Problem understanding Differentiation and Integration
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Dr.Peterson
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I see that this image comes from
Your annotations twist this (rather weird) analogy for the Taylor series into an explanation of derivatives and integrals; I would not say that is valid (and certainly it is not what was intended by the author). Integration does not produce "an entire function", and differentiation does not produce "a piece of a function".
Do you mind then explaining what an integral and derivative does (in an intuitive manner)?
From my understanding, An integral finds the area under a curve (if it is an definite integral).
A derivative on the other hand, just calculates the rate of change, rate of change of what? (this definition also does not make sense to me)
From my understanding, An integral finds the area under a curve (if it is an definite integral).
A derivative on the other hand, just calculates the rate of change, rate of change of what? (this definition also does not make sense to me)
Dr.Peterson
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Can you tell us why the definition of derivative doesn't make sense to you? That may lead to a more helpful dialogue than just giving you an explanation that we find useful.
A derivative gives the rate of change of the given function as its input varies; an indefinite integral (antiderivative) finds a function whose rate of change is represented by a given function; and the Fundamental Theorem of Calculus relates those to the definite integral, which gives the area under a function: Essentially, the rate of change of the area under a function is given by that function itself.
But this will only make sense when you understand the derivative; we'll have to say more based on your personal difficulties with it.
EngineersAreComedians
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When I started off with Calculus, I also imagined a similar relation as to what you are trying to describe.
Yes, integrating the differentiation of a function will give you the function itself. However, I do not understand what you mean when you say piece of a function.... do you mean a value at a point or a term? Or is it something different?
Yes, integrating the differentiation of a function will give you the function itself. However, I do not understand what you mean when you say piece of a function.... do you mean a value at a point or a term? Or is it something different?
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Here is the most intuitive description as far as I am concerned.
Imagine a very small number, so small it is no longer finite and is indistinguishable from zero but is not equal to zero. The difference is too small to measure.
Differential calculus is about dividing two such infinitessimals. We are dividing what is indistinguishable from zero by something that is also indistinguishable from zero.
Integral calculus is about multiplying an infinitessimal by infinity.
Multiplication and division are inverse operations.
That seems to have been how the inventors of calculus actually thought about what they were doing. It has been formalized in non-standard analysis, which defines infinitessimals and provides an arithmetic for them. A much more common formalization deals with limits. It says if you divide a very small but finite number by another very small but finite number and the quotient stabilizes as the numbers get smaller and smaller in a systematic way, the division has a limiting value. Similarly, if you multiply a number of very small magnitude by a number of very great magnitude and the product stabilizes as the one number gets smaller and the other number gets bigger ina systematic way, the product has a limit. The deriviative and the integral are simply formulas for these limits with respect to certain functions.
That is super-informal. Mathematicians have, however, developed these ideas rigorously. I am just giving my personal intuition.
Geometrically, the definite integral is the area between a curve, the x-axis, and a starting and ending point. The derivative is the slope of the tangent to the curve at a point.
Rate of change comes in when the x-axis represents time and the y-axis represents something dependent on time. The slope of the tangent is defined to be the instantaneous rate of change. It is not an intuition but rather a definition that gives useful empirical results.
Now the other helpers will explain why I am totally wrong.
Imagine a very small number, so small it is no longer finite and is indistinguishable from zero but is not equal to zero. The difference is too small to measure.
Differential calculus is about dividing two such infinitessimals. We are dividing what is indistinguishable from zero by something that is also indistinguishable from zero.
Integral calculus is about multiplying an infinitessimal by infinity.
Multiplication and division are inverse operations.
That seems to have been how the inventors of calculus actually thought about what they were doing. It has been formalized in non-standard analysis, which defines infinitessimals and provides an arithmetic for them. A much more common formalization deals with limits. It says if you divide a very small but finite number by another very small but finite number and the quotient stabilizes as the numbers get smaller and smaller in a systematic way, the division has a limiting value. Similarly, if you multiply a number of very small magnitude by a number of very great magnitude and the product stabilizes as the one number gets smaller and the other number gets bigger ina systematic way, the product has a limit. The deriviative and the integral are simply formulas for these limits with respect to certain functions.
That is super-informal. Mathematicians have, however, developed these ideas rigorously. I am just giving my personal intuition.
Geometrically, the definite integral is the area between a curve, the x-axis, and a starting and ending point. The derivative is the slope of the tangent to the curve at a point.
Rate of change comes in when the x-axis represents time and the y-axis represents something dependent on time. The slope of the tangent is defined to be the instantaneous rate of change. It is not an intuition but rather a definition that gives useful empirical results.
Now the other helpers will explain why I am totally wrong.
Steven G
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Derivative: Suppose f(x) = x^2 and f'(x)=2x. f'(x) is the derivative of f(x).
If you plug in a value for x into f'(x) what you will get is the slope of the tangent line of f(x) at that x-value.
For example, f'(3) = 2*3=6. This means if you go to the point (3,9) on f(x)=x^2 and draw the tangent line its slope will be 6.
There are other ways to think of the derivative but imo you MUST at least know the above definition
If you plug in a value for x into f'(x) what you will get is the slope of the tangent line of f(x) at that x-value.
For example, f'(3) = 2*3=6. This means if you go to the point (3,9) on f(x)=x^2 and draw the tangent line its slope will be 6.
There are other ways to think of the derivative but imo you MUST at least know the above definition
Steven G
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Being a friend of yours I will not leave a more detailed comment than just what I've written so far.