# 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

### Building Blocks

Limits
Reasoning with Limits and Continuity
• Connecting Limits and Graphical Behavior
• Computing Limits
• Defining and Reasoning with Continuity
• Applying the Existence Theorems
Derivatives
Optimization
• 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

## Year-Round Support for Teachers and Students

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.