Algebra Foundations for Calculus

Name: Algebra Foundations for Calculus
Rating: 4.5 (25 reviews)

Learn exactly what you need to in order to be successful in Calculus

Created byKatie Kormanik

Last updated 5/2020

English

What you'll learn

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

Course content

5 sections • 84 lectures • 2h 2m total length

Welcome to Algebra Foundations for Calculus!2:10
Meet your instructors and hear what you'll learn in the course.

About the course0:44
Read how you’ll learn, tips to get the most out of the course, and learning objectives.
What is calculus?1:05
This course prepares you for advanced math courses, namely Calculus. But what is Calculus? Arun and Katie describe this special field of mathematics.
Write neatly3:41
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.
Exercises: Compute expressions
Solutions: Compute expressions
See the solutions and check your answers.
Graphing lines and calculating slope4:40
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.
Exercises: Find equations of lines and calculate slope0:18
In the following exercises, practice graphing lines, finding the equations of lines, and calculating the slope.
Solutions: Find equations of lines and calculate slope
See the solutions and check your answers.
Why is the slope important in Calculus?6:13
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.
Exercises: Graph the derivative0:14
Practice visualizing and making conjectures about the derivative with the following exercises.
Solutions: Graph the derivative
See the solutions and check your answers.
Simplifying rational expressions4:13
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.
Exercises: Simplifying rational expressions0:02
Practice simplifying and/or rewriting the following rational expressions.
Solutions: Simplifying rational expressions
See the solutions and check your answers.
Simplifying logarithmic expressions5:22
Logarithms (logs) are an important type of function that you’ll work with often. Katie describes what logs are and how they’re written.
Exercises: Simplifying logarithmic expressions0:03
In the following exercises, practice using properties of logs to find the value of x.
Solutions: Simplifying logarithmic expressions
See the solutions and check your answers.
Trig basics and the unit circle5:04
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(?), cos(?), and tan(?), where ? is the angle measurement for common angles, and how to describe these angles in degrees or radians.
Exercises: Find sine, cosine, and tangent0:02
Use the unit circle to compute the following.
Solutions: Find sine, cosine, and tangent
See the solutions and check your answers.

Why solve equations?0:59
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.
Solving equations: Quadratics4:24
Arun and Katie describe different methods for solving quadratic equations and provide examples.
Exercises: Solve quadratic equations0:08
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.
Solutions: Solve quadratic equations
See the solutions and check your answers.
Solving equations: Absolute values7:06
Arun and Katie explain how to solve equations involving absolute values and highlight misconceptions that can lead you to the wrong conclusions.
Exercises: Solve equations involving absolute value0:04
Practice solving equations involving absolute values. You may use the techniques you learned in the video on solving quadratics.
Solutions: Solve equations involving absolute value
See the solutions and check your answers.
Solving equations: Polynomials5:37
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).
Exercises: Solve polynomial equations0:08
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.
Solutions: Solve polynomial equations
See the solutions and check your answers.
Solving equations: Rational expressions2:54
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.
Exercises: Solve equations involving rational expressions0:04
Practice solving the following equations, keeping in mind the original problems and ensuring your derived answers make sense.
Solutions: Solve equations involving rational expressions
See the solutions and check your answers.
Solving equations: Exponents4:01
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.
Exercises: Solve exponential equations0:03
Practice solving exponential equations using the techniques you learned in the last video.
Solutions: Solve exponential equations
See the solutions and check your answers.
Solving equations: Logs4:37
Now, you’ll apply the properties of logs and exponents toward solving logarithmic equations.
Exercises: Solve logarithmic equations0:02
Practice solving the following equations that involve logarithmic equations.
Solutions: Solve logarithmic equations
See the solutions and check your answers.
Solving equations: Trigonometric functions4:24
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.
Trig identities0:06
The following lists useful trig identities. If you want to challenge yourself, try proving each of them.
Exercises: Solve trigonometric equations0:03
Find all real solutions to the following equations that involve trigonometric functions.
Solutions: Solve trigonometric equations
See the solutions and check your answers.
Why solve inequalities?0:29
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.
Solving inequalities4:32
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.
Exercises: Solve inequalities
Find the intervals of x for which the following inequalities are true.
Solutions: Solve inequalities
See the solutions and check your answers.

Introduction to graphing0:17
Welcome to Lesson 3: Graphing! In this module, read about what you’ll learn in this lesson.
Parent functions5:05
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.
Exercises: Graph each translation0:03
Sketch each of the following functions by visualizing the parent function and applying the translation.
Solutions: Graph each translation
See the solutions and check your answers.
Domain3:54
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.
Exercises: Find the domain0:01
Find the domain of the following functions.
Solutions: Find the domain
See the solutions and check your answers.
Graphing polynomials3:54
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.
Exercises: Graph polynomials0:14
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.)
Solutions: Graph polynomials
See the solutions and check your answers.
Graphing rational functions5:52
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.
Exercises: Graph rational functions0:05
Graph the following rational functions by first finding their domain (where they have holes or vertical asymptotes) and any horizontal asymptotes.
Solutions: Graph rational functions
See the solutions and check your answers.
Logarithmic and exponential functions2:21
Become more familiar with log and exponential parent functions and see examples of how to graph more complicated functions of these types.
Exercises: Graph logarithmic and exponential functions0:01
Graph the following logarithmic and exponential functions.
Solutions: Graph logarithmic and exponential functions
See the solutions and check your answers.
Graph trigonometric functions2:31
You can now use what you learned about the Unit Circle in Lesson 1 to be able to graph trigonometric functions.
Exercises: Graph trigonometric functions
Use what you know about graphing f(x) = sin(x) and f(x) = cos(x) to solve the following problems.
Solutions: Graph trigonometric functions
See the solutions and check your answers.
Piecewise functions
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.

Introduction to limits and series0:16
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.
Limits3:12
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.
Use limits to find the derivative3:18
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.
Exercises: Find the derivative
Use limits to find the derivative (f’(x)) or the slope of the function at a certain point (f’(x*)).
Solutions: Find the derivative
See the solutions and check your answers.
Introduction to sequences and series
The final modules in this course are on series. Read about the two most common types of series: arithmetic and geometric.
Arithmetic series7:26
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.
Exercises: Arithmetic series
Use what you learned about arithmetic series in the last video to solve the following problems.
Solutions: Arithmetic series
See the solutions and check your answers.
Finite geometric series4:30
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).
Exercises: Finite geometric series
Use what you’ve learned about geometric series to calculate the sums of the following finite geometric series.
Solutions: Finite geometric series
See the solutions and check your answers.
Infinite geometric series4:18
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.
Exercises: Infinite geometric series
Practice solving infinite geometric series.
Solutions: Infinite geometric series
See the solutions and check your answers.
Course close0:44
Congratulations on completing Algebra Foundations for Calculus!
Before you go...0:38

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

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

Who this course is for:

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)

Algebra Foundations for Calculus

What you'll learn

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Course content

Welcome to Algebra Foundations for Calculus!1 lecture • 2min

The basics19 lectures • 32min

Solving equations and inequalities27 lectures • 40min

Graphing20 lectures • 24min

Limits and series17 lectures • 24min

Requirements

Description

Who this course is for: