# I want a deep understanding of Limits, Differentiation and integration

#### wolfrose

Hello

I'm a teacher in electronics department, I did a light basics of limits, differentiation and integration last semester in just solving problems.

But I want a deep understanding of what these topics actually are? Where can I use limits? Is limits still valuable these days?

What is differentiation? I know it's useful in drawing the actual graph of a certain function. For example, the movement of a running man. Scientists can develop a mathematical function that approximates the movement of the running man, but these functions are actually the integration of this movement! Is it? This is my guess. And the differentiation is the functions that draw how the man actually is running, then the original integration functions are the final results of each movement for the running man.

#### JeffM1

First, I'd avoid trying to correlate the common meanings of "differentiation" and "integration" and the technical, mathematical meanings of the same words. I strongly suspect that nothing but confusion will result. The mathematical meanings arose during the development of calculus, which was a long, awkward process involving mathematicians whose common language was French or Latin, not English.

Second, calculus did indeed receive its initial impetus from physics and astronomy, and trying to develop a deep understanding of the concepts behind calculus from examples where calculus is applied to the physical sciences may be a promising approach. I know very little about physics and even less about astronomy so I cannot suggest those examples.

Third, the study of the concepts behind calculus is called analysis. There are two versions, standard and non-standard, both of which are considered logically rigorous. I'd not venture to study either one on my own.

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#### skipjack

Forum Staff
Differentiating gives rate of change. If a particular function gives the distance you have moved, differentiating it would give the corresponding velocity function for that movement, and differentiating again would give the corresponding acceleration function.

#### Joppy

Iâ€™m not sure what you mean by â€˜electrical departmentâ€™. Are you at a university or some kind of apprenticeship program for electricians (Iâ€™m not sure calculus would even be considered in this situation..)?

Either way, there are plenty of examples to draw from within your own area which Iâ€™m sure you already know about. Then there are some higher level concepts which you can possibly offer as motivation. Laplace and Z transforms for instance involve a lot of calculus (limits, integration etc.) and are fairly typical topics that an electrical engineer would learn in a control theory or signals class.

#### SDK

It is not clear from your post what level you are at so I will only make a brief recommendation. I am happy to expand with relevant help if you include more details about what level your math education is at.

In my opinion, the moment when derivatives "click" in some big picture sense is when you realize that a derivative is not really the slope of a tangent line. A derivative is a linear transformation (or linear mapping) which is the focus of a branch of math called linear algebra. Intuitively, a linear transformation is a function which "preserves straightness" by satisfying some nice properties. The importance of this class of functions is that we have a complete global understanding of them.

On the other hand, a typical differentiable function is nonlinear, and in general this makes global analysis of such functions impossible. However, the property of being differentiable means it is "locally linear". In other words, if a function is differentiable at a point in its domain, it means it can be approximated arbitrarily well by a linear transformation and this is what the derivative is.

This means that a derivative is not a slope or even a number. It isn't even a vector. Rather, it is the unique linear mapping which best approximates the function locally at a point.

The upshot is that if you want to understand calculus, then you should study linear algebra. Oddly enough, this is true of so many fields in math I can't understand why linear algebra isn't covered much earlier in the curriculum.

Edit: To avoid confusion, let me add that when I say a derivative is not a slope, I don't mean it literally. It so happens that every linear transformation on a one-dimensional domain has the form $x \mapsto mx$ so the definition of a derivative as a slope is true. What I mean by my earlier comments is that you should avoid thinking about the derivative as a slope since this falls apart in any more general case.

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#### wolfrose

First, I'd avoid trying to correlate the common meanings of "differentiation" and "integration" and the technical, mathematical meanings of the same words.
Yes, that's just was a quick brief expression for my understanding to "differentiation" and "integration". You're right, I think it's a more complicated process for many examples that deal with "differentiation" which also don't call for "integration". Also the same for "integration", in situation that process integration applications that don't need differentiation intervention.

Second, calculus did indeed receive its initial impetus from physics and astronomy, and trying to develop a deep understanding of the concepts behind calculus from examples where calculus is applied to the physical sciences may be a promising approach. I know very little about physics and even less about astronomy so I cannot suggest those examples.
Yes of course, big part of mathematics deals with the physical world.

Iâ€™m not sure what you mean by â€˜electrical departmentâ€™. Are you at a university or some kind of apprenticeship program for electricians (Iâ€™m not sure calculus would even be considered in this situation..)?
Sorry didn't explain that clearly! I'm an electronics trainer at a technical college. We have pretty basic calculus courses, not deep topics.

The new plan includes:
1. Some easy topics in trigonometry.
2. 1st, 2nd order differentiation equations.
3. Laplace transform.
4. Fourier transform.

>> All in easy simple orientation. Not much detailed topics, it's just this easy.

Either way, there are plenty of examples to draw from within your own area which Iâ€™m sure you already know about. Then there are some higher level concepts which you can possibly offer as motivation. Laplace and Z transforms for instance involve a lot of calculus (limits, integration etc.)
That requires an intensive study to get more information, but I think the best I can do is to explain simple examples as I go through the semester, because I have other courses; like, electronics projects which take a lot of time for preparation, implementation, ... etc. I thought to keep the calculus class as simple as possible as focus more on the other courses.

I opened this thread because I wanted as you mentioned to get more examples for motivation.

I think motivation is the main reason for this thread, because mostly the trainees have to solve mathematical problems any way, motivation isn't much for them, but I like some motivation
It is not clear from your post what level you are at so I will only make a brief recommendation. I am happy to expand with relevant help if you include more details about what level your math education is at.
As I mentioned above the situation. But my math education is not intensive, the same 2 basic courses I took in my diploma study which include similar basic, simple calculus and pre-calculus.

In my opinion, the moment when derivatives "click" in some big picture sense is when you realize that a derivative is not really the slope of a tangent line. A derivative is a linear transformation (or linear mapping) which is the focus of a branch of math called linear algebra. Intuitively, a linear transformation is a function which "preserves straightness" by satisfying some nice properties. The importance of this class of functions is that we have a complete global understanding of them.

On the other hand, a typical differentiable function is nonlinear, and in general this makes global analysis of such functions impossible. However, the property of being differentiable means it is "locally linear". In other words, if a function is differentiable at a point in its domain, it means it can be approximated arbitrarily well by a linear transformation and this is what the derivative is.

This means that a derivative is not a slope or even a number. It isn't even a vector. Rather, it is the unique linear mapping which best approximates the function locally at a point.

The upshot is that if you want to understand calculus, then you should study linear algebra. Oddly enough, this is true of so many fields in math I can't understand why linear algebra isn't covered much earlier in the curriculum.

Edit: To avoid confusion, let me add that when I say a derivative is not a slope, I don't mean it literally. It so happens that every linear transformation on a one-dimensional domain has the form $x \mapsto mx$ so the definition of a derivative as a slope is true. What I mean by my earlier comments is that you should avoid thinking about the derivative as a slope since this falls apart in any more general case.
I'm sorry, I'm really tired now, I'm planning to go through this part of your answer later, because I know it requires a lot of focus and some web-searches

Pick up the Larson or Stewart Calculus textbook. At the end of every section/chapter, you will find a few application problems that will apply to various real-world fields.

For example, you will find power dissipation and calculating the current in a wire as time approaches infinity in the limits chapter.

Since you are in an applied field, it would be best to stay away from pure mathematician responses. It sounds like analytics and mappings are outside the scope of your particular course.

Use the textbooks mentioned to teach a single concept, then use the example problem at the end of the chapter to show how that concept is used in the real world. Short, simple, and to the point.

Hope that helps!

#### wolfrose

Yes, of course that should help a lot.

#### wolfrose

What does the integration example at the bottom of the slide is called?

And is this the actual case for integration applications in real life?

I don't think that it's the case of integrating one function over the flat x-axis; like the one in the middle of the slide.

#### skipjack

Forum Staff
I don't recall a name for it. Real life applications vary, and some would be quite simple.

1 person