Is calculus hard? Sure. If it wasn’t, we’d have more math majors or more high school students completing it before they graduated. But we don’t. Getting into algebra is a bit frightening but most students can handle it. It’s when you advance to the higher levels of math where calculus resides that you quickly become aware of the fact that calculus is definitely a new level of difficulty. But that’s OK…just because it’s hard doesn’t mean it’s unattainable. We are here to make it just a little easier! Calculus I Essentials is offered here at Udemy to give you that extra edge no matter where you are in your calculus journey.
What is Calculus?
Webster’s definition of “calculus” is “an advanced branch of mathematics that deals mostly with rates of change and with finding lengths, areas, and volumes.” OK, good start. So basically calculus deals with change and measurements of stuff. Another important aspect you need to understand about calculus is its ability to allow you to predict how things will change. Engineers and mathematicians everywhere will argue (quite correctly) that the use of calculus affects so much of the material in our everyday lives.
For a thorough video and text explanation of beginning calculus, try our course: Become a Calculus I Master.
Calculus gives you the ability to find the effects of changing conditions on the system being investigated. By studying these, you can learn how to control the system to make it do what you want it to do. Calculus allows you to model and control systems which have extraordinary power over the material world.
The development of calculus and its applications to physics and engineering is probably the most significant factor in the development of modern science beyond where it was in the days of Archimedes. And this was responsible for the industrial revolution and everything that has followed from it including almost all the major advances of the last few centuries.
Where do we start?
The study of calculus begins with understanding single variable calculus. You first have to have a framework for describing such notions as position speed and acceleration.
Single variable calculus, which is what we begin with, deals with the motion of an object along a fixed path. The more general problem, when motion takes place on a surface, or in space, can be handled by multivariable calculus. We study multivariable calculus by finding clever tricks for using the one dimensional ideas and methods to handle the more general problems. So single variable calculus is the key to the general problem as well.
When we deal with an object moving along a path, its position varies with time. We can describe its position at any time by a single number, which can be the distance in some units from some fixed point on that path, called the “origin” of our coordinate system. The motion of the object is then characterized by the set of its numerical positions at relevant points in time.
The set of positions and times that we use to describe motion is what we call a function. And similar functions are used to describe the quantities of interest in all the systems to which calculus is applied.
What does a Calculus course look like?
A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the “derivative”) of various functions. (The process of doing so is called “differentiation”.)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called “integration”.)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.
There are a few other standard topics depending on the course. These include a description of functions in terms of power series, and the study of when an infinite series “converges” to a number. But sometimes courses cut off early and you never reach the more intense topics.
Some Key Concepts
Numbers are numbers.
Functions are sets of (argument, value) pairs of numbers. They are often described by formulae which tell us how to compute the value from the argument. Only one value is allowed for each argument. These formula usually start with the identity function, the exponential function and the sine function, and are defined by applying arithmetic operations, substitution and inversion in some manner to them.
The derivative of a function at any argument is the slope of the straight line it resembles near that argument, if that slope is finite. The straight line it resembles near that argument is called the tangent line to the function at that argument and the function describing that line is called the linear approximation to the function at that argument. If the function does not look like a straight line near an argument, (has a kink or a jump or crazy behavior there) it is not differentiable at that argument.
There are straightforward rules for calculating derivatives of the identity, sine and exponential functions, and for computing derivatives of combinations of these obtained by applying arithmetic operations, substitution and inversion in some manner to them.
Thus we have means to obtain formulae for the derivative of all functions of the kind described above. Armed with a spreadsheet, you can plot functions and determine their derivatives with great accuracy, most of the time, with little effort.
In a Nutshell …
Learning calculus is just like learning anything else. You have to break it down to its smaller parts and spend some quality time with areas you have trouble with. That’s it. Keep doing the problems for those rough areas over and over again until you completely understand how to work them. Calculus, in the end, is more intimidating than it is hard. Happy derivatives!