Get a detailed introduction to functions, graphs, limits, continuity, and derivatives, and explore the relationship between derivatives and graphs.
Calculus can be thought of as the mathematics of change. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently constant. Solving an algebra problem, like \( y = 2x + 5 \), merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate \( R \), such as \( Y = X_0+Rt \), where \( t \) is elapsed time and \( X_0 \) is the initial deposit. With compound interest, things get complicated for algebra, as the rate \( R \) is itself a function of time with \( Y = X_0 + R(t)t \). Now we have a rate of change that itself is changing. Calculus came to the rescue, as Isaac Newton introduced the world to mathematics specifically designed to handle those things that change.
Calculus is among the most important and useful developments of human thought. Even though it is over 300 years old, it is still considered the beginning and cornerstone of modern mathematics. It is a wonderful, beautiful, and useful set of ideas and techniques. You will see the fundamental ideas of this course over and over again in future courses in mathematics as well as in all of the sciences (like physics, biology, social sciences, economics, and engineering). However, calculus is an intellectual step up from your previous mathematics courses. Many of the ideas you will gain in this course are more carefully defined and have both a functional and a graphical meaning. Some of the algorithms are quite complicated, and in many cases, you will need to make a decision as to which appropriate algorithm to use. Calculus offers a huge variety of applications, and many of them will be saved for courses you might take in the future.
This course is divided into five learning sections, or units, plus a reference section, or appendix. The course begins with a unit that provides a review of algebra, specifically designed to help and prepare you for the study of calculus. The second unit discusses functions, graphs, limits, and continuity. Understanding limits could not be more important, as that topic really begins the study of calculus. The third unit introduces and explains derivatives. With derivatives, we are now ready to handle all of those things that change, as mentioned above. The fourth unit makes visual sense of derivatives by discussing derivatives and graphs. The fifth unit introduces and explains antiderivatives and definite integrals. Finally, the reference section provides a large collection of reference facts, geometry, and trigonometry that will assist you in solving calculus problems long after the course is over.
- Unit 1: Preview and Review
- Unit 2: Functions, Graphs, Limits, and Continuity
- Unit 3: Derivatives
- Unit 4: Derivatives and Graphs
- Unit 5: The Integral
- Calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and L'Hopital's Rule;
- State whether a function given by a graph or formula is continuous or differentiable at a given point or on a given interval, and justify the answer;
- Calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically;
- Calculate derivatives of polynomial, rational, and common transcendental functions, compositions thereof, and implicitly defined functions;
- Apply the ideas and techniques of derivatives to solve maximum and minimum problems and related rate problems, and calculate slopes and rates for functions given as parametric equations;
- Find extreme values of modeling functions given by formulas or graphs;
- Predict, construct, and interpret the shapes of graphs;
- Solve equations using Newton's method;
- Find linear approximations to functions using differentials;
- Restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer;
- State which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions;
- Find antiderivatives by changing variables and using tables; and
- Calculate definite integrals.
Learn how to use algebra for a variety of everyday tasks like calculating the change you will get back after a purchase, analyzing relationships by graphing, and describing real-world situations in business, accounting, and science.
Algebra allows us to solve many different types of calculations by using basic formulas and steps that work regardless of the specific numbers in the problem. For example, we can use an algebraic formula to calculate a monthly payment or to pay off a loan of any size using a given interest rate.
In this course, we explore how to use standard mathematical and business formulas and how to translate verbal descriptions into mathematical equations to establish relationships and create predictions. In the later units, we explore how to use graphs to make these same predictions. You can apply the problem-solving strategies we discuss in this course to business, science, and many other fields. To succeed in this introductory course, you should know how to perform operations with real numbers, including negative numbers, fractions, and decimals. Be sure to review our Foundations of Real World Math course if you need a refresher!
- Unit 1: Number Properties
- Unit 2: Linear Equations
- Unit 3: Word Problems
- Unit 4: Inequalities
- Unit 5: Graphs of Linear Equations and Inequalities
- Unit 6: Systems of Linear Equations and Inequalities
- Unit 7: Operations with Monomials
- Unit 8: Operations with Polynomials
- Unit 9: Factoring Polynomials
- Evaluate and simplify algebraic expressions;
- Solve linear equations and inequalities in one variable;
- Solve systems of linear equations and inequalities;
- Solve literal equations for a given variable;
- Translate verbal phrases into algebraic (variable) expressions;
- Define percent, and solve basic percent problems;
- Apply simple interest formula to problems involving loans and savings accounts;
- Apply uniform motion formula to problems involving the motion of one or two objects;
- Solve word problems by identifying a variable and creating an equation or an inequality;
- Solve word problems by identifying two or more variables and creating a system of equations or inequalities;
- Plot points on the coordinate plane, and determine the coordinates of any point on the coordinate plane;
- Graph linear equations and inequalities in two variables on the coordinate plane;
- Calculate the slope of a line passing through two given points;
- Write an equation of a straight line in point-slope or slope-intercept form;
- Solve word problems by creating a graph of a straight line and interpreting the meaning of the slope and intercepts of the line in the context of a problem;
- Perform operations with algebraic, exponential expressions using the rules of exponents;
- Perform operations with polynomials;
- Identify polynomials that can be factored and determine the appropriate factoring strategy; and
- Identify the proper method to factor a given polynomial and use it to solve a quadratic equation.
Study the relationships between lines and angles, learn to calculate how much space an object covers, determine how much space is inside of a three-dimensional object, and explore other relationships between shapes, objects, and the mathematics that govern them.
In geometry, we study the rules of the spaces and objects in our world. Geometry allows us to make accurate predictions about the sizes of triangles, circles, and rectangles, which lets us calculate, design, and build. Geometry helps architects design studios, farmers buy the right amount of seeds for their land, engineers build houses, and pilots calculate the amount of time they need to fly to reach another city. We use geometry to calculate how much paint we need to buy to cover a wall and the exact angle we should use to launch a rocket to hit a distant target. Artists also use geometry, as you can see in this picture by M. C. Esher.
In this course, we also study the relationships that exist between lines and angles. Urban planners study lines and angles to efficiently arrange houses, buildings, roads, and highways. Our street maps, water supply, and electrical connections all depend on these precise geometric calculations. How much space is inside a three-dimensional object? You may not realize you are using principles of geometry when you are getting ready for a trip, and you need to calculate how to fit two adults, three kids, four suitcases, and a dog into your car.
- Unit 1: The Basics of Geometry
- Unit 2: Parallel Lines
- Unit 3: Triangles, Congruence, and Other Relationships
- Unit 4: Triangle Relationships
- Unit 5: Polygons and Quadrilaterals
- Unit 6: Similarity
- Unit 7: Right Triangle Trigonometry
- Unit 8: Circles
- Unit 9: Perimeter and Area
- Unit 10: Surface Area and Volume
- Identify angles, rays, line segments, and points, based on the undefined notions of point, line, distance along a line, and distance around a circular arc;
- Determine if figures are congruent;
- Calculate angles created by parallel lines;
- Determine if two given figures are similar;
- Use trigonometric ratios and the Pythagorean theorem to solve right triangles;
- Use relationships among inscribed angles, radii, and chords to find missing angle measurements;
- Find the volume of cylinders, pyramids, cones, and spheres;
- Describe objects using geometric shapes, their measures, and their properties;
- Use the formulas for the area and circumference of a circle to solve problems;
- Solve simple equations for an unknown angle in a figure using facts about supplementary, complementary, vertical, and adjacent angles; and
- Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Explore intermediate-level algebraic operations and learn how algebraic methods are used in real-world applications.
In this course, you will explore intermediate-level algebraic operations and learn how algebraic methods are used in real-world applications. Topics include systems of linear equations and inequalities, quadratic equations, exponential and logarithmic functions, and operations with rational expressions. Using algebra involves graphing functions and relating functions' properties to their graphs. You will apply these skills to create mathematical models for word problems. Each unit will include many application problems that will draw on your knowledge of the concepts presented in that unit. This course requires prior knowledge of basic algebra.
- Unit 1: Equations and Inequalities
- Unit 2: Introduction to Functions
- Unit 3: Algebraic Operations on Functions
- Unit 4: Linear Functions
- Unit 5: Polynomial Functions
- Unit 6: Rational Functions
- Unit 7: Exponential and Logarithmic Functions
- Unit 8: Exponential and Logarithmic Equations
- Unit 9: Systems of Equations and Inequalities
- Unit 10: Introduction to Conic Sections
- Unit 11: Introduction to Sequences and Series
- Solve linear, rational, quadratic, absolute value, exponential and logarithmic equations, and inequalities involving linear and nonlinear components;
- Summarize the fundamental properties of the basic functions and their inverses using standard function notation, graphing, and algebraic operations;
- Describe how transformations and combinations affect the fundamental properties of the basic functions using standard function notation, graphing, and algebraic operations;
- Model observed behavior, data, and real-life scenarios involving the basic functions using standard function notation, graphs, and algebraic operations;
- Illustrate the properties of rational and polynomial inequalities using standard notation, graphs, and algebraic operations;
- Apply the fundamental theorem of algebra to identify local properties of polynomials;
- Interpret solutions to systems of linear and nonlinear equations and inequalities constructed from real-world scenarios;
- Identify the basic properties of arithmetic and geometric sequences and series; and
- Summarize the fundamental properties of the basic conic sections using graphing and algebraic operations.
Examine the properties behind the concepts of probability and statistics by learning how to investigate the relationships between various characteristics of data.
If you invest in financial markets, you may want to predict the price of a stock in six months from now based on company performance measures and other economic factors. As a college student, you may be interested in knowing the dependence of the mean starting salary of a college graduate, based on your GPA. These are just some examples that highlight how statistics are used in our modern society. To figure out the desired information for each example, you need data to analyze. The purpose of this course is to introduce you to the subject of statistics as a science of data. Data abounds in this information age; extracting useful knowledge and gaining a sound understanding of complex data sets has been more of a challenge. In this course, we will focus on the fundamentals of statistics, broadly described as the techniques to collect, clarify, summarize, organize, analyze, and interpret numerical information.
This course will begin with a brief overview of the discipline of statistics and will then quickly focus on descriptive statistics, introducing graphical methods of describing data. You will learn about combinatorial probability and random distributions, which are the foundation for statistical inference. With inference, we will focus on estimation and hypothesis testing issues. We will also examine the techniques to study the relationship between two or more variables, known as regression. By the end of this course, you should understand what statistics represent, how to use statistics to organize and display data, and how to draw valid inferences based on data by using appropriate statistical tools.
- Unit 1: Statistics and Data
- Unit 2: Elements of Probability and Random Variables
- Unit 3: Sampling Distributions
- Unit 4: Estimation with Confidence Intervals
- Unit 5: Hypothesis Test
- Unit 6: Linear Regression
- Describe the meaning and importance of descriptive and inferential statistics;
- Distinguish between a population and a sample;
- Calculate measures of location, variability, and skewness;
- Apply simple principles of probability;
- Compute probabilities related to both discrete and continuous random variables;
- Analyze sampling distributions for statistical inferences;
- Analyze confidence intervals for means and proportions;
- Analyze data sets using descriptive statistics, parameter estimation, and hypothesis testing;
- Explain how the central limit theorem applies in inference;
- Interpret the results of hypothesis tests;
- Analyze the relationships between two variables using simple linear regression; and
- Use regression equations to make predictions.
Discover the fundamentals of math using scenarios you will come across every day that involve number properties, order of operations, fractions, fractions in other forms (decimals, ratios, proportions, and percents), and graphs.
A hungry professor walks into your aunt's coffee shop, where you sometimes work as a cashier and manager. After some indecision, your math professor customer notices she can buy a 6-pack or an 11-pack of delicious, coffee-infused chocolate cannolis. She would like to buy 49 pastries. How many 6-packs and how many 11-packs should you put together for her? Is it even possible to fulfill her order? (Take out a few sheets of paper and explore; your customer will wait patiently.)
Our cannoli story has a fascinating answer that speaks to some deep mathematics first explored in the late 1800s and re-explored in the 1980s (thanks to some chicken nuggets from McDonald's). We use math in our daily activities, and it plays an important role in nearly every career you can imagine, from business to cooking to farming to medicine and beyond. It is no surprise that many call math a "universal" language: people across the globe use the same numbers, formulas, and equations to help them navigate the world.
In this course, we study essential math concepts that will enrich your understanding of the world and illuminate a larger, mathematically rich universe. The three courses in Saylor Academy's Real World Math series not only discuss basic algebra and geometry topics but also show you how to apply these concepts to everyday life.
The material focuses on how math relates to common "real world" situations, transactions, and phenomena, such as personal finance, business, and the sciences. This "real world" focus will help you grasp the importance of the mathematical concepts you encounter in these courses and understand why you need quantitative and algebraic skills to succeed in college and in your day-to-day life.
For example, fractions allow us to tell interesting and useful stories that involve measurement, ratios, and proportions. Decimals and percentages are "fractions in disguise". They help us make financial decisions and measure or compare various types of data. This course will help clarify the different ways we represent data visually, such as with a bar or line graph.
We also examine how to interpret data, no matter how it is presented. This skill will help you read a chart that outlines the current mortgage interest rate or make sense of the latest statistics for your fantasy football league. Let's not forget our coffee-infused chocolate cannolis. You will use addition, subtraction, and multiplication to answer her question. However, we will not help you deliver the bad news to your professor-customer.
A Note on Numbers – Before we begin, let's clarify what we mean by the word numbers. We are usually referring to a quantity, such as five, seven, or 10. But mathematicians have created four different categories or types of numbers.
Here is a summary of four types of numbers (with the fancy symbols mathematicians use to refer to them):
- \(\mathbb{N}\) Natural numbers: all positive whole numbers: \(\mathbb{N}=\left \{ 1,2,3,4,5,... \right \}\)
- \(\mathbb{Z}\) Integers: all positive and negative whole numbers (and zero, too!): \(\mathbb{Z}=\left \{..., -3,-2,-1,0,1,2,3,... \right \}\)
- \(\mathbb{Q}\) Rational numbers: all positive and negative fractions (including integers): \( \mathbb{Q}=\left \{a/b\mathrm{,\: where\:}a\:\mathrm{and\:}b\mathrm{\:are\:integers} \right \}\)
- \(\mathbb{R}\) Real numbers: all possible positive and negative numbers: \(\mathbb{R}\)
Since it is difficult to present the entire set of real numbers as a collection of quantities, we usually visualize this set as a solid line of points that stretches on forever in both directions. This set includes all four number types in our list above, in addition to some more interesting expressions we won't discuss here. In Unit 1, we discuss the mathematical operations we can perform using all of these numbers: our number properties apply to all real numbers. However, most of our examples will focus on positive whole numbers (integers). In later units, we will explore negative whole numbers (integers) and fractions (also called rationals). You do not need to memorize these symbols, but know that integers can refer to positive and negative whole numbers. Fractions can also be positive or negative.
- Unit 1: Number Properties
- Unit 2: Common Multiples and Common Factors
- Unit 3: The Order of Operations
- Unit 4: Fractions and Rational Numbers
- Unit 5: Decimals
- Unit 6: Ratios and Proportions
- Unit 7: Percentages
- Unit 8: Graphs and Charts
- Use properties of the four basic operations (addition, subtraction, multiplication, and division) to set up and solve real-world and mathematical problems;
- Manipulate expressions involving fractions to set up and solve real-world and mathematical problems;
- Manipulate expressions involving ratios to set up and solve real-world and mathematical problems; and
- Manipulate expressions involving decimals to set up and solve real-world and mathematical problems.
