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Welcome to Algebra Foundations for Calculus! In this course, you'll learn the foundational Algebra principles needed to be successful in Calculus.
Instructors:
In this course, you'll watch videos in which Arun and Katie explain concepts and work through examples. We also provide many practice exercises so you can solidify what you're learning. Remember, learning math is just like learning a language, and the goal of this course is to make you fluent. Or, at least conversational. :)
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Lecture 1  02:10  
Meet your instructors and hear what you'll learn in the course. 

Section 1: The basics  

Lecture 2  00:49  
Read how you’ll learn, tips to get the most out of the course, and learning objectives.  
Lecture 3  01:05  
This course prepares you for advanced math courses, namely Calculus. But what is Calculus? Arun and Katie describe this special field of mathematics. 

Lecture 4  03:41  
One of the first steps to being successful in mathematics is the ability to properly notate mathematical ideas. Katie and Arun show examples of how you should write different mathematical expressions, especially using symbols such as parentheses. 

Lecture 5  00:14  
To do the following exercises correctly, it’s essential that you use parentheses to properly evaluate the expressions. When you’re done, check your answers on the next page.  
Lecture 6  00:01  
See the solutions and check your answers. 

Lecture 7  04:40  
Lines are the simplest type of function. It’s important to be able to graph them and calculate the slope (i.e., the rate of change), which is one of the most fundamental concepts in Calculus. While lines have a constant slope, the slope of other functions continuously changes. Understanding lines is the first step toward working with curves.  
Lecture 8  00:18  
In the following exercises, practice graphing lines, finding the equations of lines, and calculating the slope.  
Lecture 9  00:01  
See the solutions and check your answers.  
Lecture 10  06:13  
Slope is one of the most fundamental ideas in Calculus. Arun and Katie describe how Calculus uses the idea of slope to solve realworld problems. 

Lecture 11  00:17  
Practice visualizing and making conjectures about the derivative with the following exercises.  
Lecture 12  00:02  
See the solutions and check your answers.  
Lecture 13  04:13  
Simplifying expressions is another fundamental skill, as this enables you to solve equations (Lesson 2). Rational expressions have a numerator and denominator. Katie presents examples of expressions that can and cannot be simplified.  
Lecture 14  00:03  
Practice simplifying and/or rewriting the following rational expressions. 

Lecture 15  00:01  
See the solutions and check your answers.  
Lecture 16  05:22  
Logarithms (logs) are an important type of function that you’ll work with often. Katie describes what logs are and how they’re written.  
Lecture 17  00:05  
In the following exercises, practice using properties of logs to find the value of x.  
Lecture 18  00:01  
See the solutions and check your answers.  
Lecture 19  05:04  
Trigonometric (trig) functions are crucial to understand as well. Unlike the other functions you’ve learned about, trig functions are used to describe the relationships between angle measurements and the sides of the triangles that contain the angles. Katie describes how to use the Unit Circle to find sin( 

Lecture 20  00:04  
Use the unit circle to compute the following. 

Lecture 21  00:02  
See the solutions and check your answers.  
Section 2: Solving equations and inequalities  
Lecture 22  01:01  
Welcome to Lesson 2: Solving equations and inequalities! In this lesson you’ll see many examples of how to solve various equations, and have ample opportunity to practice. Read about the importance of being able to solve equations and realworld examples where you might need to.  
Lecture 23  04:24  
Arun and Katie describe different methods for solving quadratic equations and provide examples.  
Lecture 24  00:10  
Practice using the shortcut (finding two numbers that multiply to get the constant term and that add to get the coefficient of x) or the quadratic formula to solve the following quadratic equations.  
Lecture 25  00:01  
See the solutions and check your answers.  
Lecture 26  07:06  
Arun and Katie explain how to solve equations involving absolute values and highlight misconceptions that can lead you to the wrong conclusions.  
Lecture 27  00:06  
Practice solving equations involving absolute values. You may use the techniques you learned in the video on solving quadratics. 

Lecture 28  00:03  
See the solutions and check your answers.  
Lecture 29  05:37  
Solving polynomials can sometimes be a bit of a headache, but there are things you can do to get to the roots of the issue (literally).  
Lecture 30  00:10  
Use the techniques described in the last video (guessing a root based on the leading coefficient and the constant term, then performing synthetic or long division to check) to solve the following equations.  
Lecture 31  00:01  
See the solutions and check your answers.  
Lecture 32  02:54  
Solving rational equations can seem daunting, but you can save a whole lot of time by understanding a few key points. Katie and Arun describe what you should and shouldn’t do to correctly solve these types of equations.  
Lecture 33  00:06  
Practice solving the following equations, keeping in mind the original problems and ensuring your derived answers make sense. 

Lecture 34  00:01  
See the solutions and check your answers.  
Lecture 35  04:01  
You can now apply what you learned about logs and exponents in Lesson 1. In order to solve equations involving these functions, you have to apply their properties. Katie walks you through several examples. 

Lecture 36  00:04  
Practice solving exponential equations using the techniques you learned in the last video. 

Lecture 37  00:09  
See the solutions and check your answers.  
Lecture 38  04:37  
Now, you’ll apply the properties of logs and exponents toward solving logarithmic equations.  
Lecture 39  00:03  
Practice solving the following equations that involve logarithmic equations. 

Lecture 40  00:01  
See the solutions and check your answers.  
Lecture 41  04:24  
In order to solve trigonometric equations, you sometimes need to apply identities (i.e., rules that state what common expressions are equal to). This video walks through some of the most important trig identities to know and uses them to solve equations.  
Lecture 42  00:08  
The following lists useful trig identities. If you want to challenge yourself, try proving each of them.  
Lecture 43  00:04  
Find all real solutions to the following equations that involve trigonometric functions.  
Lecture 44  00:02  
See the solutions and check your answers.  
Lecture 45  00:30  
Solving inequalities is the last concept you’ll learn in this lesson. Solving them is very similar to solving equations (and you apply largely the same techniques), except now your solutions will involve a range of values. To find this range, you first have to be able to solve equalities, which is why you learned that first. Before diving into some examples, read why being able to solve inequalities is so important. 

Lecture 46  04:32  
Being able to solve inequalities is important to be able to graph functions because you’ll often want to know when the derivative f’(x) is greater than 0 (meaning the original function f(x) is increasing) or less than 0 (meaning the original function f(x) is decreasing). Arun articulates this important concept, and Katie walks you through examples of solving inequalities.  
Lecture 47  00:01  
Find the intervals of x for which the following inequalities are true.  
Lecture 48  00:01  
See the solutions and check your answers.  
Section 3: Graphing  
Lecture 49  00:16  
Welcome to Lesson 3: Graphing! In this module, read about what you’ll learn in this lesson. 

Lecture 50  05:05  
Oftentimes you’ll base your graph of a function off of the “parent function,” which is the simplest form of that type of function. Arun and Katie describe common parent functions and how you would change the graph of them to depict a more complicated function of the same type.  
Lecture 51  00:04  
Sketch each of the following functions by visualizing the parent function and applying the translation.  
Lecture 52  00:09  
See the solutions and check your answers.  
Lecture 53  03:54  
To properly graph all kinds of functions (many of which don’t have a parent function you can easily translate), you need to first find the domainall xvalues at which the functions exist. Some functions don’t exist at a certain point; some don’t exist at a range of points. Arun and Katie describe the domain and how to find it, and present examples of functions with different domains.  
Lecture 54  00:03  
Find the domain of the following functions.  
Lecture 55  00:02  
See the solutions and check your answers.  
Lecture 56  03:54  
One nice thing about polynomials is that their domain is all real numbers. However, they’re still difficult to graph in that you need to know where the function is increasing and decreasing. Katie describes how Calculus is very useful in finding this out.  
Lecture 57  00:15  
Graph the following polynomials given the equation of each, as well as the equation of the derivative f’(x). (Note: The point is not to graph them perfectly, but to be able to roughly visualize it, particularly where it’s increasing, decreasing, positive, negative, and where it intersects the xaxis. Remember that f’(x) tells us where the function is increasing and decreasing.) 

Lecture 58  00:04  
See the solutions and check your answers.  
Lecture 59  05:52  
Similar to polynomials, you need to know where rational functions are increasing and decreasing in order to properly graph them. In addition, you also need to know the domain, since it no longer necessarily consists of all real numbers. In this video, Katie describes how to find horizontal asymptotes, vertical asymptotes, and holes.  
Lecture 60  00:06  
Graph the following rational functions by first finding their domain (where they have holes or vertical asymptotes) and any horizontal asymptotes. 

Lecture 61  00:09  
See the solutions and check your answers.  
Lecture 62  02:21  
Become more familiar with log and exponential parent functions and see examples of how to graph more complicated functions of these types.  
Lecture 63  00:02  
Graph the following logarithmic and exponential functions.  
Lecture 64  00:11  
See the solutions and check your answers. 

Lecture 65  02:31  
You can now use what you learned about the Unit Circle in Lesson 1 to be able to graph trigonometric functions.  
Lecture 66  00:01  
Use what you know about graphing f(x) = sin(x) and f(x) = cos(x) to solve the following problems.  
Lecture 67  00:07  
See the solutions and check your answers.  
Lecture 68  00:02  
Piecewise functions are composed of several different functions with different equations. We will not go deep into it in this course, but it’s important to know what they are.  
Section 4: Limits and series  
Lecture 69  00:16  
Welcome to the final lesson in this course, Lesson 4: Limits and series! You’ll now learn some of the more complicated concepts that are actually part of Calculus. Read an introduction to limits and series.  
Lecture 70  03:12  
Limits are an abstract concept used to determine how a function is behaving near a certain point or as the function approaches positive and negative infinity. In this video, Arun and Katie describe what limits are and how they are used. 

Lecture 71  03:18  
You learned in Lesson 1 that the slope, or rate of change, is one of the fundamental ideas of Calculus. Limits are essential in order to find the rate of change at any given point on a curve. Katie describes how to use limits to find the rate of change of a function at a specific point.  
Lecture 72  00:00  
Use limits to find the derivative (f’(x)) or the slope of the function at a certain point (f’(x*)). 

Lecture 73  00:00  
See the solutions and check your answers. 

Lecture 74  00:00  
The final modules in this course are on series. Read about the two most common types of series: arithmetic and geometric.  
Lecture 75  07:26  
Arun tells a story about how an 8yearold boy added the numbers 1 through 100 in a matter of seconds using the principles of arithmetic series. Then, Katie describes how you can quickly add any pattern of numbers that increase by a set amount each term.  
Lecture 76  00:00  
Use what you learned about arithmetic series in the last video to solve the following problems.  
Lecture 77  00:00  
See the solutions and check your answers.  
Lecture 78  04:30  
Now you’ll learn about geometric series: a pattern of numbers in which you multiply a particular number to each term (instead of adding, as in arithmetic series). Hear a story that showcases how quickly a pattern of numbers can increase (essentially, exponential growth).  
Lecture 79  00:00  
Use what you’ve learned about geometric series to calculate the sums of the following finite geometric series.  
Lecture 80  00:00  
See the solutions and check your answers.  
Lecture 81  04:18  
Geometric series don’t necessarily explode to infinity. In fact, sometimes as you add each new term, the entire sum approaches a finite number. And if you add an infinite number of terms (theoretically), the sum will equal this number. This video describes these cases, called converging series.  
Lecture 82  00:00  
Practice solving infinite geometric series.  
Lecture 83  00:00  
See the solutions and check your answers.  
Lecture 84  00:44  
Congratulations on completing Algebra Foundations for Calculus! 

Lecture 85 
Before you go...

00:40 
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