Name: AP Calculus AB and BC Book for Elite Students
Author: VerityPrep

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Cover AP Calculus AB and BC from limits and the chain rule to slope fields, polar area and Taylor series, with 86 questions and calculator revision.

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AP Calculus BC

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This course book is designed for strong students preparing for AP Calculus AB and BC. It opens with a fundamentals chapter on special functions, transformations, inverse trigonometric, exponential and logarithmic functions, and BC topics such as parametric and polar functions.

The main chapters follow the AP Calculus course: limits and continuity including the squeeze theorem and the intermediate value theorem; differentiation from the definition through the product, quotient and chain rules, implicit differentiation and inverse functions; contextual applications such as related rates, linearisation and L'Hospital's rule; and analytical applications including the mean value theorem, extrema, concavity and optimisation.

Integration chapters cover Riemann sums, the Fundamental Theorem of Calculus, substitution, integration by parts, partial fractions and improper integrals, followed by differential equations with slope fields, Euler's method and logistic models, then area and volume applications. The BC chapters address parametric equations, vectors and polar functions, and infinite series up to Taylor and Maclaurin series. Most chapters end with calculator-allowed and no-calculator revision, and the book contains 86 questions.

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11 chapters · 266 topics

00-Fundamentals + Special Functions · AB + BC
1. Chapter 0 Overview Sample
2. 0.1 Function Representations and Function Tests
3. 0.2 Zeros, Intercepts, and Factor Reasoning
4. 0.3 Special Functions: Absolute Value and Greatest Integer
5. 0.4 Graph Transformations: |f(x)|, f(|x|), -f(x), and f(-x)
6. 0.5 Polynomial and Rational Functions
7. 0.6 Trigonometric Periodicity and Amplitude
8. 0.7 Inverse Trigonometric Functions
9. 0.8 Exponential and Logarithmic Functions
10. 0.9 Parametrically Defined Functions (BC)
11. 0.10 Parametric Graphing Challenges (BC)
12. 0.11 Polar Functions (BC): Coordinates and Common Curves
13. 0.12 Polar-Cartesian Conversion Challenges (BC)
14. Chapter 0 Final Summary
15. REVISION
01-Limits and Continuity
1. 1.1 Introducing Calculus - Can Change Occur at an Instant? - Part 1
2. 1.1 Introducing Calculus - Can Change Occur at an Instant? - Part 2
3. 1.2 Defining Limits and Using Limit Notation - Part 1
4. 1.2 Defining Limits and Using Limit Notation - Part 2
5. 1.3 Estimating Limit Values from Graphs - Part 1
6. 1.3 Estimating Limit Values from Graphs - Part 2
7. 1.4 Estimating Limit Values from Tables - Part 1
8. 1.4 Estimating Limit Values from Tables - Part 2
9. 1.5 Determining Limits Using Algebraic Properties of Limits - Part 1
10. 1.5 Determining Limits Using Algebraic Properties of Limits - Part 2
11. 1.6 Determining Limits Using Algebraic Manipulation - Part 1
12. 1.6 Determining Limits Using Algebraic Manipulation - Part 2
13. 1.7 Selecting Procedures for Determining Limits - Part 1
14. 1.7 Selecting Procedures for Determining Limits - Part 2
15. 1.8 Determining Limits Using the Squeeze Theorem - Part 1
16. 1.8 Determining Limits Using the Squeeze Theorem - Part 2
17. 1.9 Connecting Multiple Representations of Limits - Part 1
18. 1.9 Connecting Multiple Representations of Limits - Part 2
19. 1.10 Exploring Types of Discontinuities - Part 1
20. 1.10 Exploring Types of Discontinuities - Part 2
21. 1.11 Defining Continuity at a Point - Part 1
22. 1.11 Defining Continuity at a Point - Part 2
23. 1.12 Confirming Continuity over an Interval - Part 1
24. 1.12 Confirming Continuity over an Interval - Part 2
25. 1.13 Removing Discontinuities - Part 1
26. 1.13 Removing Discontinuities - Part 2
27. 1.14 Connecting Infinite Limits and Vertical Asymptotes - Part 1
28. 1.14 Connecting Infinite Limits and Vertical Asymptotes - Part 2
29. 1.15 Connecting Limits at Infinity and Horizontal Asymptotes - Part 1
30. 1.15 Connecting Limits at Infinity and Horizontal Asymptotes - Part 2
31. 1.16 Working with the Intermediate Value Theorem (IVT) - Part 1
32. 1.16 Working with the Intermediate Value Theorem (IVT) - Part 2
33. Revision (Calc Not Allowed)
34. 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. Revision (Calc Not Allowed)
23. 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. Revision (Calc Not Allowed)
15. 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, Vectors, and Polar 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 + Special Functions · AB + BC

Chapter 0 Overview

AP Calculus Book · Chapter 0: Fundamentals + Special Functions · AB + BC

This expanded Chapter 0 now incorporates the uploaded Fundamentals Special Functions material: zeros, absolute value, greatest-integer functions, transformations such as \(|f(x)|\) and \(f(|x|)\), polynomial and rational functions, trigonometric periodicity and amplitude, inverse trigonometric functions, exponential/logarithmic functions, parametrically defined functions, and polar functions.

How this revision uses the uploaded document

The added material follows the document’s visual flow: inverse graphs as reflections, zeros of polynomial functions, absolute value and greatest-integer graphs, graph transformations, polynomial/rational definitions, trig period/amplitude examples, inverse trig restricted domains and ranges, exponential/logarithmic inverse relationship, BC parametric curves, and BC polar curves.

0.1 Function Representations and Function Tests

Deep textbook explanation

Functions can appear as formulas, tables, graphs, and verbal descriptions. A graph represents a function of \(x\) only if it passes the vertical line test. In AP Calculus, students must translate between representations and interpret units, domains, and outputs.

Worked example

Worked example. If \(f(3)=8\), then \((3,8)\) is on the graph of \(y=f(x)\), a table would pair input \(3\) with output \(8\), and in words the output is \(8\) when the input is \(3\).

Q01 · 0.1 Function Representations and Function Tests

Which statement is equivalent to \(f(6)=2\)?

A\((2,6)\) lies on \(y=f(x)\)
B\((6,2)\) lies on \(y=f(x)\)
C\(f'(6)=2\)
D\(f(x)=6x+2\)
E\(f(2)=6\)

Q02 · 0.1 Function Representations and Function Tests

A graph fails the vertical line test. What does that mean?

AIt has no x-intercepts
BIt has no y-intercept
CSome input has more than one output
DIt is always discontinuous
EIt is a polynomial

FRQ Readiness Task

A verbal model says the cost is \(50\) dollars plus \(12\) dollars per hour. Write a function \(C(h)\).

0.2 Zeros, Intercepts, and Factor Reasoning

Zero graph check

A zero is not just a point near the curve. A zero must satisfy \(f(x)=0\), so its point on the graph is \((x,0)\). Therefore every zero marker must lie exactly on the x-axis, and the curve must cross or touch the x-axis at that marker.

Deep textbook explanation

Zeros of a function are the \(x\)-values for which \(f(x)=0\). On a graph, they are x-intercepts. Algebraically, factoring is usually the fastest way to find zeros of polynomial functions. Multiplicity matters: an odd-multiplicity zero crosses the x-axis, while an even-multiplicity zero usually touches and turns.

Worked example

Worked example. For \(f(x)=x^4-2x^2\), factor \(x^2(x^2-2)\). The zeros are \(x=0,\sqrt2,-\sqrt2\). The zero \(x=0\) has multiplicity 2.

Q03 · 0.2 Zeros, Intercepts, and Factor Reasoning

Find all zeros of \(f(x)=x^4-2x^2\).

A\(0,2,-2\)
B\(0,\sqrt2,-\sqrt2\)
C\(\sqrt2\) only
D\(0\) only
E\(2,-2\)

Q04 · 0.2 Zeros, Intercepts, and Factor Reasoning

If \(f(x)=(x-3)^2(x+1)\), which zero has even multiplicity?

A\(-1\)
B\(0\)
C\(1\)
D\(3\)
E\(-3\)

FRQ Readiness Task

Explain graphically how a zero of even multiplicity differs from a zero of odd multiplicity.

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