# Theory behind Calculus

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Hello! My son is doing calculus through CTC math, and it is going well. He completed the differential part of the course in the spring and the plan was to cover integrals this fall. However, coming back to it after the summer, he feels like he wants to understand the "why" behind working the problems. He asked his older brother, who was a math major in college, and was told that he was asking about something that covered two semesters of college classes, lol.

My question is what to do to give him some of the theory. Is there anything concise that could possibly provide that for him? I thought about the Art of Problem Solving Calculus book or Calculus Made Easy by Thompson and Gardner, but I'm actually looking for something more concise. Although my son is a strong math student, it is probably his least favorite subject. His quest to understand the theory is suprising me, actually!

I would appreciate any input! Thank you!

Edited by Shelly in VA
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I don’t know if this book will help (we just put it on hold at our local library), but it looks good.

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2 hours ago, Porridge said:

I don’t know if this book will help (we just put it on hold at our local library), but it looks good.

That looks interesting! I haven't seen it before. Thanks!

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Personally I think the easiest way to see the why of Calculus (the whys of Calculus?) is through physics.

Quote

# 0.2 What Is Calculus and Why do we Study it?

Calculus is the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.

I have been around for a while, and know how things change, more or less. What can calculus add to that?

I am sure you know lots about how things change. And you have a qualitative notion of calculus. For example the concept of speed of motion is a notion straight from calculus, though it surely existed long before calculus did and you know lots about it.

So what does calculus add for me?

It provides a way for us to construct relatively simple quantitative models of change, and to deduce their consequences.

Integrals is the area under the curve and differentiation is the rate of change.

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5 hours ago, Porridge said:

I don’t know if this book will help (we just put it on hold at our local library), but it looks good.

I've listened to it! Highly recommend it. I really, really wish I could have read it when I took calc. I never understood the "why" of any of it, and this book explains it really well.

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Does he want to know why it works, or why one would want to solve those problems? For the latter, take physics. It becomes obvious why physicists would have invented calculus - rates if change ( derivatives) and summation of tiny parts of a continuum (integral)

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11 hours ago, regentrude said:

Does he want to know why it works, or why one would want to solve those problems? For the latter, take physics. It becomes obvious why physicists would have invented calculus - rates if change ( derivatives) and summation of tiny parts of a continuum (integral)

He wants to know why it works. He has taken a conceptual physics class (not calculus based), and he does seem to understand how calculus is used in practical applications, but he wants to understand why the math of calculus works.

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1 hour ago, Shelly in VA said:

He wants to know why it works. He has taken a conceptual physics class (not calculus based), and he does seem to understand how calculus is used in practical applications, but he wants to understand why the math of calculus works.

Don't they *prove* that in any decent calculus text? Like, derive for any function the derivative through a limit of the quotient of differences? They didn't do that in his class???

Edited by regentrude
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1 hour ago, regentrude said:

Don't they *prove* that in any decent calculus text? Like, derive for any function the derivative through a limit of the quotient of differences? They didn't do that in his class???

This is what I was thinking as well.

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Yes, they prove it in any decent calculus text. No, they don't spend enough time on it for the proof to make sense to 98% of kids . . .

What exactly does he want to understand, @Shelly in VA? Could you ask him if there's a specific thing he's stuck on?

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On 9/19/2023 at 9:53 PM, regentrude said:

Don't they *prove* that in any decent calculus text? Like, derive for any function the derivative through a limit of the quotient of differences? They didn't do that in his class???

I doubt CTC proves much, and most modern books tend to leave proofs of more advanced stuff to the appendices (e.g.)

OP, if he's looking for mathematical theory, you have a few options: Spivak, which is very tough, Apostol, which is more reasonable, and the AoPS book, which is the shortest of the three and probably somewhere between the two in terms of difficulty.

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On 9/19/2023 at 12:53 PM, regentrude said:

Don't they *prove* that in any decent calculus text? Like, derive for any function the derivative through a limit of the quotient of differences? They didn't do that in his class???

I assumed they did, but after you posted the question I asked him and also looked at the CTC lessons. It turns out that no, they don't, not at all. It's pretty plug and chug, which is why he was able to do the math but now he is frustrated that he doesn't understand it. I guess it's a good thing that he is questioning it now.

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8 minutes ago, Shelly in VA said:

I assumed they did, but after you posted the question I asked him and also looked at the CTC lessons. It turns out that no, they don't, not at all. It's pretty plug and chug, which is why he was able to do the math but now he is frustrated that he doesn't understand it.

Omg, that's horrible. Get any basic calc textbook. Aops is great, but a more theoretical approach than he might want. But even a standard text like Stewart should derive those things.

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On 9/18/2023 at 12:23 PM, Shelly in VA said:

Is there anything concise that could possibly provide that for him? I thought about the Art of Problem Solving Calculus book , but I'm actually looking for something more concise.

I'm not sure what part of AoPS is not concise.

I've pasted below the proof for the derivative of a polynomial.  It's all of 4 lines.

The AoPS calculus textbook 320 pages, much slimmer than any other standard calculus text.

(Let me know if you can't view the image below.)

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58 minutes ago, regentrude said:

Omg, that's horrible. Get any basic calc textbook. Aops is great, but a more theoretical approach than he might want. But even a standard text like Stewart should derive those things.

I just checked Stewart.  It derives everything.

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1 hour ago, daijobu said:

I'm not sure what part of AoPS is not concise.

I've pasted below the proof for the derivative of a polynomial.  It's all of 4 lines.

The AoPS calculus textbook 320 pages, much slimmer than any other standard calculus text.

(Let me know if you can't view the image below.)

Thanks for posting that image! AoPS is certainly concise, you're right. What I was trying to say was that I was not looking for an entirely new curriculum to use, just looking for one source for a proof of derivatives, if that makes sense.

Edited by Shelly in VA
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I own Foerster's Calculus and it also includes the proofs. Rarely, they are assigned as homework but there is an answer key.  The book and solution manual were not expensive. I am sure some of the others named upthread would work well too!

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On 9/19/2023 at 8:04 PM, Not_a_Number said:

Yes, they prove it in any decent calculus text. No, they don't spend enough time on it for the proof to make sense to 98% of kids . . .

What exactly does he want to understand, @Shelly in VA? Could you ask him if there's a specific thing he's stuck on?

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

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I have AoPS and Saxon on my shelf, along with an ancient college calculus textbook of mine. As easy as CTC was to use from my perspective, I think I'm going to have to switch to a traditional paper and pencil textbook at this point.

Thank you all for the help!

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4 minutes ago, Shelly in VA said:

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

Why anyone thought to do this is because the rate of change of a function is an incredibly important quantity in physics. Newton invented calculus (ok, Leibnitz too, almost simultaneously)

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24 minutes ago, Shelly in VA said:

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

Give him a curve and ask him to estimate the area under it. He will naturally break it up into sub shapes and add them up.  Have him cut the shape into 5 rectangles and calculate the area, then 10 rectangles, and then 20 rectangles, and he will understand that he is getting more accurate. It is then quite an easy step to a limit as x approaches 0. All my students can understand this idea quite easily and can see how you could invent it.

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You can do the same thing with gradients, it is just a bit messier. Use an example, show him how you can estimate a gradient at a point by picking 2 points on either side and getting the gradient of that line. Then pick points closer and closer together. Keep drawing the lines on the curve until it becomes clear to him that a tangent would obviously be the most accurate. Then pick 2 numbers only 1/1000 apart and calculate the gradient. Then do it with calculus to show how accurate your estimate is. It should be clear to him that someone could have thought of this. Only after you go through this process do you bring up the proof and walk him through it.

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The other thing I do is differentiate between 'magic' and math.  The way calculus is done in high school is 'magic'. I call it 'retrofitted patterns' which are based on math that most calculus kids never internalize.  I make it clear, that taking a derivative using a retrofitted pattern is not 'math', it is memorising.  So when doing a question, I make them tell me if they are doing an algebra line (which is mathematical and logical) or a calculus line (which is 'magic' and memorized and nonsensical). This distinction really helps kids. They have been told all their life not to memorize math, but to really understand it. And then they are given all this stuff to memorize and they think there should be some logic to it. And without limits, the patterns are nonsensical, but diligent kids think they are supposed to be logical and so get very confused.

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29 minutes ago, Shelly in VA said:

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

There is a documentary "The Story of Maths" it's a 4 part series and the 3rd part covers Calculus. It's fun and doesn't really derive the things, but it talks about why and the progression of how we (humanity) derived the mathematics we use today.

I can see why someone would want to know this knee-deep in Calculus. I know for me I went through a lot of math where the bulk of the lessons you have concrete real life problems like "I have 2 apples but I need to feed 6 people. How many more apples do I need?" Calculus was the first math subject where that isn't the case. Sure I want to find the area of something, but there isn't a clear reason why I'd want to find the area under a curve which is described by some crazy function of sine, cosine and polynomials. A simplistic answer is that the real life problems aren't always trying to find the area for some jug that is the crazy function of sine, cosine, and polynomials. Generally speaking the "area" may be an abstraction of some physical properties that can be described by some crazy functions of sine, cosine and polynomials.

For example if you are doing signal processing (say auto-tuning) you are manipulating sound which can be described as a bunch of sines and cosines combined together. You aren't necessarily finding the area under the sound. integration and derivation manipulates the curves to get the auto-tuning that you want. Which then translates into the code/circuitry you build to make that happen. The answer to can't you just build it and check, is yes you can but using the math is less tedious and allows you to tell a machine how to figure out what manipulations are needed themselves and in real time.

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1 hour ago, lewelma said:

Give him a curve and ask him to estimate the area under it. He will naturally break it up into sub shapes and add them up.  Have him cut the shape into 5 rectangles and calculate the area, then 10 rectangles, and then 20 rectangles, and he will understand that he is getting more accurate. It is then quite an easy step to a limit as x approaches 0. All my students can understand this idea quite easily and can see how you could invent it.

50 minutes ago, lewelma said:

You can do the same thing with gradients, it is just a bit messier. Use an example, show him how you can estimate a gradient at a point by picking 2 points on either side and getting the gradient of that line. Then pick points closer and closer together. Keep drawing the lines on the curve until it becomes clear to him that a tangent would obviously be the most accurate. Then pick 2 numbers only 1/1000 apart and calculate the gradient. Then do it with calculus to show how accurate your estimate is. It should be clear to him that someone could have thought of this. Only after you go through this process do you bring up the proof and walk him through it.

43 minutes ago, lewelma said:

The other thing I do is differentiate between 'magic' and math.  The way calculus is done in high school is 'magic'. I call it 'retrofitted patterns' which are based on math that most calculus kids never internalize.  I make it clear, that taking a derivative using a retrofitted pattern is not 'math', it is memorising.  So when doing a question, I make them tell me if they are doing an algebra line (which is mathematical and logical) or a calculus line (which is 'magic' and memorized and nonsensical). This distinction really helps kids. They have been told all their life not to memorize math, but to really understand it. And then they are given all this stuff to memorize and they think there should be some logic to it. And without limits, the patterns are nonsensical, but diligent kids think they are supposed to be logical and so get very confused.

Thank you!

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22 minutes ago, Clarita said:

There is a documentary "The Story of Maths" it's a 4 part series and the 3rd part covers Calculus. It's fun and doesn't really derive the things, but it talks about why and the progression of how we (humanity) derived the mathematics we use today.

I can see why someone would want to know this knee-deep in Calculus. I know for me I went through a lot of math where the bulk of the lessons you have concrete real life problems like "I have 2 apples but I need to feed 6 people. How many more apples do I need?" Calculus was the first math subject where that isn't the case. Sure I want to find the area of something, but there isn't a clear reason why I'd want to find the area under a curve which is described by some crazy function of sine, cosine and polynomials. A simplistic answer is that the real life problems aren't always trying to find the area for some jug that is the crazy function of sine, cosine, and polynomials. Generally speaking the "area" may be an abstraction of some physical properties that can be described by some crazy functions of sine, cosine and polynomials.

For example if you are doing signal processing (say auto-tuning) you are manipulating sound which can be described as a bunch of sines and cosines combined together. You aren't necessarily finding the area under the sound. integration and derivation manipulates the curves to get the auto-tuning that you want. Which then translates into the code/circuitry you build to make that happen. The answer to can't you just build it and check, is yes you can but using the math is less tedious and allows you to tell a machine how to figure out what manipulations are needed themselves and in real time.

I'll look for that documentary. Thank you for the input and for the examples of calculus applications.

Edited by Shelly in VA
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2 hours ago, Shelly in VA said:

When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place.

As for why people thought to do this--the answer is physics.  The Mechanical Universe develops this idea accessibly.  There is also a series of videos that goes with it.

If he is wondering how people figured out how to do these things, that is something I am always wondering myself!

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5 hours ago, Shelly in VA said:

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

Thought to do what, do you know? 😊 Figure out the slope of the tangent or an area under a curve? Or something else?

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5 hours ago, Shelly in VA said:

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

Infinite Powers by Strogatz, mentioned above, would be good for this purpose.

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6 hours ago, Shelly in VA said:

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

It's a good question, and it's the reason we remember the greats like Newton and Leibniz.  I'm impressed people were able to come with a formal way to describe concepts such as "infinitely large" and "infinitely small" that actually result in correct usable solutions to actual problems.

I certainly couldn't tell you how these great minds work!

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23 hours ago, Shelly in VA said:

I have now shown him a proof, and gone through it with him. When I asked what, exactly, he is stuck on, he says that he still feels like he doesn't understand why anyone thought to do this in the first place. I'm not sure what he means by that. I can't decide if this is teen debating or a genuine question from him!

http://library.lol/main/2D18697B5D446867191B30BE113249E7 (this book is probably the most modern/accesible)

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22 hours ago, Clarita said:

There is a documentary "The Story of Maths" it's a 4 part series and the 3rd part covers Calculus. It's fun and doesn't really derive the things, but it talks about why and the progression of how we (humanity) derived the mathematics we use today.

We watched this episode this morning - very interesting!

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