So, you came here because you were curious about what calculus is. Calculus is essentially the study of instantaneous change, infinitesimals, and areas under curves. It is a major subject in fields such as engineering, for example, where algebra alone will not suffice. It is a difficult subject, but it helps with many problems we face in mathematics. Without calculus we would have never had a formula for the area of a circle.
If you were looking for information pertaining to the history of calculus visit this link: http://en.wikipedia.org/wiki/History_of_calculus
The main definition of calculus
This is very similar to what I previously said. We are just adding names to the previous ideas. Calculus involves studying and applying the derivative to determine instantaneous change, using integrals to take the area under curves, and using infinitesimals, which we now call limits, to do all of these things. Limits are probably the simplest thing in calculus, but the most useful. They are a must when taking a derivative.
What are limits?
Limits are one of the most important ideas in calculus. So many things in the subject, such as derivatives, require them. Limits are the restraints of a function given how far you let it go. Limits are often expressed as lim x > h. In this example “x” is the variable in the function and “h” is what it is approaching. You would pronounce it as “the limit as x approaches h.”
For example, the limit of 2x as x approaches 4 is 8. This is because if we keep plugging in values for x that get closer and closer to 4, but are not 4, the function will never reach 8. Therefore, it is a limit. Oftentimes for limits you can just plug in what x is approaching with no problems, but this does not work every time.
What are derivatives and differentiation?
All in all, derivative is just a fancy word for slope, and differentiation is the process of finding the derivative. If we examine linear functions such as 2x + 1 we know the slope is 2. The derivative of that function also happens to be 2.
What is the slope of the function x^2. You are probably realizing that it forms a curve and does not have a constant slope. The slope can be taken by finding what the slope of any tangent line to the graph is. To do this we have to find a slope of a secant line, or a line that is touching two points on the graph, and take the limit of the slope as the change from the second point on our secant line approaches the first.
The slope, or derivative, can be represented as 2x. This means that the slope at any particular x-coordinate is two times that x-coordinate. The slope at x = 4, for example, is 8. How did I get this? I used what we call the definition of derivative. The definition of derivative states the following.
f ‘(x) = lim h > 0 (f(x + h) – f(x)) / h
f ‘(x) is pronounced “f prime of x” and is the derivative.
Now lets plug our x^2 into the equation and do this.
f ‘(x^2) = lim h > 0 ((x + h)^2 – x^2)) / h
f ‘(x^2) = lim h > 0 (x^2 + 2xh + h^2 – x^2) / h (expand the (x + h)^2)
f ‘(x^2) = lim h > 0 (2xh + h^2) / h (the positive and negative x^2 cancel)
f ‘(x^2) = lim h > 0 (2x + h) (now we can divide by h)
This enables us to take the limit as h goes to 0 and the answer is 2x. That is a basic derivative for you. They are an important concept in calculus and are very helpful in many careers you may want to pursue.
What are integrals?
Integrals are probably one of the hardest concepts in calculus to understand. They can be rigorous because there is no single way to do them. Every last one is done a little differently. Integrals are split up into two categories called the definite and indefinite integrals.
Integrals are the area under curves of functions between certain intervals. Since it would be very difficult to count all the little boxes on a graph that were under a curve we have integrals.
Indefinite integrals are a way to find any definite integral on a particular curve. Indefinite integrals are found by calculating the antiderivative of a function. Lets take the function x^2 for simplicity. The antiderivative, or indefinite integral, is 1/3x^3 + C. It is simply the inverse of the derivative. If you were to take the derivative of 1/3x^3 you would end up with x^2. I simply added the “C” because the derivative of any constant is 0. Therefore, there is no way of knowing what number, if any, was there. We call that the constant of integration.
Definite integrals are a much easier concept. It is simply applying the indefinite integral. If we want the area under the curve of the graph x^2 from 2 to 5, we just plug it into the antiderivative. The antiderivative as explained before is 1/3x^3. First, we plug in our second interval which is 5.
1/3(5)^3 = 41.66666….
Then, we plug in our first interval which is 2.
1/3(2)^3 = 2.66666….
We then subtract what our first yielded from our second.
41.6666…. – 2.6666…. = 39
The definite integral, or area under the curve, is 39.
That is what integrals are. They can be rigorous, but are a must for engineering.
The Fundamental Theorem Of Calculus:
This theorem simply states that differentiation and integration are inverses. We already somewhat explained that with the term antiderivative, but this theorem proves that it is true for all cases. It is the foundation for how we study calculus today, and it is important to know what it is.
I hope you are now informed about calculus. Whether you were just curious, or you were thinking about taking the course; I hope you got something out of this.