## Topic outline

- General
- Course Introduction
### Course Introduction

Calculus2 is primarily concerned with developing your understanding of the concepts of calculus and providing you with its methods and applications. The course emphasizes a multi-representational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. Broad concepts and widely applicable methods are also emphasized. The focus of the course is neither manipulation nor memorization of an extensive taxonomy of functions, curves, theorems, or problem types, but rather, the course uses the unifying themes of derivatives, integrals, limits, approximation, and applications and modeling to become a cohesive whole.

- Unit 1: Functions, Graphs, and Limits
### Unit 1: Functions, Graphs, and Limits

This unit begins with a review of functions that should have been learned in a previous course. Specifically, we will examine the relationship between formulas and graphs of functions, as well as general properties of their graphs. After completing the first part of this unit, you should feel comfortable explaining general patterns that occur in certain types of functions, as well as be able to give specific information about a graph given its formula. After studying functions, we will examine the first "big" topic in calculus, the limit. As you will see, much of calculus is based on this concept, so it is important to have a good understanding of its fundamental meaning now. We will study limits as they relate to graphs and formulas, and we will use formal limit notation to explain certain behaviors of graphs that you have studied before (such as continuity), but without the concept of limits. This is a very exciting unit and it is important to understand limits well before moving on to the next part of the course.

**Completing this unit should take you approximately 15 hours.** - 1.1: Analysis of Graphs
### 1.1: Analysis of Graphs

Functions can model real-life scenarios to determine the relationship among variables. One example is linear modeling, which can compare supply and demand models. This concept is used during the Christmas shopping season to see if the supply of the hot new toy is available to meet consumer demand. Nonlinear modeling occurs when the relationship fluctuates between the dependent and independent variables. An example is the number of bacteria growing in a petri dish, which can occur at an exponential rate. Functions can also be compared with the parent function by looking at the transformations of the function.

Subunit 1.1 contains a review of properties of functions and their graphs, including intercepts, symmetry, domain, and range. The properties discussed in this subunit will be applied to functions throughout the course.

Read this section on equations and graphs, stopping at "Review Questions." This reading discusses finding solutions to equations of graphs, intercepts, symmetry of graphs, points of intersection for the graphs, and linear models. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to identify intercepts and symmetry of the graph and represent linear modules. The solutions to these questions are located here.

Take notes as you watch these videos. Listen to the presentation carefully until you are able to explain the library of parent functions and relations. The functions include quadratics and logarithms and trigonometric and conic sections. You should also be able to explain how a function is graphed and discuss its properties using transformations.

Read this section on relations and functions. This reading discusses eight basic functions as well as the domain and the range for each function. There is also a review of material from the previous section. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to recognize the domain and range as well graph functions applying transformations. The solutions to these problems are located here.

Read this material describing an initial model. Watch all of the videos in this section and run each of the applets to see examples of functions. This literacy component will allow you to explore an application of a growth model function.

Read this material describing a rational function model. Watch all of the videos in this section and run each of the applets to see examples of functions. This literacy component will allow you to explore a realistic example of a rational function.

- 1.2: Limits of Functions (including one-sided limits)
### 1.2: Limits of Functions (including one-sided limits)

There are limits to many things around us, including algebraic functions. Limits can help us find values that are not possible in other aspects of mathematics, including the velocity of a falling object as well as the area under curves. Other examples include the population of certain animals, depending on predators and food sources, as well as the instant speed of an object. Limits allow us to explore what occurs with a function when the dependent variable approaches a certain value.

Subunit 1.2 discusses the fundamental concept of a limit, which is a critical definition for calculus. This subunit will require you to find limits, including one-sided and infinite limits, graphically and algebraically. This subunit also helps you identify the properties of limits and cases where limits do not exist.

- 1.2.1: Finding Limits
### 1.2.1: Finding Limits

Read this section on approximations to start understanding the concepts of limits. This reading discusses the tangent line to a graph and approximate area under a curve. Then, complete review questions 1-8 toward the bottom of the page. These exercises will provide you with the opportunity to apply concepts of a point tangent to a graph and approximate the area under curves. The solutions to these problems are located here.

Take notes as you watch these videos. Listen to the presentation carefully until you are able to understand the concept of a limit in calculus and the formal definition of a limit.

Read this section on finding limits. This reading discusses how to find a limit numerically, using a graph, identify nonexisting limits, and use the formal definition of a limit. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to find limits algebraically, graphically, and using tables. The solutions to these problems are located here.

- 1.2.2: Evaluating Limits
### 1.2.2: Evaluating Limits

Take notes as you watch this video. Listen to the presentation carefully until you are able to recall and apply limit properties.

Read this section on finding limits. This reading discusses how to find a limit of various functions, the properties of limits, and using the squeeze theorem to find special limits.

Complete exercises 1-2 and 7-26. These exercises will provide you with the opportunity to find limits algebraically, graphically, and using tables. The solution for each problem can be found by clicking on the gray triangle beside each problem.

- 1.2.3: Infinite Limits
### 1.2.3: Infinite Limits

Take notes as you watch these videos. Listen to the presentation carefully until you are able to recall and apply properties of limits dealing with infinity and to find limits of positive and negative infinity.

Read this section on finding limits. Watch all of the videos in the section as well. This reading discusses how to find infinite limits of functions and the properties of infinite limits, identify asymptotes of functions, and analyze end behavior of functions. Then, complete review questions 8-10 toward the bottom of the page. These exercises will provide you with the opportunity to analyze the end behavior of the graph and find the asymptotes. The solutions to these problems are located here.

Complete exercises 27-36 toward the bottom of the page. These exercises will provide you with the opportunity to find limits algebraically, graphically, and using tables. The solution for each problem can be found by clicking on the gray triangle beside each problem.

- 1.3: Continuity as a Property of Functions
### 1.3: Continuity as a Property of Functions

Some functions are continuous, while others are not, which is evident when looking at a graph of a function. Discontinuity can occur when there are rational functions, and examples are cost of a product and medical interactions. The speed of an automobile during heavy traffic is where one can see continuity in action because at times you are moving at a constant speed and at other times the car has stopped.

Subunit 1.3 explores the continuity as a property of functions. Continuity is evident in various calculus theorems as well as in one-sided limits. This concept will be applied when looking at functions within unit 2.

Read this section on finding limits. Watch the videos in the section as well. This reading discusses how to understand continuous function properties, one-sided limits, and solving problems using the min-max and intermediate value theorem. Then, complete review questions 7-10 toward the bottom of the page. These exercises will provide you with the opportunity to solve problems using the max-min and intermediate value theorems. The solutions to these problems are located here.

Complete exercises 3-6 and 37-39. These exercises will provide you with the opportunity to find one-sided limits. The solution for each problem can be found by clicking on the gray triangle beside each problem. The solution includes detailed work on how to find the given limit.

- Unit 2: Derivatives
### Unit 2: Derivatives

In this unit, you will learn first the concept of a derivative by explaining it through limits. You will see that the limit definition and computation are drawn out, and often difficult, so you will learn shortcuts to compute derivatives. You will learn to compute derivatives for all different types of functions, as well as for combinations of functions. After you learn how to compute derivatives, you will look at what derivatives tell you specifically at any point in a function, as well as what they tell you about the function as a whole. You will finish this unit by looking at the computation and meaning of second derivatives, as well as real-life applications of derivatives. The main applications to be reviewed in this unit will be related rates problems, optimization problems, and problems involving displacement, velocity, and acceleration.

**Completing this unit should take you approximately 54 hours.** - 2.1: Concept of the Derivative
### 2.1: Concept of the Derivative

The velocity of an airplane is not always consistent due to various factors, including the fact that it does not travel in a straight line. So, most people look at an average velocity to determine the speed of the plane. Sometimes, there is an instantaneous velocity used to determine the speed at a specific time. This is an example of how derivatives are used to help determine specific and average amounts.

The idea of derivatives may be challenging at first, but it makes concepts easier throughout the unit. You will start by using concepts that you know, such as slope of a line and limits. These concepts will be expanded further to help define a derivative.

Subunit 2.1 begins by defining the slope and slope of a tangent line. Then, the slope of a tangent line for a point on a graph is used to define a derivative. The derivative of a function is defined in this section, also using the concept of limits.

- 2.1.1: Tangent Lines and Rates of Change
### 2.1.1: Tangent Lines and Rates of Change

Take notes as you watch this video. Listen to the presentation carefully until you are able to apply concepts of slope to the tangent line of a graph. You will also need to understand that the derivative is the slope of a curve at a point.

Read this section on tangent lines and rates of change. Watch the videos in the section as well. This instruction discusses how to understand the slope of the tangent line of a graph as well as the average and the instantaneous rates of change. Then, complete review questions 1-6 at the bottom of the page. These exercises will provide you with the opportunity to solve problems using the slope of a tangent line as well as the average and the instantaneous rates of change. The solutions to these problems are located here.

- 2.1.2: The Derivative
### 2.1.2: The Derivative

Take notes as you watch this video. Listen to the presentation carefully until you are able to understand how to find the slope (or derivative) of a curve at a particular point.

Read this section on finding derivatives, stopping at "Review Questions." Watch the videos in the section as well. This instruction discusses how to understand the derivative of a function as a slope of the tangent line. It also includes a discussion on the relationship between continuity and differentiability.

Complete Exercises 1-9 in the section titled "Find the Derivative by Definition." These exercises will provide you with the opportunity to find the derivative by definition. The solution for each problem can be found by clicking on the gray triangle beside each problem. The solution is the derivative of the function.

- 2.2: Computation of Derivatives
### 2.2: Computation of Derivatives

Computing derivatives can be a challenging task using the definition of a limit to find each derivative. Even though each problem can use this definition, the rules of differentiation help make computation of derivatives easier. Understanding and applying rules, differentiation is simpler as well as applicable throughout calculus.

Subunit 2.2 contains the various derivative rules ranging from power, product, and quotient to special rules with some functions. These rules will be applied throughout this and remaining sections; they will also be used for various applications.

- 2.2.1: Basic Derivative Rules
### 2.2.1: Basic Derivative Rules

Take notes as you watch these videos. Listen to the presentation carefully until you are able to apply the power rule for derivatives and the rules for constant, power rule, power rule with a constant, and addition and subtraction derivatives.

Read this section on finding derivatives. This section discusses techniques of differentiation, and at the end of the reading, you should be able to apply the basic rules and the product and quotient rules of differentiation as well as understand higher order derivatives.

Take notes as you watch this video, stopping at the 5:03 mark. Listen to the presentation carefully until you are able to apply the product rule of derivatives.

Complete problems 11-30 and 77. These exercises will provide you with the opportunity to apply the power, product, and quotient rules of derivatives. The solution for each problem can be found by clicking on the gray triangle beside each problem. The solution is the derivative of the function.

Complete the activity that tests your knowledge on derivatives using the definition with slope and limits. You can review the concepts associated with these questions with the Khan Academy videos in the "Stuck? Watch a Video" section (or review other content within the section). Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again. Work through all of the problems.

- 2.2.2: Trigonometry Derivatives
### 2.2.2: Trigonometry Derivatives

Read this section on derivatives of trig functions. Watch the video in the section as well. This instruction discusses the derivatives of trig functions, and at the end of the reading, you should be able to apply the six trig rules. Then, complete review questions 1-10 at the bottom of the page. These exercises will provide you with the opportunity to find the derivatives of trig functions. The solutions to these problems are located here.

- 2.2.3: Exponential and Log Derivatives
### 2.2.3: Exponential and Log Derivatives

Read this section on derivatives of exponential and logarithm functions, stopping at Example 3. This instruction discusses the derivatives of exponential and logarithm functions, and at the end of the reading, you should be able to apply these rules.

Complete problems 42-50 and 64-68. These exercises will provide you with the opportunity to apply exponential and logarithm rules of derivatives. The solution for each problem can be found by clicking on the gray triangle beside each problem. The solution is the derivative of the function.

Complete the activity that tests your knowledge on derivatives using the definition with slope and limits. You can review the concepts associated with these questions with the Khan Academy videos in the "Stuck? Watch a Video" section (or review other content within the section). Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again. Work through all of the problems.

Complete the activity that tests your knowledge on derivatives using the definition with slope and limits. You can review the concepts associated with these questions with the Khan Academy videos in the "Stuck? Watch a Video" section (or review other content within the section). Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again. Work through all of the problems.

- 2.2.4: Chain Rule
### 2.2.4: Chain Rule

Take notes as you watch these videos. Listen to the presentations carefully until you are able to apply the chain rule using this rule and the other rules of differentiation.

Read this section on derivatives of trig functions. Watch all of the videos within the section. This instruction discusses the chain rule as well as all of the other rules of differentiation. At the end of this reading, you should be able to apply the chain rule as well as the rules of differentiation discussed earlier.

Complete problems 31-41. These exercises will provide you with the opportunity to apply the chain rule for derivatives. The solution for each problem can be found by clicking on the gray triangle beside each problem. The solution is the derivative of the function.

Complete this activity, which tests your knowledge on derivatives using the definition with slope and limits. You can review the concepts associated with these questions with the Khan Academy videos in the "Stuck? Watch a Video" section (or review other content within the section). Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again. Work through all of the problems.

- 2.2.5: Implicit Differentiation
### 2.2.5: Implicit Differentiation

Take notes as you watch this video. Listen to the presentation carefully until you are able to understand and apply implicit differentiation.

Take notes as you watch this video. Listen to the presentation carefully until you are able to understand and apply implicit differentiation.

Take notes as you watch this video. Listen to the presentation carefully until you are able to understand finding the slope of a tangent line using implicit differentiation.

Read this section on implicit differentiation, stopping at "Review Questions." Watch all of the videos within the section. This instruction discusses implicit differentiation as well as all of the other rules of differentiation. At the end of this instruction, you should be able to apply implicit differentiation. Then, complete review questions 1-10 at the bottom of the page. These exercises will provide you with the opportunity to use implicit differentiation. The solutions to these problems are located here.

Read the material on describing an application of a derivative. This literacy component will allow you to further explore the concepts of a derivative.

- 2.3: Mean Value Theorem
### 2.3: Mean Value Theorem

Identifying maximum and minimum can be a challenging aspect of calculus, especially with complex polynomial and trigonometric functions. These points are valuable to determine when there is a maximum profit or height as well as a minimum loss. There is a specific theorem that can help us determine these values in an efficient manner.

Subunit 2.3 contains the mean value theorem, which is an important theorem in calculus, where a number exists on the interval that satisfies this theorem. Another important part of this theorem is determining when functions are continuous and differentiable.

Take notes as you watch this video. Listen to the presentation carefully until you are able to understand when a function is continuous and differentiable. The mean value theorem is also a point of focus that you need to understand and apply as well.

Read this section on extrema and the mean value theorem. Watch all of the videos within the section. At the end of the instruction, you should be able to apply and understand concepts of differentiability and continuous functions as well as theorems related to these concepts. Complete review questions 1-9 at the bottom of the page. These exercises will provide you with the opportunity to find extrema and apply Rolle's and mean value theorems. The solutions to these problems are located here.

Please click on the link above, and read the material on describing an application of a derivative and the mean value theorem, stopping at "Summary of Questions." This literacy component will allow you to further explore the concepts of a derivative and the mean value theorem. Write a one-page summary that discusses the key components of the article.

- 2.4: L'Hopital's Rule
### 2.4: L'Hopital's Rule

There are some instances in calculus where a limit cannot be determined from the given information, but the limit does exist. There is a rule that can be used along with previously learned concepts that can find these unique limits.

Section 2.4 uses L'Hopital's rule to explain limits at infinity when the limit is in an indeterminate form. This rule can help define these limits by applying rules of derivatives.

Take notes as you watch these videos. Listen to the presentation carefully until you are able to understand and apply L'Hopital's rule to various problems.

Please click on the link, and read the section on limits at infinity. Watch the video within the section. At the end of the instruction, you should be able to apply L'Hopital's rule and examine end behavior on infinite intervals and infinite limits at infinity. Then, complete review questions 1-10 at the bottom of the page. These exercises will provide you with the opportunity to apply L'Hopital's rule. The solutions to these problems are located here.

Complete this activity, which tests your knowledge on derivatives using the definition with slope and limits. You can review the concepts associated with these questions with the Khan Academy videos in the "Stuck? Watch a Video" section (or review other content within the section). Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again. Work through all of the problems.

- 2.5: Derivative as a Function
### 2.5: Derivative as a Function

The graph of a function has many key components that can be useful in various scenarios. A graph can tell a company when it is making a profit, when it is experiencing a loss, or when its revenues fluctuate between profit and loss. A sine wave is a natural occurring pattern in ocean, sound, and light waves. The graph of a function can also determine maximum and minimum amount as well.

Subunit 2.5 highlights derivatives and their relationship to the graphs of various functions. The first and second derivatives tests determine important points on the graph as well as increasing and concavity intervals. Other key components of graphing that are explored include these: domain/range, asymptotes, and x- and y- intercepts.

- 2.5.1: First Derivative Test
### 2.5.1: First Derivative Test

Take notes as you watch the video, stopping at the 4:35 mark. Listen to the presentation carefully until you are able to understand how the first derivative test is used to find maximum and minimum values.

Take notes as you watch the video. Listen to the presentation carefully until you are able to apply the first derivative test to determine minima, maxima, and critical points.

Read this section on the first derivative test. Watch all videos within the section. At the end of the instruction, you should be able to apply the first derivative test to determine extrema and the increasing and decreasing intervals of the function. Then, complete review questions 1-10 at the bottom of the page. These exercises will provide you with the opportunity to apply the first derivative test. The solutions to these problems are located here.

- 2.5.2: Second Derivative Test
### 2.5.2: Second Derivative Test

Take notes as you watch the video, starting at the 4:35 mark. Listen to the presentation carefully until you are able to understand how the second derivative test is used to find inflection points.

Read this section on the second derivative test. Watch the video within the section. At the end of the instruction, you should be able to apply the second derivative test to determine inflection point and intervals of concave up and concave down of the function. Complete review questions 1-9 at the bottom of the page. These exercises will provide you with the opportunity to apply the second derivative test. The solutions to these problems are located here.

Complete this activity, which tests your knowledge on derivatives using the definition with slope and limits. You can review the concepts associated with these questions with the Khan Academy videos in the "Stuck? Watch a Video" section (or review other content within the section). Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again. Work through all of the problems.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to understand all of the parts of the problem from the Calculus AB Exam.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to understand all of the parts of the problem from the Calculus AB Exam.

- 2.5.3: Analyzing the Graph of a Function
### 2.5.3: Analyzing the Graph of a Function

Take notes as you watch these videos. Listen to the presentation carefully until you are able to understand and apply the first and second derivatives test as well as find extrema and points of inflection for various functions.

Read this section on analyzing the graph of a function. Watch the first and last video within the section, omitting "Higher Order Derivatives: Part 1 of 2." At the end of the instruction, you should be able to apply the second derivatives test to determine inflection point and intervals of concave up and concave down of the function. Then, complete review questions 1-7 at the bottom of the page. These exercises will provide you with the opportunity to apply the first and second derivative tests as well as find all parts of a function. The solutions to these problems are located here.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to understand all of the parts of the problem from the Calculus AB Exam.

- 2.6: Applications of Derivatives
### 2.6: Applications of Derivatives

Have you wondered how a measure is impacted by the rate of another value? Have you wondered how to maximize the profits of selling an item or determine the minimum amount of cost to make this item? Those in the business of developing apps for smartphones analyze how much they can sell an app for as well as the cost of creating the app. These ideas determine whether the developer can proceed with the app development and sale of the product. There are many applications of derivatives that explore these concepts.

Subunit 2.6 highlights the applications of derivatives. The rate of change and related rates are problems where a given rate is changing; you are to find the rate at the value related to the changing rate. Optimization problems determine the maximum and/or minimum amount of a given scenario.

- 2.6.1: Rates of Change
### 2.6.1: Rates of Change

Take notes as you watch these videos. Listen to the presentations carefully until you are able to understand how to find the rate of change between the radius and the area of a circle using derivatives, the rate of change of the height of a balloon using derivatives, and the rate of change between approaching cars using derivatives.

- 2.6.2: Related Rates
### 2.6.2: Related Rates

Take notes as you watch these videos. Listen to the presentations carefully until you are able to understand how to use the related rate to find the amount of water pouring into the cone using derivatives and how to find falling ladder related rates using derivatives.

Read this section on related rates. Watch all videos with the section. At the end of the instruction, you should be able to apply the derivatives to related rate problems. Then, complete review questions 1-8 (omitting the two subquestions before problem no. 1 and including the two subquestions beneath problem no. 8) at the bottom of the page. These exercises will provide you with the opportunity to apply related rates to various problems. The solutions to these problems are located here.

- 2.6.3: Optimization Problems
### 2.6.3: Optimization Problems

Take notes as you watch this video. Listen to the presentation carefully until you are able to solve optimization problems using derivatives.

Take notes as you watch these videos. Listen to the presentations carefully until you are able to understand and apply optimization of a box both graphically and analytically.

Read this section on optimization. Watch all videos with the section. At the end of the instruction, you should be able solve optimization applications using the first and second derivative tests. Then, complete review questions 1-8 at the bottom of the page. These exercises will provide you with the opportunity to apply concepts of optimization to various problems. The solutions to these problems are located here.

Complete the word problems 22-24. These exercises will provide you with the opportunity to practice each type of derivative word problem. The solution for each problem can be found by clicking on the gray triangle beside each problem.

- Unit 3: Integrals
### Unit 3: Integrals

Integrals are the final unit in this course. In a similar way to derivatives, you will approach integrals first by learning their formal definition, then by learning techniques by which to compute them, and then lastly learning about their applications. In this unit, you will also learn the FTC (fundamental theorem of calculus), learn to solve basic differential equations, and also learn some cool approximation techniques. One of the reasons this unit is really interesting is because of all the cool images that can be generated using integrals. You will see what this means very soon!

**Completing this unit should take you approximately 65 hours.** - 3.1: Techniques of Antidifferentiation (Integration)
### 3.1: Techniques of Antidifferentiation (Integration)

Antidifferentiation (integration) can model real-life applications to determine the relationship among functions or a group of functions. Integration rules can help you eventually solve problems in multiple dimensions. Antidifferentiation is the opposite operation of differentiation, and this allows you to find values, such as areas under a curve, that are challenging without these concepts.

Subunit 3.1 focuses on the various techniques of antidifferentiation (integration) used in calculus. The subunit starts with the basic rules of antidifferentiation and then moves on to specialized cases such as trigonometry, integration by parts, integration by substitution, and partial fractions. These methods are critical to understanding definite integrals and additional calculus-related applications.

- 3.1.1: Basic Antidifferentiation Rules
### 3.1.1: Basic Antidifferentiation Rules

Take notes as you watch these videos. Listen to the presentations carefully until you are able to understand the basic rules of an antiderivative (indefinite integral).

Read the material on indefinite integrals. This reading discusses eight basic indefinite integrals using the rules of derivatives to explain each step. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to find antiderivatives as well as indefinite integrals of various functions. The solutions to these problems are located here.

Complete exercises 1-5 and 10-12. These exercises will provide you with the opportunity to apply basic rules of integration. The solution for each problem can be found by clicking on the gray triangle beside each problem.

Read this material on describing the indefinite integral. Also run each of the applets to see examples. This literacy component will allow you to explore a further understanding of the indefinite integral.

- 3.1.2: Integration by Substitution
### 3.1.2: Integration by Substitution

Take notes as you watch the videos. Listen to the presentations carefully until you are able to understand the basic rules of integration by substitution.

Read this material on trigonometric integration. Then, complete review questions 1-7 toward the bottom of the page. These exercises will provide you with the opportunity to find antiderivatives as well as indefinite integrals of various functions using the substitution method. The solutions to these problems are located here.

Complete review questions 1, 3, and 7-9 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals of various functions using the substitution method. The solutions to these problems are located here.

Read this article on various functions and integration. This literacy component will allow you to explore a further understanding of function properties and integration. The blog also contains information on how a specific function cannot be integrated. Write a one-page summary that discusses the key components of the article.

- 3.1.3: Trigonometry Integration
- 3.1.3.1: Trigonometry Integrands
### 3.1.3.1: Trigonometry Integrands

Read this material on trigonometric integration, starting at "Trigonometric Integrands" and stopping at "Using Substitution on Definite Integrals." Then, continue on to watch the video entitled "Math Video Tutorials by James Sousa, Integration by Substitution, Part 2 of 2." This video discusses the process of integration by substitution (or the u-substitution method) and the basic trigonometric integrals. Then, complete review questions 8-12 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals of trigonometric functions using the substitution method. The solutions to these problems are located here.

Complete exercises 6-9. These exercises will provide you with the opportunity to apply basic rules of integration. The solution for each problem can be found by clicking on the gray triangle beside each problem.

- 3.1.3.2: Trigonometry Integrals
### 3.1.3.2: Trigonometry Integrals

Read this material on trigonometric integration. Then, continue on to watch all videos within the section. This material discusses the process of integration of powers of sines and cosines as well as secants and tangents. Then, complete review questions 1-7 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals of trigonometric functions using the substitution method and trigonometric integrals. The solutions to these problems are located here.

- 3.1.3.3: Trigonometry Substitutions
### 3.1.3.3: Trigonometry Substitutions

Take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to apply rules of trigonometric substitutions to find specific indefinite integrals.

Read this material on trigonometric substitutions. Then, continue on to watch both videos within the section. This material discusses the process of integration using trigonometric integrals, substitution method for integrals, and other techniques. Then, complete review questions 1-7 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals using trigonometric integrals, substitution method for integrals, and other techniques. The solutions to these problems are located here.

- 3.1.4: Integration by Parts
### 3.1.4: Integration by Parts

Take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to apply the formula for integration by parts to various functions.

Read this material on integration by parts. Then, continue on to watch all videos within the section. This material discusses the process of integration by parts for various functions. Then, complete review questions 1-9 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals using integration by parts. The solutions to these problems are located here.

- 3.1.5: Integration by Partial Fractions
### 3.1.5: Integration by Partial Fractions

Take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to apply the formula for integration by parts to various functions.

Read this material on integration by partial fractions. Then, continue on to watch the first two videos within the section, omitting the "MIT Courseware" video. This material discusses the process of integration by using partial fractions. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to find indefinite integrals using integration by using partial fractions. The solutions to these problems are located here.

- 3.2: Interpretations and Properties of Definite Integrals
### 3.2: Interpretations and Properties of Definite Integrals

Definite integrals allow us to find the exact area underneath a curve. Without these integrals, it would be very difficult to even estimate the area under a curve that is not a linear function. The rules of integration allow us to find the area underneath any function.

Subunit 3.2 focuses on finding definite integrals for various functions. The rules of integration learned in subunit 3.1 will be explored further as we work to find the exact values of integrals over a given region.

Take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to apply the rules of integration and then find the area of a specific region.

Read this material on evaluating definite integrals. This reading discusses the process of finding definite integrals and incorporates the rules of integrals to solve problems. Then, complete review questions 1-8 and 10 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals using integration rules. The solutions to these problems are located here.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to apply the rule of integrations to find a definite integral.

Complete review questions 2, 4, 6, and 10 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

Complete review questions 13-15 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

Complete review questions 10-11 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

Complete review questions 5-6 toward the bottom of the page. These exercises will provide you with the opportunity to find definite integrals of various functions using the various integral rules. The solutions to these problems are located here.

- 3.3: Fundamental Theorem of Calculus
### 3.3: Fundamental Theorem of Calculus

How are a derivative and an integral related? The fundamental theorem of calculus ties these concepts together and helps us to quickly find the area underneath the curve of a function. This theorem allows us to perform complex calculations at a faster rate using a very simple rule.

The fundamental theorem of calculus provides us with another approach to finding definite integrals and is used when multiple functions are involved in finding a specific area.

Take notes as you watch these videos. Listen to the presentations carefully until you are able to understand how integrals and derivatives are use to prove the fundamental theorem of calculus and are able to apply the rule of integrations to find a definite integral.

Read this material on the fundamental theorem of calculus. Please continue on to watch the video in this section. This material discusses the process of using the fundamental theorem of calculus to evaluate definite integrals. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to apply the fundamental theorem of calculus to evaluate definite integrals. The solutions to these problems are located here.

- 3.4: Improper Integrals
### 3.4: Improper Integrals

What happens when an integral is undefined? What can we use when one of the values on the interval is infinite? Improper integrals help us define integrals when the fundamental theorem of calculus cannot be used because the interval is not continuous. Since not all functions are continuous within each interval, other approaches need to be used to find the value of the definite integral.

This subunit defines how to find the values of improper integrals using a variety of techniques. One approach is used when infinity is contained in the interval and limits are used. Another approach is used when there is discontinuity within the integral.

Take notes as you watch the video. Listen to the presentation carefully until you are able to understand the concept of integrals involving infinity.

Read this material on improper integrals. Then continue on to watch both videos in this section. This material discusses the process of using integration techniques to solve improper integrals. Then, complete review questions 1-6 toward the bottom of the page. These exercises will provide you with the opportunity to solve problems that contain improper integrals. The solutions to these problems are located here.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to apply the rules of integrals to solve a problem with an improper integral.

- 3.5: Applications of Integrals
### 3.5: Applications of Integrals

Applications of integrals are plentiful and allow one to find calculations to complex functions. With rules of integrals, we can find the area under a curve, and by expanding this rule, we can find the volume between curves. One example is determining the amount of work required to lift a bucket attached to a cable. Finding distance, velocity, and acceleration of an object being shot in the area is another application of integrals.

Subunit 3.5 explores the many applications of integrals. These include the area between curves, the volumes of various figures, the length of plane curves, and the area of various surfaces. There are also many applications within other disciplines that will be explored in this subunit.

- 3.5.1: Area between Two Curves
### 3.5.1: Area between Two Curves

Take notes as you watch the video. Listen to the presentation carefully until you are able to understand how to find the area between two curves.

Read this material on the area between two curves. Then continue on to watch the two videos in this section. This material discusses the process of finding the area between two curves with respect to the x-axis and y-axis. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to solve problems using integrals to find the area between two curves. The solutions to these problems are located here.

- 3.5.2: Volumes
- 3.5.2.1: Disc, Washer, and Shell Methods Around an Axis
### 3.5.2.1: Disc, Washer, and Shell Methods Around an Axis

Take notes as you watch the video, stopping at 15:33. Listen to the presentation carefully until you are able to find the volume of figures using disks and washers.

Read this material on volumes. This reading discusses the process of finding the volume of objects using the disk and washer methods. Then, complete review questions 1-12 toward the bottom of the page. These exercises will provide you with the opportunity to find the object of volumes using the disk, washer, and shell methods. The solutions to these problems are located here.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to find the volume of a figure rotated around an axis using the disk and washer methods.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to find the volume of a figure rotated around an axis using the shell method.

- 3.5.2.2: Disc, Washer, and Shell Methods Around a Non-Axis Line
### 3.5.2.2: Disc, Washer, and Shell Methods Around a Non-Axis Line

Take notes as you watch the video, starting at 15:33 and watching until the end. Listen to the presentation carefully until you are able to find the volume of figures using discs and washers rotating around a non-axis line.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to find the volume of a figure rotated around a horizontal and a vertical line using the disk method.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to find the volume of a figure rotated around a horizontal and a vertical line using the washer method.

Take notes as you watch the videos. Listen to the presentations carefully until you are able to find the volume of a figure rotated around a horizontal and a vertical line using the shell method.

- 3.5.3: The Length of a Plane Curve
### 3.5.3: The Length of a Plane Curve

Read this material on the area between two curves. Then move on to watch the two videos in this section. This material discusses the process of finding the length of a plane curve for a given function. Then, complete review questions 1-4 toward the bottom of the page. These exercises will provide you with the opportunity to solve problem using integrals to find the length of a plane curve. The solutions to these problems are located here.

- 3.5.4: Area of a Surface of Revolution
### 3.5.4: Area of a Surface of Revolution

Read this material on the area between two curves. Then move on to watch the two videos in this section. This material discusses the process of finding the area of surface of revolution. Then, complete review questions 1-6 toward the bottom of the page. These exercises will provide you with the opportunity to solve problems using integrals to find the area of a surface of revolution. The solutions to these problems are located here.

- 3.5.5: Other Applications
### 3.5.5: Other Applications

Read this material on the area between two curves. Then move on to watch all videos in this section. This material discusses the uses of integral applications from various disciplines. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity use integrals to solve problems from other disciplines. The solutions to these problems are located here.

- 3.6: Numerical Approximations to Definite Integrals
### 3.6: Numerical Approximations to Definite Integrals

There are times when the antiderivative is very challenging to find or may not even exist, which means the definite integral cannot be evaluated. What can one do in this case when the integral cannot be found? Three approaches can be taken to find the approximate value of the region under a curve.

Subunit 3.6 looks at various ways to use numerical approximations to find definite integrals. The Riemann sum, trapezoidal rule, and Simpson's rule can all be used to find an accurate approximation of an area under a curve.

- 3.6.1: Riemann Sums
### 3.6.1: Riemann Sums

Take notes as you watch the videos. Listen to the presentations carefully until you are able to use Riemann approximation to estimate areas under curves.

Read this material on the area between two curves. Then move on to watch the two videos in this section. This material discusses how to use Riemann sums to approximate the area under a curve. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to use Riemann sums to solve various integral problems. The solutions to these problems are located here.

Read this material on Riemann sums. In this reading, also run each of the applets to provide examples of calculating the area. This literacy component will allow you to explore an application of Riemann sums. Information in this reading pertains to finding the area under the curve using Riemann sums. Write a one-page summary that discusses the key components of the article.

- 3.6.2: Trapezoidal Sums
### 3.6.2: Trapezoidal Sums

Take notes as you watch the video. Listen to the presentations carefully until you are able to use trapezoidal approximation to estimate areas under curves.

Read this material on the area between two curves. Then move on to watch the two videos in this section. This material discusses how to use the trapezoidal and Simpson rules to approximate the area under a curve. Then, complete review questions 1-10 toward the bottom of the page. These exercises will provide you with the opportunity to use the trapezoidal and Simpson rules to solve various integral problems. The solutions to these problems are located here.