I have a question on integrals.

SusanCoutu

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My question is if the derivative of a function is only a very close approximation to the actual marginal change of the function, provided dx is very small compared to dy, then shouldn't any integrals arrived from the derivative by using integration be only an approximation as well of the function the derivative was originally supposed to represent? My textbook seems to suggest that an integral arrived at by using antidifferentiation is indeed the true antiderivative and thus it's a perfect description of any measurement of total change, but I just find it hard to wrap my head around how it should be possible to arrive at the original by reversing only an approximation. Maybe my understanding of this is flawed.
Thanks for any help!
 

Subhotosh Khan

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My question is if the derivative of a function is only a very close approximation to the actual marginal change of the function, provided dx is very small compared to dy, then shouldn't any integrals arrived from the derivative by using integration be only an approximation as well of the function the derivative was originally supposed to represent? My textbook seems to suggest that an integral arrived at by using antidifferentiation is indeed the true antiderivative and thus it's a perfect description of any measurement of total change, but I just find it hard to wrap my head around how it should be possible to arrive at the original by reversing only an approximation. Maybe my understanding of this is flawed.
Thanks for any help!
This is a topic of face-to-face discussion. It is very hard to explain through "posts" - at least for me.

For the time being - just take it for granted.

Ask this question to your teacher and have a face-to-face discussion with the teacher.

Before that brush-up your concept of limit - thoroughly.
 

tkhunny

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This was a common difficulty prior to the emergence of the calculus. It was thought that if we chopped things up in small enough pieces, then we could measure them exactly. Eventually, we figured out that Real Numbers are dense. There is no space small enough so that you can get from one Real Number to the next. (Like you can with Natural Numbers).

Let's abandon that concept and truly attempt to understand the Calculus. If the derivative or the integral is just an approximation, then you do not understand the limit - you stopped before you actually found the limit. We sometimes use words like "in the limit" or "at the limit" or "take the limit". These are okay, but not perfect. The simplest example to understand is the common question:

Is 0.99999.... EQUAL to one (1), or is it just an approximation of one (1)? Maybe really, REALLY close?

Well, if "..." means you just keep going, then it IS EQUAL to one (1). If you stop, somewhere, anywhere, then it is only an approximation. I like to think of it with this question: If 0.999..... is NOT EQUAL to one (1), then how far from one (1) is it? You cannot supply a meaningful answer.

Definitely a question for a face-to-face. Giving up finite thinking isn't usually easy. You'll get it.
 

SusanCoutu

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I should have mentioned this in my 1st post. This question comes from my son who is in prison in GA. He's teaching himself. I'll email the answer to him.To tell you the truth, I don't even know if this is calculus, it was my guess. lol
 

Dr.Peterson

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My question is if the derivative of a function is only a very close approximation to the actual marginal change of the function, provided dx is very small compared to dy, then shouldn't any integrals arrived from the derivative by using integration be only an approximation as well of the function the derivative was originally supposed to represent? My textbook seems to suggest that an integral arrived at by using antidifferentiation is indeed the true antiderivative and thus it's a perfect description of any measurement of total change, but I just find it hard to wrap my head around how it should be possible to arrive at the original by reversing only an approximation. Maybe my understanding of this is flawed.
Thanks for any help!
I would reverse what you say: an actual marginal [rate of] change is only an approximation to the derivative!

The derivative, thought of as the slope of a curve at a point, is the limit of the slopes of secant lines. It is exact, though anything we could actually calculate or meaasure would be the slope of a secant of the curve, just an approximation to the slope of the curve itself.

Similarly, the definite integral, thought of as the area under a curve, is the limit of the areas of sums of rectangles; the former is exact, while the latter are only approximations.

This is why calculus is so powerful!
 

tkhunny

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Excellent guess.

Funny story: Stoners keep stealing Mile Marker "420" from highways. I've seen stories from at least Oregon and Colorado where they have replaced "420" with "419.99" or "419.99..." in an effort to end the theft. Someone in the highway department is definitely thinking about the problem. Their notation might not always be right, but I think it's pretty funny. Math in the real word I think we call that. :)
 

lev888

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Excellent guess.

Funny story: Stoners keep stealing Mile Marker "420" from highways. I've seen stories from at least Oregon and Colorado where they have replaced "420" with "419.99" or "419.99..." in an effort to end the theft. Someone in the highway department is definitely thinking about the problem. Their notation might not always be right, but I think it's pretty funny. Math in the real word I think we call that. :)
If they put "419.99" at 419.99, then it's exactly right.
 

tkhunny

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If they put "419.99" at 419.99, then it's exactly right.
Indeed. I have not seen any "we also moved he sign 52.8 ft from it's previous location" or any follow up to suggest the technique was effective in preventing theft.
 
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