Name: AP Calculus AB & BC Course Book: Step by Step
Author: VerityPrep

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Learn AP Calculus AB and BC step by step: limits and continuity, derivative rules, the chain rule, related rates and optimization, with full revision sets.

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AP Calculus BC

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VerityPrep Calculus, Step by Step AP Classroom Edition for AP Calculus AB & BC cover

About this book

This course book is written for students taking AP Calculus AB or BC and follows the AP Classroom sequence one topic at a time. A Fundamentals chapter reviews the algebra and precalculus the course relies on: inequalities and absolute value, exponents and factoring, functions and their graphs, polynomial, exponential and logarithmic functions, and trigonometry from the unit circle to identities.

Limits and Continuity builds limits from graphs, tables and algebra, the squeeze theorem, types of discontinuity, asymptotes and the Intermediate Value Theorem. The differentiation chapters define the derivative, then cover the power, product and quotient rules, trigonometric, exponential and logarithmic derivatives, the chain rule, implicit differentiation, inverse functions and higher-order derivatives.

The applications chapters cover straight-line motion, related rates, linearization and L'Hospital's rule, followed by the Mean Value Theorem, extrema, concavity, curve sketching and optimization. Every topic is split into two parts, and every chapter ends with calculator and no-calculator revision.

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11 chapters · 281 topics

00-Fundamentals
1. Chapter 1 Overview Sample
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)
01-Limits and Continuity
1. Chapter 1 Overview
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)
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 Exploring Accumulations of Change - Part 1
3. 6.1 Exploring Accumulations of Change - Part 2
4. 6.2 Approximating Areas with Riemann Sums - Part 1
5. 6.2 Approximating Areas with Riemann Sums - Part 2
6. 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation - Part 1
7. 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation - Part 2
8. 6.4 The Fundamental Theorem of Calculus and Accumulation Functions - Part 1
9. 6.4 The Fundamental Theorem of Calculus and Accumulation Functions - Part 2
10. 6.5 Interpreting the Behavior of Accumulation Functions Involving Area - Part 1
11. 6.5 Interpreting the Behavior of Accumulation Functions Involving Area - Part 2
12. 6.6 Applying Properties of Definite Integrals - Part 1
13. 6.6 Applying Properties of Definite Integrals - Part 2
14. 6.7 The Fundamental Theorem of Calculus and Definite Integrals - Part 1
15. 6.7 The Fundamental Theorem of Calculus and Definite Integrals - Part 2
16. 6.8 Finding Antiderivatives and Indefinite Integrals - Basic Rules and Notation - Part 1
17. 6.8 Finding Antiderivatives and Indefinite Integrals - Basic Rules and Notation - Part 2
18. 6.9 Integrating Using Substitution - Part 1
19. 6.9 Integrating Using Substitution - Part 2
20. 6.10 Integrating Functions Using Long Division and Completing the Square - Part 1
21. 6.10 Integrating Functions Using Long Division and Completing the Square - Part 2
22. 6.11 Integrating Using Integration by Parts - Part 1
23. 6.11 Integrating Using Integration by Parts - Part 2
24. 6.12 Integrating Using Linear Partial Fractions - Part 1
25. 6.12 Integrating Using Linear Partial Fractions - Part 2
26. 6.13 Evaluating Improper Integrals - Part 1
27. 6.13 Evaluating Improper Integrals - Part 2
28. 6.14 Selecting Techniques for Antidifferentiation - Part 1
29. 6.14 Selecting Techniques for Antidifferentiation - Part 2
30. Revision (Calc Not Allowed)
31. 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 Approximating Solutions Using Euler's Method - Part 1
11. 7.5 Approximating Solutions Using Euler's Method - Part 2
12. 7.6 Finding General Solutions Using Separation of Variables - Part 1
13. 7.6 Finding General Solutions Using Separation of Variables - Part 2
14. 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables - Part 1
15. 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables - Part 2
16. 7.8 Exponential Models with Differential Equations - Part 1
17. 7.8 Exponential Models with Differential Equations - Part 2
18. 7.9 Logistic Models with Differential Equations - Part 1
19. 7.9 Logistic Models with Differential Equations - Part 2
20. Revision (Calc Not Allowed)
21. 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. 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled - Part 1
27. 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled - Part 2
28. Revision (Calc Not Allowed)
29. Revision (Calc Allowed)
09-Parametric Equations Polar Coordinates and Vector-Valued Functions
1. Chapter 9 Overview
2. 9.1 Defining and Differentiating Parametric Equations - Part 1
3. 9.1 Defining and Differentiating Parametric Equations - Part 2
4. 9.2 Second Derivatives of Parametric Equations - Part 1
5. 9.2 Second Derivatives of Parametric Equations - Part 2
6. 9.3 Finding Arc Lengths of Curves Given by Parametric Equations - Part 1
7. 9.3 Finding Arc Lengths of Curves Given by Parametric Equations - Part 2
8. 9.4 Defining and Differentiating Vector-Valued Functions - Part 1
9. 9.4 Defining and Differentiating Vector-Valued Functions - Part 2
10. 9.5 Integrating Vector-Valued Functions - Part 1
11. 9.5 Integrating Vector-Valued Functions - Part 2
12. 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions - Part 1
13. 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions - Part 2
14. 9.7 Defining Polar Coordinates and Differentiating in Polar Form - Part 1
15. 9.7 Defining Polar Coordinates and Differentiating in Polar Form - Part 2
16. 9.8 Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve - Part 1
17. 9.8 Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve - Part 2
18. 9.9 Finding the Area of the Region Bounded by Two Polar Curves - Part 1
19. 9.9 Finding the Area of the Region Bounded by Two Polar Curves - Part 2
20. Revision (Calc Not Allowed)
21. Revision (Calc Allowed)
10-Infinite Sequences and Series
1. Chapter 10 Overview
2. 10.1 Defining Convergent and Divergent Infinite Series - Part 1
3. 10.1 Defining Convergent and Divergent Infinite Series - Part 2
4. 10.2 Working with Geometric Series - Part 1
5. 10.2 Working with Geometric Series - Part 2
6. 10.3 The nth Term Test for Divergence - Part 1
7. 10.3 The nth Term Test for Divergence - Part 2
8. 10.4 Integral Test for Convergence - Part 1
9. 10.4 Integral Test for Convergence - Part 2
10. 10.5 Harmonic Series and p-Series - Part 1
11. 10.5 Harmonic Series and p-Series - Part 2
12. 10.6 Comparison Tests for Convergence - Part 1
13. 10.6 Comparison Tests for Convergence - Part 2
14. 10.7 Alternating Series Test for Convergence - Part 1
15. 10.7 Alternating Series Test for Convergence - Part 2
16. 10.8 Ratio Test for Convergence - Part 1
17. 10.8 Ratio Test for Convergence - Part 2
18. 10.9 Determining Absolute or Conditional Convergence - Part 1
19. 10.9 Determining Absolute or Conditional Convergence - Part 2
20. 10.10 Alternating Series Error Bound - Part 1
21. 10.10 Alternating Series Error Bound - Part 2
22. 10.11 Finding Taylor Polynomial Approximations of Functions - Part 1
23. 10.11 Finding Taylor Polynomial Approximations of Functions - Part 2
24. 10.12 Lagrange Error Bound - Part 1
25. 10.12 Lagrange Error Bound - Part 2
26. 10.13 Radius and Interval of Convergence of Power Series - Part 1
27. 10.13 Radius and Interval of Convergence of Power Series - Part 2
28. 10.14 Finding Taylor or Maclaurin Series for a Function - Part 1
29. 10.14 Finding Taylor or Maclaurin Series for a Function - Part 2
30. 10.15 Representing Functions as Power Series - Part 1
31. 10.15 Representing Functions as Power Series - Part 2
32. Revision (Calc Not Allowed)
33. Revision (Calc Allowed)

Sample · 00-Fundamentals

Chapter 1 Overview

VerityPrep-AP-Calculus-Light-Chapter-00-Fundamentals

Calculus is built on algebra, functions and trigonometry. Most difficulties in a calculus class are not really calculus difficulties: they come from a fraction that was added wrongly, an exponent law that was forgotten, or a sine value that was not remembered. This chapter collects the fundamentals that the rest of the book uses every day: inequalities and absolute value, exponents and factoring, rational expressions and equations, functions and their graphs, lines and slope, polynomial and rational functions, composite and inverse functions, exponentials and logarithms, and trigonometry. It is not part of the AP Calculus CED; it is a VerityPrep foundation chapter for AB and BC students. Every topic explains the ideas step by step, starts each Part with easy Honey Badger warm-up questions, and has a Real-Life Application box and a Career Investigation box. 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

Mira wanted to study engineering, but the first calculus lesson felt like a foreign language. Her teacher asked her to simplify \( \dfrac{x^2-4}{x-2} \), and she realized the problem was not the new ideas but the old ones. After two weeks of practising fundamentals, the calculus examples suddenly made sense: each new step used tools she already owned.

That is the purpose of this chapter: to make sure every tool is sharp before the climb begins. The people below use these same tools every working day. Which row will be yours?

Career	What they do	Topics used	Example university programs
Quality control engineer	keeps parts within tolerance using absolute value	0.1	Industrial Engineering, Istanbul Technical University; Manufacturing Engineering, University of Michigan
Data storage engineer	sizes storage in powers of two	0.2	Computer Engineering, Bogazici University; Computer Science, University of Waterloo
Electrical engineer	adds fractions to find resistances	0.3	Electrical and Electronics Engineering, METU; Electrical Engineering, Imperial College London
Logistics pricing analyst	writes price rules as piecewise functions	0.4	Industrial Engineering, Koc University; Supply Chain Management, MIT
Motion graphics designer	moves and stretches curves with transformations	0.5	Animation, Bilkent University; Motion Design, Savannah College of Art and Design
Civil engineer	designs road grades as slopes	0.6	Civil Engineering, Istanbul Technical University; Civil Engineering, TU Delft
Packaging engineer	models box volumes with polynomials	0.7	Mechanical Engineering, Sabanci University; Packaging Engineering, Michigan State University
Cybersecurity analyst	encodes and decodes with inverse functions	0.8	Computer Engineering, Hacettepe University; Information Security, Royal Holloway University of London
Seismologist	measures earthquakes on a logarithmic scale	0.9	Geophysical Engineering, Istanbul Technical University; Geophysics, Caltech
Wind turbine engineer	models turning blades with sine and cosine	0.10	Energy Systems Engineering, Izmir Institute of Technology; Wind Energy, DTU Denmark
Land surveyor	turns measured ratios into angles	0.11	Geomatics Engineering, Yildiz Technical University; Geomatics, University College London

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