ap calculus sequences homework

AP Precalculus

Ap precalculus course and exam description.

This is the core document for this course.

Course Overview

AP Precalculus prepares students for other college-level mathematics and science courses. Through regular practice, students build deep mastery of modeling and functions, and they examine scenarios through multiple representations. The course framework delineates content and skills common to college precalculus courses that are foundational for careers in mathematics, physics, biology, health science, social science, and data science.

The Benefits of AP Precalculus

AP Precalculus gives any student ready for high school precalculus the opportunity to earn college credit and/or placement and stand out to colleges. 

To learn more about the benefits of this new AP subject and find out how to offer it at your school, visit Adopt AP Precalculus.

Course and Exam Description

This is the core document for this course. Unit guides clearly lay out the course content and skills and recommend sequencing and pacing for them throughout the year.

AP Precalculus Course and Exam Description Clarification

This document provides teachers with clarifications, corrections, and guidance for the AP Precalculus Course and Exam Description.

Course Resources

Ap precalculus course overview.

This resource provides a succinct description of the course and exam.

AP Precalculus Course at a Glance

Excerpted from the AP Precalculus Course and Exam Description, the Course at a Glance document outlines the topics and skills covered in the AP Precalculus course, along with suggestions for sequencing.

Course Content

The course framework included in the AP Precalculus Course and Exam Description is organized into four commonly taught units of study that offer one possible sequence for the course. 

Units 1, 2, and 3 are assessed on the end-of-course AP Exam and describe what students should know and be able to do to qualify for college credit or placement. Unit 4  is not assessed on the exam and  describes additional topics you might include based on state or local requirements.

You have the flexibility to organize the course content as you like. You can also augment the framework to meet state and local requirements.    

Mathematical Practices

The AP Precalculus framework outlines distinct skills, associated with three mathematical practices, that students should practice throughout the year—skills that will help them learn to think and act like mathematicians.

AP and Higher Education

Higher education professionals play a key role in developing AP courses and exams, setting credit and placement policies, and scoring student work.

The AP Higher Education section features research about AP, resources on evaluating AP courses for credit and placement, and information on how to get involved.

This chart  shows recommendations for what cut score should be demonstrated to earn college credit and how many semesters of credit should be awarded. Your students can look up credit and placement policies for colleges and universities on the  AP Credit Policy Search .

AP Precalculus Course Development

Every AP course is designed in consultation with college faculty and experienced high school teachers. In an ongoing effort to maintain alignment with best practices in college-level learning, AP courses and exams emphasize research-based curricula aligned with higher education expectations. College faculty and experienced high school teachers guide the development of the AP course framework, which defines what students must know and be able to do to earn a qualifying score on the AP Exam, thus conferring college credit or placement.

As part of the course development process for AP Precalculus, the AP Program gathered course research through examination of college syllabi, analysis of textbooks and pedagogical research, and content advisory sessions with college faculty. Then, an advisory board and writing team collaborated on the course framework based on these research inputs.

Meet the AP Precalculus Development Committee

The AP Program is unique in its reliance on Development Committees. These committees, made up of an equal number of college faculty and experienced secondary AP teachers from across the country, are essential to the preparation of AP course curricula and exams.

AP Precalculus Development Committee

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AP®︎/College Calculus AB

Meet an ap®︎ teacher who uses ap®︎ calculus in his classroom, unit 1: limits and continuity, unit 2: differentiation: definition and basic derivative rules, unit 3: differentiation: composite, implicit, and inverse functions, unit 4: contextual applications of differentiation, unit 5: applying derivatives to analyze functions, unit 6: integration and accumulation of change, unit 7: differential equations, unit 8: applications of integration, unit 9: ap calculus ab solved free response questions from past exams, unit 10: ap®︎ calculus ab standards mappings.

Course Progress

I will keep this page updated as we complete our sessions so that you can see where we stand in terms of content.

Unit 1: Limits and Continuity

• 1.1 Introducing Calculus: Can Change Occur at an Instant?
• 1.2 Defining Limits and Using Limit Notation
• 1.3 Estimating Limit Values from Graphs
• 1.4 Estimating Limit Values from Tables
• 1.5 Determining Limits Using Algebraic Properties of Limits
• 1.6 Determining Limits Using Algebraic Manipulation
• 1.7 Selecting Procedures for Determining Limits
• 1.8 Determining Limits Using the Squeeze Theorem
• 1.9 Connecting Multiple Representations of Limits
• 1.10 Exploring Types of Discontinuities
• 1.11 Defining Continuity at a Point
• 1.12 Confirming Continuity over an Interval
• 1.13 Removing Discontinuities
• 1.14 Connecting Infinite Limits and Vertical Asymptotes
• 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
• 1.16 Working with the Intermediate Value Theorem (IVT)

Unit 2: Differentiation: Definition and Fundamental Properties

• 2.1 Defining Average and Instantaneous Rates of Change at a Point
• 2.2 Defining the Derivative of a Function and Using Derivative Notation
• 2.3 Estimating Derivatives of a Function at a Point
• 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
• 2.5 Applying the Power Rule
• 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
• 2.7 Derivatives of ( \cos x ), ( \sin x ), ( e^x ), and ( \ln x )
• 2.8 The Product Rule
• 2.9 The Quotient Rule
• 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

• 3.1 The Chain Rule
• 3.2 Implicit Differentiation
• 3.3 Differentiating Inverse Functions
• 3.4 Differentiating Inverse Trigonometric Functions
• 3.5 Selecting Procedures for Calculating Derivatives
• 3.6 Calculating Higher-Order Derivatives

Unit 4: Contextual Applications of Differentiation

• 4.1 Interpreting the Meaning of the Derivative in Context
• 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
• 4.3 Rates of Change in Applied Contexts Other Than Motion
• 4.4 Introduction to Related Rates
• 4.5 Solving Related Rates Problems
• 4.6 Approximating Values of a Function Using Local Linearity and Linearization
• 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms

Unit 5: Analytical Applications of Differentiation

• 5.1 Using the Mean Value Theorem
• 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
• 5.3 Determining Intervals on Which a Function is Increasing or Decreasing
• 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
• 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
• 5.6 Determining Concavity of Functions over Their Domains
• 5.7 Using the Second Derivative Test to Determine Extrema
• 5.8 Sketching Graphs of Functions and Their Derivatives
• 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
• 5.10 Introduction to Optimization Problems
• 5.11 Solving Optimization Problems
• 5.12 Exploring Behaviors of Implicit Relations

Unit 6: Integration and Accumulation of Change

• 6.1 Exploring Accumulations of Change
• 6.2 Approximating Areas with Riemann Sums
• 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
• 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
• 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
• 6.6 Applying Properties of Definite Integrals
• 6.7 The Fundamental Theorem of Calculus and Definite Integrals
• 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
• 6.9 Integrating Using Substitution
• 6.10 Integrating Functions Using Long Division and Completing the Square
• 6.11 Integrating Using Integration by Parts (BC Only)
• 6.12 Integrating Using Linear Partial Fractions (BC Only)
• 6.13 Evaluating Improper Integrals (BC Only)
• 6.14 Selecting Techniques for Antidifferentiation

Unit 7: Differential Equations

• 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 Only)
• 7.6 Finding General Solutions Using Separation of Variables
• 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
• 7.8 Exponential Models with Differential Equations
• 7.9 Logistic Models with Differential Equations

Unit 8: Applications of Integration

• 8.1 Finding the Average Value of a Function on an Interval
• 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
• 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
• 8.4 Finding the Area Between Curves Expressed as Functions of (x)
• 8.5 Finding the Area Between Curves Expressed as Functions of (y)
• 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
• 8.7 Volumes with Cross Sections: Squares and Rectangles
• 8.8 Volumes with Cross Sections: Triangles and Semicircles
• 8.9 Volume with Disc Method: Revolving Around the (x)- or (y)-Axis
• 8.10 Volume with Disc Method: Revolving Around Other Axes
• 8.11 Volume with Washer Method: Revolving Around the (x)- or (y)-Axis
• 8.12 Volume with Washer Method: Revolving Around Other Axes
• 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC Only)

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC Only)

• 9.1 Defining and Differentiating Parametric Equations
• 9.2 Second Derivatives of Parametric Equations
• 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
• 9.4 Defining and Differentiating Vector-Valued Functions
• 9.5 Integrating Vector-Valued Functions
• 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
• 9.7 Defining Polar Coordinates and Differentiating in Polar Form
• 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
• 9.9 Finding the Area of the Region Bounded by Two Polar Curves

Unit 10: Infinite Sequences and Series (BC Only)

• 10.1 Defining Convergent and Divergent Infinite Series
• 10.2 Working with Geometric Series
• 10.3 The (n)th Term Test for Divergence
• 10.4 Integral Test for Convergence
• 10.5 Harmonic Series and (p)-Series
• 10.6 Comparison Tests for Convergence
• 10.7 Alternating Series Test for Convergence
• 10.8 Ratio Test for Convergence
• 10.9 Determining Absolute or Conditional Convergence
• 10.10 Alternating Series Error Bound
• 10.11 Finding Taylor Polynomial Approximations of Functions
• 10.12 Lagrange Error Bound
• 10.13 Radius and Interval of Convergence of Power Series
• 10.14 Finding Taylor or Maclaurin Series for a Function
• 10.15 Representing Functions as Power Series