation function. Today …
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Session Focus. An accumulation function \(g(x) = \displaystyle\int_a^x f(t)\,dt\) is a function like any other, so all of Unit 5 applies to it. The twist …
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Session Focus. Definite integrals obey a small set of algebraic rules. These rules let you combine, split, and rearrange integrals, and they appear on nearly …
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Session Focus. Riemann sums and geometry can only take you so far. Today you meet the result that makes integrals easy to compute: the Fundamental …
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Session Focus. Session 51 used antiderivatives to evaluate definite integrals. Today you study antiderivatives in their own right. You will write indefinite integrals with the …
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Session Focus. The chain rule produces derivatives such as \(2x\cos(x^2)\). To integrate them, you need to run the chain rule backward. That technique is called …
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Session Focus. Some rational integrands do not match any basic rule or substitution as written. Two algebraic tools fix this: long division, when the numerator's …
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Session Focus. You now have every integration technique in AP Calculus AB. The hard part on an exam is not carrying out a technique, but …
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Session Focus. This session closes Unit 6, the most heavily weighted unit on the AP Calculus AB exam. You will review accumulation, Riemann sums, the …
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Session Focus. Unit 7 studies equations that involve derivatives. A differential equation describes how a quantity changes, and solving it tells you the quantity itself. …
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Course Syllabus
This self-paced AP Calculus AB course follows a 72-session sequence over 18 weeks, with four sessions per week. Work through the units in order, because each unit builds directly on the one before it.
Course overview
AP Calculus AB is equivalent to a first-semester college calculus course. The sequence follows the eight units of the College Board AP Calculus AB Course and Exam Description (CED). Most schools teach this content over a full year, so this one-semester version moves quickly: each session carries roughly the content of two standard class periods. Plan for about 90 minutes per session, plus independent review.
Prerequisites
Students should have completed Precalculus or an equivalent course, such as Grade 11 Math: Advanced Algebra and Precalculus Foundations. Required skills include function notation and transformations; polynomial, rational, exponential, and logarithmic functions; the unit circle and trigonometric identities; and fluent algebraic manipulation.
Materials
- A graphing calculator (for example, TI-84 Plus CE or TI-Nspire CX) or the Desmos graphing calculator. A graphing calculator is required on the calculator-active parts of the AP exam.
- A notebook for full written solutions. AP free-response scoring rewards correct reasoning, notation, and units, not only final answers.
Course sequence
Unit 1 - Limits and Continuity (Sessions 1-12)
- Week 1, Session 1: Introducing Calculus: Change and the Tangent Problem
- Week 1, Session 2: Limit Notation and Estimating Limits from Graphs
- Week 1, Session 3: Estimating Limits from Tables and One-Sided Limits
- Week 1, Session 4: Algebraic Properties of Limits
- Week 2, Session 1: Algebraic Manipulation: Factoring and Rationalizing
- Week 2, Session 2: The Squeeze Theorem and Trigonometric Limits
- Week 2, Session 3: Types of Discontinuity and Continuity at a Point
- Week 2, Session 4: Removing Discontinuities and Continuity over Intervals
- Week 3, Session 1: Infinite Limits and Vertical Asymptotes
- Week 3, Session 2: Limits at Infinity and Horizontal Asymptotes
- Week 3, Session 3: The Intermediate Value Theorem
- Week 3, Session 4: Unit 1 Mastery Assessment: Limits and Continuity
Unit 2 - Differentiation: Definition and Fundamental Properties (Sessions 13-20)
- Week 4, Session 1: Average and Instantaneous Rates of Change
- Week 4, Session 2: The Derivative as a Limit of a Difference Quotient
- Week 4, Session 3: Estimating Derivatives, Differentiability, and Continuity
- Week 4, Session 4: Power, Constant, Sum, and Difference Rules
- Week 5, Session 1: Derivatives of sin x, cos x, e^x, and ln x
- Week 5, Session 2: The Product Rule
- Week 5, Session 3: The Quotient Rule and Remaining Trigonometric Derivatives
- Week 5, Session 4: Unit 2 Mastery Assessment: Differentiation Fundamentals
Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions (Sessions 21-27)
- Week 6, Session 1: The Chain Rule
- Week 6, Session 2: Chain Rule with Multiple Layers and Mixed Rules
- Week 6, Session 3: Implicit Differentiation
- Week 6, Session 4: Derivatives of Inverse Functions
- Week 7, Session 1: Derivatives of Inverse Trigonometric Functions
- Week 7, Session 2: Selecting Procedures and Higher-Order Derivatives
- Week 7, Session 3: Unit 3 Mastery Assessment: Composite, Implicit, and Inverse Functions
Unit 4 - Contextual Applications of Differentiation (Sessions 28-33)
- Week 7, Session 4: Interpreting the Derivative in Context and with Units
- Week 8, Session 1: Straight-Line Motion: Position, Velocity, and Acceleration
- Week 8, Session 2: Introduction to Related Rates
- Week 8, Session 3: Solving Related Rates Problems
- Week 8, Session 4: Local Linearity, Linear Approximation, and L'Hospital's Rule
- Week 9, Session 1: Unit 4 Mastery Assessment: Contextual Applications
Unit 5 - Analytical Applications of Differentiation (Sessions 34-42)
- Week 9, Session 2: The Mean Value Theorem
- Week 9, Session 3: The Extreme Value Theorem and Critical Points
- Week 9, Session 4: Increasing and Decreasing Functions and the First Derivative Test
- Week 10, Session 1: The Candidates Test for Absolute Extrema
- Week 10, Session 2: Concavity and the Second Derivative Test
- Week 10, Session 3: Connecting the Graphs of f, f', and f''
- Week 10, Session 4: Optimization Problems I
- Week 11, Session 1: Optimization Problems II and Behavior of Implicit Relations
- Week 11, Session 2: Unit 5 Mastery Assessment: Analytical Applications
Midterm (Sessions 43-44)
- Week 11, Session 3: Midterm Review: Units 1-5
- Week 11, Session 4: Midterm Exam: Units 1-5
Unit 6 - Integration and Accumulation of Change (Sessions 45-56)
- Week 12, Session 1: Accumulation and Area under a Rate Graph
- Week 12, Session 2: Left, Right, Midpoint, and Trapezoidal Sums
- Week 12, Session 3: Riemann Sums, Summation Notation, and the Definite Integral
- Week 12, Session 4: The Fundamental Theorem of Calculus and Accumulation Functions
- Week 13, Session 1: Interpreting the Behavior of Accumulation Functions
- Week 13, Session 2: Properties of Definite Integrals
- Week 13, Session 3: Evaluating Definite Integrals with Antiderivatives
- Week 13, Session 4: Indefinite Integrals and Basic Antiderivative Rules
- Week 14, Session 1: Integration by Substitution
- Week 14, Session 2: Integrating with Long Division and Completing the Square
- Week 14, Session 3: Selecting Integration Techniques: Mixed Practice
- Week 14, Session 4: Unit 6 Mastery Assessment: Integration and Accumulation
Unit 7 - Differential Equations (Sessions 57-60)
- Week 15, Session 1: Modeling with Differential Equations and Verifying Solutions
- Week 15, Session 2: Slope Fields
- Week 15, Session 3: Separation of Variables and Particular Solutions
- Week 15, Session 4: Exponential Models and Unit 7 Mastery Check
Unit 8 - Applications of Integration (Sessions 61-68)
- Week 16, Session 1: Average Value of a Function
- Week 16, Session 2: Position, Velocity, and Acceleration Using Integrals
- Week 16, Session 3: Accumulation and Net Change in Applied Contexts
- Week 16, Session 4: Area between Curves (with respect to x and y)
- Week 17, Session 1: Area of Regions with More than Two Intersections
- Week 17, Session 2: Volumes with Known Cross Sections
- Week 17, Session 3: Volumes by the Disc and Washer Methods
- Week 17, Session 4: Unit 8 Mastery Assessment: Applications of Integration
AP Exam Review and Final (Sessions 69-72)
- Week 18, Session 1: Multiple-Choice Strategies and Calculator-Active Review
- Week 18, Session 2: Free-Response Practice: Rates, Tables, and Accumulation
- Week 18, Session 3: Free-Response Practice: Graph Analysis and Differential Equations
- Week 18, Session 4: Final Exam: Full-Length AP-Style Practice Test
Assessment and grading
- Lesson practice and self-checks: 20%
- Unit mastery assessments (Units 1-8): 40%
- Midterm exam (Units 1-5): 15%
- Final exam (full-length AP-style practice test): 25%
AP exam format (per the CED)
- Section I: 45 multiple-choice questions (Part A: 30 questions, no calculator; Part B: 15 questions, calculator required), 50% of the exam score.
- Section II: 6 free-response questions (Part A: 2 questions, calculator required; Part B: 4 questions, no calculator), 50% of the exam score.
- Check AP Central for the current exam date and delivery format before the review weeks.
Practice and completion
- Complete each session's worked examples before attempting independent practice.
- After every unit assessment, correct each wrong answer in writing and explain why the original method failed.
- Use the midterm and final as diagnostics: revisit the sessions linked to any weak skill before moving on.
Course Outcomes
By the end of this course, learners will be able to:
- Determine limits graphically, numerically, and analytically, and use limits to justify continuity and asymptotic behavior.
- Define the derivative as the limit of a difference quotient and interpret it as an instantaneous rate of change with correct units.
- Differentiate polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, composite, and implicitly defined functions.
- Apply derivatives to straight-line motion, related rates, linear approximation, L'Hospital's Rule, and optimization problems.
- Use the Mean Value Theorem, the Extreme Value Theorem, and the first and second derivative tests to analyze and justify the behavior of functions.
- Approximate definite integrals with Riemann and trapezoidal sums, and evaluate them using the Fundamental Theorem of Calculus and substitution.
- Interpret accumulation functions and definite integrals of rates of change in context.
- Solve separable differential equations, sketch and read slope fields, and model exponential growth and decay.
- Compute average value, net change, area between curves, and volumes by cross sections, discs, and washers.
- Communicate solutions with correct notation and written justification in the style required by AP free-response questions.
📝 Practice Questions
615 interactive questions with instant feedback and explanations.
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