AP Insight: Calculus

A Road Map to AP Success

AP educators leveraged years of teaching experience, data, and student work to identify key challenge areas – the concepts and skills most foundational to success in college and AP Calculus (AB and BC). Teachers use AP Insight to create a road map in their own syllabus, target challenge areas, and help students connect building blocks to master the course. 

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Strategically Focus Instruction

AP Calculus Overview of Challenge Areas

Big Ideas

Challenge Areas

Building Blocks

Reasoning with Limits and Continuity
  • Connecting Limits and Graphical Behavior
  • Computing Limits
  • Defining and Reasoning with Continuity
  • Applying the Existence Theorems
  • Optimization: Setting
    Up the Problem
  • Optimization: Finding
    Critical Points
  • Optimization: Justifying
    Extreme Values
Derivatives and Properties of Functions
  • The First Derivative Test
  • Extrema
  • Concavity and Inflection
  • Analysis of Derivative Functions
Understanding and Calculating Derivatives
  • Definition of Derivative
  • Differentiating Combined Algebraic Functions
  • Differentiating Transcendental Functions
  • Understanding the Derivative as a Slope
  • Understanding the Derivative as a Rate of Change
Related Rates of Change
  • Geometric Problems: One Quantity Varying
  • Geometric Problems: Several Quantities Varying
  • Non-Geometric Related Rates Problems
Integrals and the Fundamental Theorem of Calculus
Using Visual Representations to Understand Area and Volume
  • Area with Vertical Slices
  • Area with Horizontal and Vertical Slices
  • Volume of Solids with Known Cross Sections
  • Volume of Solids of Revolution
Applications of Definite Integrals
  • Calculating Average Value of a Function
  • Understanding Accumulation of Rates of Change
  • Interpreting Notation in the Context of a Problem
  • Calculating Quantities Associated with Motion
Calculating Antiderivatives and Evaluating Integrals 
  • Antiderivatives of Algebraic Functions
  • Finding Antiderivatives Using Substitution
  • Antiderivatives of Transcendental Functions
  • Antidifferentiation Techniques
  • Evaluate Definite Integrals
Functions Defined by Integrals
  • Derivatives of Functions Defined by Integrals
  • Analyze g(x) Through Accumulation of Signed Areas
  • Interpreting Behavior of g(x) from a Graph of f(t)
  • Finding Local Extrema or Inflection Points for g
Riemann Sums and Definite Integrals
  • Geometric Connections
  • Approximating Area with Riemann Sums
  • The Definite Integral as the Limit of Riemann Sums


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AP teachers and higher education faculty developed classroom-tested tools and resources to help you focus, gain insight, and increase impact on students as you prepare, teach, assess, and act on the challenge areas in AP.


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