Name: AP Calculus AB Textbook: Climb to 5
Author: VerityPrep

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AP Calculus AB textbook and AP Calc study guide: limits, derivatives, related rates, optimization, slope fields and volumes, plus revision in every chapter.

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AP Calculus AB

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Climb to 5 is an AP Calculus AB textbook for students who want to work through the whole course one topic at a time. It opens with a Fundamentals chapter that reviews what calculus relies on: inequalities and absolute value, exponents and factoring, functions and their graphs, composite and inverse functions, exponential and logarithmic functions, and trigonometry from the unit circle to identities.

The eight main chapters follow the AP Calculus AB units. Limits and Continuity builds limits from graphs, tables and algebra, the squeeze theorem, discontinuities, asymptotes and the Intermediate Value Theorem. Two differentiation chapters cover the definition of the derivative, the power, product and quotient rules, the chain rule, implicit differentiation and inverse functions. The applications chapters cover motion, related rates, linearization, L'Hospital's rule, the Mean Value Theorem, extrema, concavity and optimization. The book ends with integration and the Fundamental Theorem of Calculus, differential equations with slope fields and separation of variables, and applications of integration from average value and area between curves to volumes by cross sections, discs and washers.

Every topic is split into two parts, and every chapter ends with calculator-allowed and no-calculator revision. The book is read online on VerityPrep, where the topic tests are solved alongside the lessons, so it works both as a course textbook and as an AP Calculus AB review book and study guide.

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9 chapters · 215 topics

01-Limits and Continuity
1. Chapter 1 Overview Sample
2. 1.1 Introducing Calculus - Can Change Occur at an Instant? - Part 1
3. 1.1 Introducing Calculus - Can Change Occur at an Instant? - Part 2
4. 1.2 Defining Limits and Using Limit Notation - Part 1
5. 1.2 Defining Limits and Using Limit Notation - Part 2
6. 1.3 Estimating Limit Values from Graphs - Part 1
7. 1.3 Estimating Limit Values from Graphs - Part 2
8. 1.4 Estimating Limit Values from Tables - Part 1
9. 1.4 Estimating Limit Values from Tables - Part 2
10. 1.5 Determining Limits Using Algebraic Properties of Limits - Part 1
11. 1.5 Determining Limits Using Algebraic Properties of Limits - Part 2
12. 1.6 Determining Limits Using Algebraic Manipulation - Part 1
13. 1.6 Determining Limits Using Algebraic Manipulation - Part 2
14. 1.7 Selecting Procedures for Determining Limits - Part 1
15. 1.7 Selecting Procedures for Determining Limits - Part 2
16. 1.8 Determining Limits Using the Squeeze Theorem - Part 1
17. 1.8 Determining Limits Using the Squeeze Theorem - Part 2
18. 1.9 Connecting Multiple Representations of Limits - Part 1
19. 1.9 Connecting Multiple Representations of Limits - Part 2
20. 1.10 Exploring Types of Discontinuities - Part 1
21. 1.10 Exploring Types of Discontinuities - Part 2
22. 1.11 Defining Continuity at a Point - Part 1
23. 1.11 Defining Continuity at a Point - Part 2
24. 1.12 Confirming Continuity over an Interval - Part 1
25. 1.12 Confirming Continuity over an Interval - Part 2
26. 1.13 Removing Discontinuities - Part 1
27. 1.13 Removing Discontinuities - Part 2
28. 1.14 Connecting Infinite Limits and Vertical Asymptotes - Part 1
29. 1.14 Connecting Infinite Limits and Vertical Asymptotes - Part 2
30. 1.15 Connecting Limits at Infinity and Horizontal Asymptotes - Part 1
31. 1.15 Connecting Limits at Infinity and Horizontal Asymptotes - Part 2
32. 1.16 Working with the Intermediate Value Theorem (IVT) - Part 1
33. 1.16 Working with the Intermediate Value Theorem (IVT) - Part 2
34. Revision (Calc Not Allowed)
35. Revision (Calc Allowed)
01-VerityPrep-AP-Calculus-AB-Light-Chapter-00-Fundamentals
1. Chapter 1 Overview
2. 0.1 Real Numbers, Intervals, Inequalities, and Absolute Value - Part 1
3. 0.1 Real Numbers, Intervals, Inequalities, and Absolute Value - Part 2
4. 0.2 Exponents, Radicals, and Factoring - Part 1
5. 0.2 Exponents, Radicals, and Factoring - Part 2
6. 0.3 Rational Expressions and Solving Equations - Part 1
7. 0.3 Rational Expressions and Solving Equations - Part 2
8. 0.4 Functions - Notation, Domain, Range, and Piecewise Functions - Part 1
9. 0.4 Functions - Notation, Domain, Range, and Piecewise Functions - Part 2
10. 0.5 Graphs and Transformations of Functions - Part 1
11. 0.5 Graphs and Transformations of Functions - Part 2
12. 0.6 Linear Functions, Slope, and Equations of Lines - Part 1
13. 0.6 Linear Functions, Slope, and Equations of Lines - Part 2
14. 0.7 Polynomial and Rational Functions - Part 1
15. 0.7 Polynomial and Rational Functions - Part 2
16. 0.8 Composite and Inverse Functions - Part 1
17. 0.8 Composite and Inverse Functions - Part 2
18. 0.9 Exponential and Logarithmic Functions - Part 1
19. 0.9 Exponential and Logarithmic Functions - Part 2
20. 0.10 Radians, the Unit Circle, and Graphs of Trigonometric Functions - Part 1
21. 0.10 Radians, the Unit Circle, and Graphs of Trigonometric Functions - Part 2
22. 0.11 Trigonometric Identities and Equations - Part 1
23. 0.11 Trigonometric Identities and Equations - Part 2
24. Revision (Calc Not Allowed)
25. Revision (Calc Allowed)
02-Differentiation Definition and Fundamental Properties
1. Chapter 2 Overview
2. 2.1 Defining Average and Instantaneous Rates of Change at a Point - Part 1
3. 2.1 Defining Average and Instantaneous Rates of Change at a Point - Part 2
4. 2.2 Defining the Derivative of a Function and Using Derivative Notation - Part 1
5. 2.2 Defining the Derivative of a Function and Using Derivative Notation - Part 2
6. 2.3 Estimating Derivatives of a Function at a Point - Part 1
7. 2.3 Estimating Derivatives of a Function at a Point - Part 2
8. 2.4 Connecting Differentiability and Continuity - Determining When Derivatives Do and Do Not Exist - Part 1
9. 2.4 Connecting Differentiability and Continuity - Determining When Derivatives Do and Do Not Exist - Part 2
10. 2.5 Applying the Power Rule - Part 1
11. 2.5 Applying the Power Rule - Part 2
12. 2.6 Derivative Rules - Constant, Sum, Difference, and Constant Multiple - Part 1
13. 2.6 Derivative Rules - Constant, Sum, Difference, and Constant Multiple - Part 2
14. 2.7 Derivatives of cos x, sin x, e^x, and ln x - Part 1
15. 2.7 Derivatives of cos x, sin x, e^x, and ln x - Part 2
16. 2.8 The Product Rule - Part 1
17. 2.8 The Product Rule - Part 2
18. 2.9 The Quotient Rule - Part 1
19. 2.9 The Quotient Rule - Part 2
20. 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions - Part 1
21. 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions - Part 2
22. 2.11 Equations of Tangent and Normal Lines - Part 1
23. 2.11 Equations of Tangent and Normal Lines - Part 2
24. Revision (Calc Not Allowed)
25. Revision (Calc Allowed)
03-Differentiation Composite Implicit and Inverse Functions
1. Chapter 3 Overview
2. 3.1 The Chain Rule - Part 1
3. 3.1 The Chain Rule - Part 2
4. 3.2 Implicit Differentiation - Part 1
5. 3.2 Implicit Differentiation - Part 2
6. 3.3 Differentiating Inverse Functions - Part 1
7. 3.3 Differentiating Inverse Functions - Part 2
8. 3.4 Differentiating Inverse Trigonometric Functions - Part 1
9. 3.4 Differentiating Inverse Trigonometric Functions - Part 2
10. 3.5 Selecting Procedures for Calculating Derivatives - Part 1
11. 3.5 Selecting Procedures for Calculating Derivatives - Part 2
12. 3.6 Calculating Higher-Order Derivatives - Part 1
13. 3.6 Calculating Higher-Order Derivatives - Part 2
14. 3.7 Tangent and Normal Lines for Composite, Inverse, and Implicit Curves - Part 1
15. 3.7 Tangent and Normal Lines for Composite, Inverse, and Implicit Curves - Part 2
16. Revision (Calc Not Allowed)
17. Revision (Calc Allowed)
04-Contextual Applications of Differentiation
1. Chapter 4 Overview
2. 4.1 Interpreting the Meaning of the Derivative in Context - Part 1
3. 4.1 Interpreting the Meaning of the Derivative in Context - Part 2
4. 4.2 Straight-Line Motion - Connecting Position, Velocity, and Acceleration - Part 1
5. 4.2 Straight-Line Motion - Connecting Position, Velocity, and Acceleration - Part 2
6. 4.3 Rates of Change in Applied Contexts Other Than Motion - Part 1
7. 4.3 Rates of Change in Applied Contexts Other Than Motion - Part 2
8. 4.4 Introduction to Related Rates - Part 1
9. 4.4 Introduction to Related Rates - Part 2
10. 4.5 Solving Related Rates Problems - Part 1
11. 4.5 Solving Related Rates Problems - Part 2
12. 4.6 Approximating Values of a Function Using Local Linearity and Linearization - Part 1
13. 4.6 Approximating Values of a Function Using Local Linearity and Linearization - Part 2
14. 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms - Part 1
15. 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms - Part 2
16. Revision (Calc Not Allowed)
17. Revision (Calc Allowed)
05-Analytical Applications of Differentiation
1. Chapter 5 Overview
2. 5.1 Using the Mean Value Theorem - Part 1
3. 5.1 Using the Mean Value Theorem - Part 2
4. 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points - Part 1
5. 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points - Part 2
6. 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing - Part 1
7. 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing - Part 2
8. 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema - Part 1
9. 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema - Part 2
10. 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema - Part 1
11. 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema - Part 2
12. 5.6 Determining Concavity of Functions over Their Domains - Part 1
13. 5.6 Determining Concavity of Functions over Their Domains - Part 2
14. 5.7 Using the Second Derivative Test to Determine Extrema - Part 1
15. 5.7 Using the Second Derivative Test to Determine Extrema - Part 2
16. 5.8 Sketching Graphs of Functions and Their Derivatives - Part 1
17. 5.8 Sketching Graphs of Functions and Their Derivatives - Part 2
18. 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative - Part 1
19. 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative - Part 2
20. 5.10 Introduction to Optimization Problems - Part 1
21. 5.10 Introduction to Optimization Problems - Part 2
22. 5.11 Solving Optimization Problems - Part 1
23. 5.11 Solving Optimization Problems - Part 2
24. 5.12 Exploring Behaviors of Implicit Relations - Part 1
25. 5.12 Exploring Behaviors of Implicit Relations - Part 2
26. Revision (Calc Not Allowed)
27. Revision (Calc Allowed)
06-Integration and Accumulation of Change
1. Chapter 6 Overview
2. 6.1 Finding Antiderivatives and Indefinite Integrals - Basic Rules and Notation - Part 1
3. 6.1 Finding Antiderivatives and Indefinite Integrals - Basic Rules and Notation - Part 2
4. 6.2 Integrating Using Substitution - Part 1
5. 6.2 Integrating Using Substitution - Part 2
6. 6.3 Integrating Functions Using Long Division and Completing the Square - Part 1
7. 6.3 Integrating Functions Using Long Division and Completing the Square - Part 2
8. 6.4 Selecting Techniques for Antidifferentiation - Part 1
9. 6.4 Selecting Techniques for Antidifferentiation - Part 2
10. 6.5 Exploring Accumulations of Change - Part 1
11. 6.5 Exploring Accumulations of Change - Part 2
12. 6.6 Approximating Areas with Riemann Sums - Part 1
13. 6.6 Approximating Areas with Riemann Sums - Part 2
14. 6.7 Riemann Sums, Summation Notation, and Definite Integral Notation - Part 1
15. 6.7 Riemann Sums, Summation Notation, and Definite Integral Notation - Part 2
16. 6.8 The Fundamental Theorem of Calculus and Accumulation Functions - Part 1
17. 6.8 The Fundamental Theorem of Calculus and Accumulation Functions - Part 2
18. 6.9 Interpreting the Behavior of Accumulation Functions Involving Area - Part 1
19. 6.9 Interpreting the Behavior of Accumulation Functions Involving Area - Part 2
20. 6.10 Applying Properties of Definite Integrals - Part 1
21. 6.10 Applying Properties of Definite Integrals - Part 2
22. 6.11 The Fundamental Theorem of Calculus and Definite Integrals - Part 1
23. 6.11 The Fundamental Theorem of Calculus and Definite Integrals - Part 2
24. Revision (Calc Not Allowed)
25. Revision (Calc Allowed)
07-Differential Equations
1. Chapter 7 Overview
2. 7.1 Modeling Situations with Differential Equations - Part 1
3. 7.1 Modeling Situations with Differential Equations - Part 2
4. 7.2 Verifying Solutions for Differential Equations - Part 1
5. 7.2 Verifying Solutions for Differential Equations - Part 2
6. 7.3 Sketching Slope Fields - Part 1
7. 7.3 Sketching Slope Fields - Part 2
8. 7.4 Reasoning Using Slope Fields - Part 1
9. 7.4 Reasoning Using Slope Fields - Part 2
10. 7.5 Finding General Solutions Using Separation of Variables - Part 1
11. 7.5 Finding General Solutions Using Separation of Variables - Part 2
12. 7.6 Finding Particular Solutions Using Initial Conditions and Separation of Variables - Part 1
13. 7.6 Finding Particular Solutions Using Initial Conditions and Separation of Variables - Part 2
14. 7.7 Exponential Models with Differential Equations - Part 1
15. 7.7 Exponential Models with Differential Equations - Part 2
16. Revision (Calc Not Allowed)
17. Revision (Calc Allowed)
08-Applications of Integration
1. Chapter 8 Overview
2. 8.1 Finding the Average Value of a Function on an Interval - Part 1
3. 8.1 Finding the Average Value of a Function on an Interval - Part 2
4. 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals - Part 1
5. 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals - Part 2
6. 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts - Part 1
7. 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts - Part 2
8. 8.4 Finding the Area Between Curves Expressed as Functions of x - Part 1
9. 8.4 Finding the Area Between Curves Expressed as Functions of x - Part 2
10. 8.5 Finding the Area Between Curves Expressed as Functions of y - Part 1
11. 8.5 Finding the Area Between Curves Expressed as Functions of y - Part 2
12. 8.6 Finding the Area Between Curves That Intersect at More Than Two Points - Part 1
13. 8.6 Finding the Area Between Curves That Intersect at More Than Two Points - Part 2
14. 8.7 Volumes with Cross Sections - Squares and Rectangles - Part 1
15. 8.7 Volumes with Cross Sections - Squares and Rectangles - Part 2
16. 8.8 Volumes with Cross Sections - Triangles and Semicircles - Part 1
17. 8.8 Volumes with Cross Sections - Triangles and Semicircles - Part 2
18. 8.9 Volume with Disc Method - Revolving Around the x- or y-Axis - Part 1
19. 8.9 Volume with Disc Method - Revolving Around the x- or y-Axis - Part 2
20. 8.10 Volume with Disc Method - Revolving Around Other Axes - Part 1
21. 8.10 Volume with Disc Method - Revolving Around Other Axes - Part 2
22. 8.11 Volume with Washer Method - Revolving Around the x- or y-Axis - Part 1
23. 8.11 Volume with Washer Method - Revolving Around the x- or y-Axis - Part 2
24. 8.12 Volume with Washer Method - Revolving Around Other Axes - Part 1
25. 8.12 Volume with Washer Method - Revolving Around Other Axes - Part 2
26. Revision (Calc Not Allowed)
27. Revision (Calc Allowed)

Sample · 01-Limits and Continuity

Chapter 1 Overview

VerityPrep-AP-Calculus-AB-Light-Chapter-01-Limits and Continuity

This chapter builds the language that the rest of calculus is written in. It covers AP Calculus Unit 1, CED topics 1.1 to 1.16: rates of change at an instant, limits from graphs, tables and formulas, the algebra and theorems used to evaluate them, continuity and discontinuities, infinite limits and limits at infinity, and the Intermediate Value Theorem. On the AP exam Unit 1 carries roughly 10 to 12 percent of the AB multiple-choice weight, and its ideas sit underneath almost every later question. The chapter ends with two mixed review sets, Revision (Calc Not Allowed) and Revision (Calc Allowed), each with multiple-choice questions and a 9-point free-response question.

🎓 Where This Chapter Can Take You

Zeynep was the student in her class who always asked "but what happens right at that moment?" A ball's speed at the top of its throw, a drug level at the exact hour of the next dose, the temperature at the instant a furnace switches off. Her teacher's answer was always the same: we never look at the moment alone, we look at what happens as we get closer and closer to it.

That answer is a limit, written \( \displaystyle \lim_{x\to a}f(x) \). It lets engineers compute a speed from position data even though a single instant has no length, lets scientists say what a model settles to as \( t\to\infty \), and lets anyone prove that a continuous process must pass through a value in between, the Intermediate Value Theorem. The people below use these exact ideas every working day. Zeynep is now studying aerospace engineering; which row will be yours?

Career	What they do	Topics used	Example university programs
Flight test engineer	turns altitude data into speeds at exact instants	1.1, 1.4	Aerospace Engineering, Technical University of Munich; Aeronautics and Astronautics, MIT
Machine-learning researcher	makes training code behave at 0 and at infinity	1.2, 1.15	Computer Science, Stanford University; Computer Engineering, Bogazici University
Pharmacokinetics scientist	reads dose tables to predict drug levels	1.4, 1.15	Pharmacology, Imperial College London; Pharmaceutical Sciences, University of Toronto
Robotics engineer	removes \( \tfrac{0}{0} \) breakdowns from motion formulas	1.6, 1.13	Robotics Institute, Carnegie Mellon University; Mechanical Engineering, ETH Zurich
Power systems engineer	classifies jumps and spikes in grid signals	1.10, 1.11	Electrical and Computer Engineering, National University of Singapore; Electrical Engineering, METU
Optical engineer	designs lenses near their vertical asymptotes	1.14	Applied Physics, Caltech; Physics, EPFL
Materials engineer	proves a furnace passes a target temperature	1.12, 1.16	Materials Science and Nano Engineering, Sabanci University; Materials, University of Oxford

1.1 Introducing Calculus - Can Change Occur at an Instant? - Part 1

CED 1.1 · interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant · AB · Part A

💼 Math at Work — Flight Test Engineer

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