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!
- Limits (Calculus Starts with Limits)
- Limit Definition (Precise Definition of Limit)
- Find the Limit (Techniques for Finding Limits)
- Continuous (Its Meaning and Applications)
- Horizontal Asymptotes
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.
- Rate of Change and Tangent Lines
- Derivative Definition (The Derivative as a Function)
- Differentiation Rules (with Examples)
- Product Rule and Quotient Rule
- Derivatives of Trigonometric Functions
- The Chain Rule
- Derivative Examples (The Role of the Derivative)
- Implicit Differentiation
- Derivatives of Inverse Functions
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!
- Indefinite Integrals
- Area and Limits of Riemann Sums
- Integral Definition (The Definite Integral)
- Fundamental Theorem of Calculus
- Integration by Substitution
- Area Between Curves (with Examples)
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Partial Fractions
- Improper Integrals
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.
- Related Rates
- Linearization and Differentials
- Extreme Value Theorem
- Mean Value Theorem
- Monotonic Functions
- First Derivative Test (and Curve sketching)
- Inflection Points and Concavity
- Optimization Problems
- Parametric Equations and Calculus
- L ‘Hospital’s Rule and Intermediate Forms
- Volumes of Solids of Revolution
- Numerical Integration
- Arc Length and Surfaces of Revolution
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”?
- Convergent Sequences and the Squeeze Theorem
- Infinite Series and Convergence
- Integral Test for Convergence
- Comparison Test
- P-Series Test
- Alternating Series Test and Conditional Convergence
- Ratio Test and Root Test
- Power Series
- Taylor Series
- Taylor Polynomials and Approximations
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