# Learn Calculus 1 (Start Here) – Enter the World of Calculus

by Dave
(DAVE)—

Calculus is a broad, well-developed subject, so where do we start? With limits! Calculus begins with limits and continuity of functions. In this article, I take the reader on a tour of the Calculus. Through derivatives and integrals, the story continues. Many applications of these concepts are also detailed. Calculus is an exciting journey, so let’s get started.

Welcome, calculus enthusiasts, and enthusiasts-to-be! This page is your introduction to the world of calculus. You may be studying it for the first time, or maybe for the first time since high school. You may be using it in your work, or maybe have just always wondered what all the fuss was about. However you found yourself here, we hope you find what you are looking for.

Have you found that mastering Calculus is tough? You’re not alone. Calculus 1  dispels the myth that Calculus can’t be friendly and proves that clear, sensible examples can actually save you study time. This page will provide an introductory map to this world. Starting with the basics of two dimensional calculus, it will continue through to modern applications like vector calculus. Here we will look at the purpose of calculus. The numbers and symbols can wait for now, but you will find help with problems on this site as well.

You shouldn’t need to spend days in the library burrowed in your Calculus textbook. We will help you save time and improve in your Calculus class with straightforward examples beyond your lectures. Without question, this book provides everything you need to study Calculus. Additionally, students can use Calculus 1 as a class supplement or as a more manageable alternative to your textbook.

Although many of us might not use calculus in our day to day lives, it has had a profound effect on the world as a whole. It’s crucial for much of the engineering, economics, and science involved in the great advancements of the last century. This means learning it can put you in a position to be a part of this change. But even if that’s not your goal, calculus and advanced mathematics in general provide a workout for your brain. In the same way football players lift weights to get stronger, you can do calculus to strengthen your mind in preparation for other mental activities. Calculus 2 is part of a series of books to introduce you to the field. Through them, you can gain important foundations for any field that interests you.

## Calculus in Two Dimensions

Calculus is the study of change. Some changes are so easy to model with calculus that a fifth grader could follow along. At other times things change in deeply complex patterns. Some of the problems in calculus took centuries to solve, despite the greatest minds working on them.

Calculus is also the study of adding things up. Finding the area of a rectangle is calculus on the simplest scale. Integrating will show us the same process with more and more clever methods.

### Calculus Starts with Limits

In Limits, we investigate one of the ideas at the foundation of calculus. Limits deal with not only very large changes, but very small ones. For example, we know from Newton’s Law’s of Motion, together with his law of gravity, that an object moving away rom the Earth will always slow down. Gravity always pulls it back towards the Earth. It seems like an object that is always slowing down will eventually have to slow down to zero. Anything might eventually turn back around, returning to the Earth, but this isn’t the case!

As of early 2019, both of the Voyager spacecraft were moving between 35 and 40 thousand miles an hour away from the Sun. They are always slowing down, but slowing down less over time. This brings them closer and closer to some final constant speed. That speed is a limit. The spacecraft will get closer and closer to that speed as they get farther and farther away, but never reach it. Thinking in terms of very small limits helped Newton and Leibniz develop calculus in the 17th century. They used their new math to solve vexing problems of the day. The debate continues on who developed calculus first!

### Derivatives

In Derivatives, we will see the fruit of this early work by Leibniz, and especially Newton. Newton wanted to understand how to find the slopes of curves, which he called “crooked lines”. Solutions had been proposed which could address very simple mathematical curves, such as parabolas, circles, and ellipses. Nonetheless, several curves lacked any simple methods to determine their slopes. While thinking about the problem in terms of limits, he found a solution that worked for many different functions. Over time, methods were found to determine the slope, or derivative, of any function.

For Newton, this was important for questions in astronomy and physics, but today, being able to find derivatives is at the heart of studying change – from changes in speed or position, to changes in barometric pressure, changes in blood flow in the body, even changes in stock prices. This section will look at the original discovery, and at all of the unexpected applications of differentiation (the process of finding derivatives) that have arisen over the last few centuries. Rather than focusing deeply on the mathematics, we will explore the full cycle of necessity, innovation, and application.

### Integrals

While Newton was deeply considering derivatives, Leibniz was more interested in integrals. Where derivatives give the slope of a curve at a given point, integrals give the area underneath a curve for a given portion of the curve. While on the surface, these sound like two completely different ideas, they are, in a sense, inverses of each other. Finding the derivative of a function has the opposite effect of finding the integral, much in the same way that addition has the opposite effect of subtraction.

Just as an example, if you have the speed of an object over time, finding the acceleration requires differentiating. If you have the acceleration over time, then finding the speed involves integrating. The two processes aren’t cleanly opposite of each other in the same way that addition and subtraction are. Why not? Jump into this section to explore this idea!

We will explore how Leibniz arrived at the calculus from this different approach, but still used this idea of limits. If we want to find the area under a curve, we can find it by adding up rectangles that fit under the curve. The smaller the rectangles, the more accurate we get, and we would be right on target if we could just add up an infinite number of rectangles. Integrating lets us do that!

### Applications of Derivatives and Integrals

It is no accident that the problems which inspired these tools are the first that we would explore, once it is time to dive into Applications. For derivatives, the most prominent example is the study of motion. Differentiating an equation for position over time gives us the velocity. Differentiating the velocity gives us acceleration. Newton needed these values as he was trying to understand how the planets moved through the solar system. Today, we realize that, while some study of motion can happen without derivatives, that this is a case where working to understand derivatives pays us back with some very simple answers to complex problems.

Leibniz was taking more of a geometrical approach, and integration has many applications in geometry. If there is a method for adding up an infinite number of things, suddenly it becomes possible to add up circles to find the area of a cone, or nearly any shape, as long as we have some way to describe its shape as a function. If we want to know how much energy it takes to complete a project, and have a way to describe how the energy changes with time, we can add up each of the infinitely tiny bits of work to get our answer through integration.

### Infinite Sequences and Series

These terms sound so similar, so they are worth defining carefully. A sequence is a list of numbers or expressions. A finite sequence contains just a few numbers, or just a few million. An infinite sequence continues indefinitely. A series is the sum of a sequence. So while (1, 2, 3, 4, 5) is a sequence, $(1+2+3+4+5)$ is a series, both of them finite. Assuming you see the pattern in those numbers, you might see that extending it infinitely would give you an infinite series that becomes… infinite. This is a divergent series because it never settles in on a final sum.

Calculus provides methods for finding the value of a series. Take a peek up above. Would it be integration or differentiation? This solved one of the greatest philosophical questions from ancient times, Zeno’s Paradox. Zeno imagined a person trying to walk from A to B. He proposed that this task involved walking halfway to the door, then half of the remaining distance, then half of that distance, and so on infinitely. If each of these distance was a finite distance, and there was an infinite number of steps, he reasoned it would take an infinite amount of time to complete this task. By applying calculus to infinite series such as this one, you will find yourself able to tell Zeno, “Not so fast!”, or perhaps, “Not so slow”?

## What You’ll Learn in Limits and Continuity

• Intuitive Definition of Limit
• One-Sided Limits
• Using Graphing Utilities to Evaluate Limits
• Computing Limits Using the Laws of Limits
• Limits of Polynomial and Rational Functions
• Limits of Trigonometric Functions
• Precise Definition of a Limit
• A Geometric Interpretation
• Continuous Functions
• Continuity at a Number
• Continuity at an Endpoint
• Continuity on an Interval
• Continuity of Composite Functions
• Intermediate Value Theorem
• An Intuitive Look at Tangent Lines and Rates of Change
• Estimating the Rate of Change of a Function from Its Graph
• More Examples Involving Rates of Change
• Defining a Tangent Line
• Tangent Lines, Secant Lines, and Rates of Change

### Chapter Description

We begin this course with an intuitive introduction to limits. First, we study limits using a table of values and or graphs to understand what a limit is. Then we look at one-sided limits and two-sided limits and especially make use of piecewise-defined functions.

Next, we begin studying how to compute limits. We learn the limit laws and the limits of polynomial and rational functions. We also study finding limits using rationalization and other algebraic approaches. Limits of trigonometric functions and squeeze theorem are also examined.

After using tables of values and graphs to get an intuitive idea of the limit, we then clearly show how that approach is ambiguous. We motivate the precise definition of a limit using epilson-deltas. Several examples are illustrated by proving the value of a limit using the precise definition of a limit. We also show how a limit doesn’t exist by considering a piecewise function and applying the limit’s precise definition. We end this lesson by proving the limit laws (using the precise definition of a limit).

After a thorough understanding of limits we will begin to study continuous functions. First, we discuss continuity at a number, and then continuity at an endpoint, and also continuity on an interval. We next explain the continuity of the composition of functions, including the sum, product, and quotients. We also consider the continuity of polynomial and rational functions and the continuity of trigonometric functions. Towards the end of this lesson, we explore the Intermediate Value Theorem, and we end with a discussion on the existence of zeros of a continuous function.

We end this course with a discussion of tangent lines and rates of change. After that, we explore estimating the rate of change of a function from its graph and explaining the tangent line. Finally, we end this course with a detailed explanation of the difference between an average rate of change and an instantaneous rate of change by working through several examples.

## What You’ll Learn in Theory of Derivatives

• What is the Derivative?
• Differentiation
• Finding the Derivative of a Function
• Using the Graph of f to Sketch the Graph of f’
• Differentiability
• Differentiability and Continuity
• Some Basic Rules
• The Derivative of the Natural Exponential
• The Product Rule
• The Quotient Rule
• Extending the Power Rule
• Higher-Order Derivatives
• Motion Along a Line
• Marginal Functions in Economics
• Derivatives of Sines and Cosines
• Derivatives of Other Trigonometric Functions
• Composite Functions
• The Chain Rule
• Applying the Chain Rule
• The General Power Rule
• The Chain Rule and Trigonometric Functions
• The Derivative of Exponentials
• Implicit Functions
• Implicit Differentiation
• Derivatives of Inverse Functions
• Derivatives of Inverse Trig Functions
• Derivatives of Rational Powers of x
• The Derivatives of Logarithmic Functions
• Logarithmic Differentiation
• The General Version of the Power Rule
• The Number e as a Limit
• Related Rates Problems
• Solving Related Rates Problems
• Increments
• Differentials
• Error Estimates
• Linear Approximations
• Error in Approximating

### Chapter Description

We begin this course with the formal definition of the derivative of a function. Then, we give both the geometric interpretation and the physical interpretation of the derivative. Next, we discuss differentiation and its notation, and we work through several examples of finding the derivative of a function. After that, we examine the differentiability of a function and the relationship between differentiability and continuity.

In the next lesson, we talk about differentiation’s basic rules, including the linearity rule and the power rule. Then, we examine the derivative of the natural exponential function. After that, we study the product and quotient rules by working through several examples. We also discuss higher order derivatives and derivative notation.

We follow those lessons with the role of the derivative in the real world. In particular, we examine motion along the line, including velocity, speed, acceleration, and jerk. An introduction to marginal analysis and other applications are also included.

In the next lesson, we discuss the derivatives of the trigonometric functions followed by the derivatives of the composition of functions. The chain rule is covered in great detail, both its statement and proof, and we work through several examples of applying the chain rule. Derivatives of trigonometric functions, exponential functions, and logarithmic functions are highlighted.

An exciting part of the course is when we cover implicit differentiation and logarithmic differentiation. The derivative of an inverse function and derivatives of inverse trigonometric functions are covered in great detail. Towards the end of the course, we study related rates. In great detail, we discuss what makes a related rates problem and guidelines for solving a related rates problem; several examples are given in various categories. In the final lesson in this course, we investigate differentials and linear approximations.

## What You’ll Learn in Applications of the Derivative

• Absolute Extrema of Functions
• Relative Extrema of Functions
• Finding the Extreme Values
• An Optimization Problem
• Rolle’s Theorem
• Mean Value Theorem
• Consequences of the Mean Value Theorem
• Determining the Number of Zeros
• Increasing and Decreasing Functions
• The First Derivative Test
• Finding the Relative Extrema of a Function
• Concavity
• Inflection Points
• The Second Derivative Test
• Determining the Shape of a Graph
• Infinite Limits
• Vertical Asymptotes
• Limits at Infinity
• Horizontal Asymptotes
• Interest Compounded Continuously
• Infinite Limits at Infinity
• Precise Definitions
• The Graph of a Function
• Guide to Curve Sketching
• Slant Asymptotes
• Formulating Optimization Problems
• The Indeterminate Forms Type I
• l’Hôpital’s Rule
• The Indeterminate Forms Type II
• The Indeterminate Forms Type III
• What is Newton’s Method?
• Applying Newton’s Method
• When Newton’s Method Does Not Work

### Chapter Description

This course begins with a thorough investigation into the extrema of functions. We study both absolute extrema and relative extrema by working through several examples. Next, we analyze Rolle’s Theorem and the Mean Value Theorem. We both motivate the theorem and provide proof. Some consequences of the Mean Value Theorem and several examples are detailed.

After that, we begin a thorough investigation into sketching the graphs of functions. Increasing and decreasing functions and the first derivative test are examined first. Following this, we study concavity, inflection points, and the second derivative test.

We then continue our investigation to the graphs of functions by studying limits involving infinity and different types of asymptotes. We cap these lessons off with a thorough guide to curve sketching, including slant asymptotes, vertical tangent, and cusps.

Now that a thorough analysis of the graph of a function is accomplished, we begin studying optimization problems. We establish guidelines for solving optimization problems and work through several examples to demonstrate how to use these guidelines.

The exciting conclusion to this course is to examine indeterminate forms. We begin by classifying seven indeterminate forms into three classes and explain the strategy behind each type. The primary tool being l’Hopital’s Rule along with algebraic manipulation.

In the end, we explore Newton’s method, how to apply it, and when it fails.

## What You’ll Learn in Theory of Integrals

• Antiderivatives
• The Indefinite Integral
• Basic Rules of Integration
• Differential Equations
• Initial Value Problems
• How the Method of Substitution Works
• The Technique of Integration by Substitution
• More Examples
• An Intuitive Look at Area
• Sigma Notation
• Summation Formulas
• The Area Problem
• Area and Distance
• Definition of the Definite Integral
• Defining the Area of the Region Under the Graph of a Function
• Geometric Interpretation of the Definite Integral
• The Definite Integral and Displacement
• Properties of the Definite Integral
• More General Definition of the Definite Integral
• The Mean Value Theorem for Definite Integrals
• The Fundamental Theorem of Calculus, Part I
• The Fundamental Theorem of Calculus, Part II
• Evaluating Definite Integrals Using Substitution
• The Definite Integral as a Measure of Net Change
• Definite Integrals of Odd and Even Functions
• Approximating Definite Integrals
• The Trapezoidal Rule
• The Error in the Trapezoidal Rule
• Simpson’s Rule
• The Error in Simpson’s Rule

### Chapter Description

This exciting course begins with a detailed look into antiderivatives. In particular, we study the basic integration formulas and discuss differential equations and initial value problems. After that, we expand our methods to add integration by substitution. We formalize this method and demonstrate how to use it through several examples. We include integrals of trigonometric functions and integrals of inverse trigonometric functions.

Next, we begin to motivate the definite integral through a serious look at the concept of area. We start with an intuitive, classical look and then proceed by studying sigma notation, finite sums and their properties, and limits of sums. After that, we end this lesson with a formal look at the area of the region under the graph of a function.

Finally, one of the main topics of calculus is reached: the Definite Integral. We begin with a formal definition of the definite integral and connect it to our previous examination of area. We illustrate the Riemann integral by using several examples, including a geometric interpretation of the definite integral and displacement. Several properties of the definite integral are proven and demonstrated through examples.

Towards the end of the course, we discuss (at length) the Fundamental Theorems of Calculus (no spoilers here). Then at the end, we study numerical integration techniques such as the trapezoidal rule and Simpson’s rule. The theorems are illustrated with many examples, and the errors involved with these methods are analyzed in detail.

## Examine a new world called Calculus 1. With these articles and videos that explain everything step-by-step, you will master calculus.

Calculus 1 is a high-school or undergraduate mathematics course. Newton and Leibniz invented calculus, and then it was significantly improved upon by Euler, Gauss, and many others. Topics center around the concepts of the derivative and the integral. However, to correctly understand calculus, an understanding of limits and continuity is essential. For this reason, students first study limits of functions, followed by the continuity of functions. Then the main topics of differentiation and integration fill the rest of the course. Towards the end, instructors surprise students with a magnificent theorem.