# AP Calculus AB

The mathematics of Calculus is based on the idea of rates of change. Topics include analyzing functions, limits, differentiation, curve sketching, extreme value problems, anti-differentiation, definite integration, areas under curves, and volumes of solids. This course prepares students for the Calculus Advanced Placement Test for college credit.

**Credits:** 1.0 (0.5 per semester)

**Estimated Completion Time:** 2 semesters/18-36 weeks

**State Course Number:**

Dual Credit

NCAA Approved

**Unit 1: Limits (9-10 days)**

- Unit Overview: Unit 1 focuses on the study of limits. You will learn to evaluate limits graphically, numerically, and analytically. You will discuss continuity and one-sided limits. You will learn how to find infinite limits and limits at infinity
**.** - 1.1 A Preview of Calculus [CR1a], [CR2b], [CR2e]
- Essential Question: What is Calculus?

- 1.2 Finding Limits Graphically and Numerically [CR1a], [CR2a], [CR2b], [CR2d], [CR3a], [CR3b], [CR3d]
- Essential Question: What is a limit and how can you determine the limit of a function as x approaches c?

- 1.3 Evaluating Limits Analytically Part 1 [CR1a], [CR2b], [CR2c], [CR2d], [CR2e]
- Essential Question: What algebraic techniques can you use to evaluate a limit?

- 1.4 Continuity and One-Sided Limits [CR1a], [CR2a], [CR2b], [CR2c], [CR2e], [CR2f]
- Essential Question: What is continuity and how does it apply to the Intermediate Value Theorem?

**Discussion Question: What does it mean to be continuous? List three ways a function can be discontinuous.**

- 1.5 Infinite Limits [CR1a], [CR2b], [CR2e], [CR3a], [CR3b], [CR3d]
- Essential Question: What is an infinite limit?

- 1.6 Limits at Infinity [CR1a], [CR2b], [CR2c], [CR2e], [CR3a], [CR3b], [CR3d]
- Essential Question: What is a limit at infinity?

- Unit 1 Review
- Unit 1 Test

**Unit 2: Differentiation (13-15 days)**

- Unit Overview: Unit 2 focuses on the study of derivatives. You will learn to differentiate functions using the limit process and derivative rules. You will discuss what a derivative is and what it means. You will learn to solve problems that use derivatives.
- 2.1 The Derivative and the Tangent Line Problem [CR1b], [CR2a], [CR2b], [CR2e]
- Essential Question: What is the derivative and what is its relationship to continuity?

- 2.2 Basic Differentiation Rules and Rates of Change [CR1b], [CR2b], [CR2c], [CR2e]
- Essential Question: How do you find the derivatives of basic algebraic functions, trigonometric functions, and exponential functions?

- 2.3 Product and Quotient Rules and Higher-Order Derivatives [CR1b], [CR2b], [CR2c], [CR2e]
- Essential Question:How do you find the derivatives of functions involving products and quotients?

- 2.4 The Chain Rule [CR1b], [CR2b], [CR2c], [CR2e], [CR2f]
- Essential Question: How do you find the derivatives of composite functions, natural logarithmic functions, and exponential functions with bases other than e?

**Chain Rule Project:**

- 2.5 Implicit Differentiation [CR1b], [CR2a], [CR2b], [CR2c], [CR2e]
- Essential Question: How do you find the derivative of implicitly defined functions?

- 2.6 Derivatives of Inverse Function [CR1b], [CR2a], [CR2b], [CR2c], [CR2e]
- Essential Question: How do you find the derivatives of inverse functions, including inverse trigonometric functions?

- 2.7 Related Rates [CR1b], [CR2b], [CR2c], [CR2e]
- Essential Question: What is a related rate and how do you find it?

- 2.8 Newton’s Method [CR1b], [CR2b], [CR2c], [CR2e], [CR3a], [CR3b], [CR3d]
- Essential Question: How can you use derivatives to approximate the zero of a function or the approximation of a square root?

- Unit 2 Review
- Unit 2 Test

**Unit 3: Applications of Derivatives (12-15 days)**

- Unit Overview: Unit 3 focuses on the analysis of functions using the first and second derivative. You will determine intervals of increasing and decreasing, along with local extrema and points of inflection. You will also learn the Mean Value Theorem and solve optimization problems.
- 3.1 Extrema on an Interval [CR1b], [CR2a], [CR2b], [CR2c], [CR3a], [CR3b], [CR3d]
- Essential Question: What are extrema and how can you find them on open and closed intervals?

- 3.2 Rolle’s Theorem and Mean Value Theorem [CR1b], [CR2a], [CR2b], [CR2c], [CR2e]
- Essential Question: What is the Mean Value Theorem and how is it used?

- 3.3 Increasing and Decreasing Functions and the First Derivative Test [CR1b], [CR2a], [CR2b], [CR2c], [CR3a], [CR3b], [CR3d]
- Essential Question: How can you determine the intervals on which a function is increasing or decreasing and the location of the function’s relative extrema?

- 3.4 Concavity and the Second Derivative Test [CR1b], [CR2a], [CR2b], [CR2c]
- Essential Question: How do you determine the concavity of a function and find its inflection points?

- 3.5 A Summary Curve Sketching [CR1b], [CR2b], [CR2c], [CR2d], [CR3a], [CR3b], [CR3d]
- Essential Question: How do you analyze a function and sketch its graph?

- 3.6 Optimization Problems [CR1b], [CR2b], [CR2c]
- Essential Question: How do you maximize or minimize quantities?

- 3.7 Differentials [CR1b], [CR2b]
- Essential Question: How are differentials used to explain the tangent line approximation?

- L’Hopitals Rule [CR1a], [CR1b], [CR2a], [CR2b], [CR2e]
- Essential Question: How do you evaluate a limit when direct substitution produces an indeterminate form?

- Unit 3 Review
- Unit 3 Test

**Unit 4: Integration (13-15 days)**

- Unit Overview: Unit 4 focuses on the study of integration. You will learn how to evaluate simple integrals through antidifferentiation. You will then learn various methods of integration to evaluate more difficult integrals. Another important concept is using Riemann Sums and the Trapezoidal rule to find the area of a plane region. These topics lead to the important discovery of the Fundamental Theorem of Calculus.
- 4.1 Antiderivatives and Indefinite Integration [CR1c], [CR2b], [CR2d], [CR2e]
- Essential Question: What are antiderivatives and how are they used?

- 4.2 Area [CR1c], [CR2d]
- Essential Question:How can you approximate the area of a plane region?

- 4.3 Riemann Sums and Definite Integrals [CR1c], [CR2a], [CR2b], [CR2d], [CR2e]
- Essential Question: How are Riemann sums similar to the Trapezoidal Rule and how are they different?

- 4.4 The Fundamental Theorem of Calculus Part 1 [CR1c], [CR2a], [CR2b], [CR2c], [CR2d], [CR2e], [CR3a], [CR3b], [CR3d]
- Essential Question: What is the Fundamental Theorem of Calculus?

- 4.4.1: Average Value of a Function Desmos [CR1c], [CR2a], [CR2b], [CR2c], [CR2e], [CR2f]
- Essential Question: How does the idea of average relate to the average value of a function?

- 4.4.2 The Fundamental Theorem of Calculus Part 2 (Mean Value Theorem, Average Value of a Function, Second Fundamental Thm of Calc) [CR1c], [CR2a], [CR2b], [CR2c], [CR2e]
- Essential Question: What is the Second Fundamental Theorem of Calculus?

- 4.4.3 Functions Defined with an Integral [CR1c], [CR2b], [CR2d], [CR2e]
- Essential Question: How can a function be defined with an integral and how do you analyze this type of function?

- 4.5 Integration by Substitution [CR1c], [CR2b], [CR2e], [CR2f]
- Essential Question: How do you integrate composite functions?

- 4.6 The Natural Logarithmic Functions: Integration [CR1c], [CR2b],[CR2c]
- Essential Question: How do you integrate rational functions and trigonometric functions other than sine and cosine?

- 4.7 Inverse Trigonometric Functions: Integration [CR1c], [CR2b], [CR2c]
- Essential Question: How can you recognize when an integral results in an inverse trigonometric function?

- Unit 4 Review
- Unit 4 Test

**Unit 5 – ****Differential Equations (6-7 days)**

- Unit Overview: Unit 5 combines the conceptual topics learned with a number of real-life applications. You will solve differential equations using slope fields as a tool for differential equations that cannot be easily solved.
- 5.1 Slope Fields and Euler’s Method (6-8 days) [CR1c], [CR2a], [CR2b], [CR2c], [CR2e], [CR3a], [CR3b], [CR3d]
- Essential Question: How do you approximate the particular solution of a differential equation?

- 5.2 Growth and Decay [CR1c], [CR2b], [CR2c]
- Essential Question: How are differential equations used in application problems, such as the exponential growth and decay model?

- 5.3 Separation of Variables [CR1c], [CR2b], [CR2c], [CR2e]
- Essential Question: How do you solve separable differential equations?

- 5.4 The Logistic Equation [CR1c], [CR2b], [CR2c]
- Essential Question: How do you solve logistic differential equations?

- Unit 5 Review
- Unit 5 Test

**Unit 6 – ****Applications of Integration (5-7 days)**

- Unit Overview: Unit 6 focuses on the application of integration. You will learn to find the area between to curves. You will also develop a method for find the volume of a solid of revolution and a volume with known-cross sections.
- 6.1 Area of a Region Between Two Curves [CR1c], [CR2b], [CR2c], [CR2f], [CR3a], [CR3b], [CR3d]
- Essential Question: How do you find the area of a region between two curves?

- 6.2 Volume: The Disk and Washer Method [CR1c], [CR2b], [CR2c], [CR3a], [CR3b], [CR3d]
- Essential Question: How can you use integrals to find the volume of a solid?

- 6.2 Volume: Known Cross-Sections [CR1c], [CR2b], [CR2c]
- Essential Question: How can you use integrals to find the volume of a solid with known cross-sections?

This course is available for dual credit through Ozarks Technical Community College. Find out more about dual credit on the institution's website or by contacting their office:

**Ozarks Technical Community College**

OTC Dual Credit

1001 E. Chestnut Expressway

Springfield, MO 65802

Phone: 417-447-8198

Email: dualcredit@otc.edu