Algebra Foundations for Calculus

Learn exactly what you need to in order to be successful in Calculus
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  • Lectures 85
  • Length 2 hours
  • Skill Level Intermediate Level
  • Languages English
  • Includes Lifetime access
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About This Course

Published 1/2016 English

Course Description

Welcome to Algebra Foundations for Calculus! In this course, you'll learn the foundational Algebra principles needed to be successful in Calculus.

Instructors:

  • Arun Sharma, PhD, Calculus professor at UC Berkeley
    After majoring in math, Arun got his PhD in pure mathematics from UC Berkeley in 2009. Since 2010 he has been teaching introductory math classes at UC Berkeley. Because of this, he intimately understands the areas in which students struggle. He feels many students entering college math classes are not as prepared in Algebra as they should be. Because of this, he is excited to offer this course to help ensure students succeed in Calculus and beyond. A fun fact about Arun is that he is highly ranked in chess and runs the US Chess League.
  • Katie Kormanik, CEO, TURN THE WHEEL
    Katie designs the courses, selects the experts and leaders the courses feature, and creates all the engaging content that you’ll see when you sign up. She is passionate about education and particularly excited about the potential of online learning. She has designed courses for Udacity, Stanford Graduate School of Business, and McKinsey Academy. Click here to see samples of her work. She has consulted for a number of education non-profits, start-ups, and for-profit companies on product development, curriculum development, pedagogy, and investments in edtech. And of course, she has been an educator in a number of capacities, from tutoring to teaching supplemental college courses, after-school programs, and summer school programs. Her two greatest passions are learning and making learning fun for others.

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. :)

A few tips as you work through this course:

  • Re-watch the videos to refresh your memory
  • Keep a pen and paper with you so you can do the exercises
  • Pause the videos occasionally to give you time to process the concepts
  • There's no need for calculators. This course is less about performing basic calculations and more about understanding the principles

For a limited time, we are offering the course half-off ($49) to students in exchange for detailed feedback in order to make the course even better. Use the coupon code ALGFOUNDATIONS-HALFOFF4FEEDBACK.

What are the requirements?

  • Students should know the order of operations, basic arithmetic, and how to evaluate functions at a particular value
  • Students should have paper and pencil handy as they work through the course so that they can attempt the exercises and check their answers

What am I going to get from this course?

  • Become fluent in the language of math
  • Solve equations and inequalities involving exponents, logarithms, trigonometric functions
  • Graph polynomials and exponential, logarithmic, and trigonometric functions
  • Define and use the derivative--the most important concept in Calculus--to better understand how functions behave
  • Use limits to graph functions and find the derivative
  • Find sums of finite and infinite series

What is the target audience?

  • This course is meant for anyone preparing to take Calculus, as well as anyone who would like a foundational understanding of pure mathematics
  • This course is NOT meant for those who seek to learn applied mathematics (for these students, I recommend our textbook Street-Smart Stats: A Friendly Introduction to Statistical Research Methods)

What you get with this course?

Not for you? No problem.
30 day money back guarantee.

Forever yours.
Lifetime access.

Learn on the go.
Desktop, iOS and Android.

Get rewarded.
Certificate of completion.

Curriculum

02:10

Meet your instructors and hear what you'll learn in the course.

Section 1: The basics
Article
Read how you’ll learn, tips to get the most out of the course, and learning objectives.
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.

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.

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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.
Article

See the solutions and check your answers.

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.
Article
In the following exercises, practice graphing lines, finding the equations of lines, and calculating the slope.
Article
See the solutions and check your answers.
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 real-world problems.

Article
Practice visualizing and making conjectures about the derivative with the following exercises.
Article
See the solutions and check your answers.
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.
Article

Practice simplifying and/or rewriting the following rational expressions.

Article
See the solutions and check your answers.
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.
Article
In the following exercises, practice using properties of logs to find the value of x.
Article
See the solutions and check your answers.
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(

Article

Use the unit circle to compute the following.

Article
See the solutions and check your answers.
Section 2: Solving equations and inequalities
Article
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 real-world examples where you might need to.
04:24
Arun and Katie describe different methods for solving quadratic equations and provide examples.
Article
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.
Article
See the solutions and check your answers.
07:06
Arun and Katie explain how to solve equations involving absolute values and highlight misconceptions that can lead you to the wrong conclusions.
Article

Practice solving equations involving absolute values. You may use the techniques you learned in the video on solving quadratics.

Article
See the solutions and check your answers.
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).
Article
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.
Article
See the solutions and check your answers.
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.
Article

Practice solving the following equations, keeping in mind the original problems and ensuring your derived answers make sense.

Article
See the solutions and check your answers.
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.

Article

Practice solving exponential equations using the techniques you learned in the last video.

Article
See the solutions and check your answers.
04:37
Now, you’ll apply the properties of logs and exponents toward solving logarithmic equations.
Article

Practice solving the following equations that involve logarithmic equations.

Article
See the solutions and check your answers.
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.
Article
The following lists useful trig identities. If you want to challenge yourself, try proving each of them.
Article
Find all real solutions to the following equations that involve trigonometric functions.
Article
See the solutions and check your answers.
Article

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.

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.
Article
Find the intervals of x for which the following inequalities are true.
Article
See the solutions and check your answers.
Section 3: Graphing
Article

Welcome to Lesson 3: Graphing! In this module, read about what you’ll learn in this lesson.

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.
Article
Sketch each of the following functions by visualizing the parent function and applying the translation.
Article
See the solutions and check your answers.
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 domain--all x-values 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.
Article
Find the domain of the following functions.
Article
See the solutions and check your answers.
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.
Article
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 x-axis. Remember that f’(x) tells us where the function is increasing and decreasing.)
Article
See the solutions and check your answers.
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.
Article

Graph the following rational functions by first finding their domain (where they have holes or vertical asymptotes) and any horizontal asymptotes.

Article
See the solutions and check your answers.
02:21
Become more familiar with log and exponential parent functions and see examples of how to graph more complicated functions of these types.
Article
Graph the following logarithmic and exponential functions.
Article

See the solutions and check your answers.

02:31
You can now use what you learned about the Unit Circle in Lesson 1 to be able to graph trigonometric functions.
Article
Use what you know about graphing f(x) = sin(x) and f(x) = cos(x) to solve the following problems.
Article
See the solutions and check your answers.
Article
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
Article
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.
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.

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.
Article

Use limits to find the derivative (f’(x)) or the slope of the function at a certain point (f’(x*)).

Article

See the solutions and check your answers.

Article
The final modules in this course are on series. Read about the two most common types of series: arithmetic and geometric.
07:26
Arun tells a story about how an 8-year-old 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.
Article
Use what you learned about arithmetic series in the last video to solve the following problems.
Article
See the solutions and check your answers.
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).
Article
Use what you’ve learned about geometric series to calculate the sums of the following finite geometric series.
Article
See the solutions and check your answers.
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.
Article
Practice solving infinite geometric series.
Article
See the solutions and check your answers.
00:44

Congratulations on completing Algebra Foundations for Calculus!

Before you go...
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Instructor Biography

TURN THE WHEEL Inc, Environment • Society • Economy

TURN THE WHEEL offers online courses that teach practical things you can do to improve the environment, society, and the economy and in doing so, live healthier, happier, and more successful lives.

Besides our online course Algebra Foundations for Calculus, we offer a friendly, colorful statistics textbook for beginners: Street-Smart Stats: A Friendly Introduction to Statistical Research Methods.

Subscribe on our website, turnthewheel dot org, to hear updates on new offerings in 2016!

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