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I'm confused.

What's the difference between f(x) and f(t)?

What's the difference between f(x) and f(t)?

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- Thread starter JamesGold
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- #1

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I'm confused.

What's the difference between f(x) and f(t)?

What's the difference between f(x) and f(t)?

- #2

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I'm confused.

What's the difference between f(x) and f(t)?

I see what you mean, that can be a little confusing.

First, when we talk about f(x), the role of 'x' is that of a dummy variable. If we said f(x) = x^2 or f(t) = t^2, we would be talking about the same function. That's why when people are being picky and pedantic, they say that

Ok so we have some function, which we

Now we want to define another function, g, by saying that for some point x, g(x) is the definite integral of f from a to x, where a is some fixed constant.

So to evaluate g(x), we compute the integral of f as f ranges between a and x But now we need a dummy variable inside the integral to integrate against. We can't call it x, because we're already using that variable to define g. So we use a new dummy variable, t. We choose an x; then we compute the definite integral of f as f ranges between a and x. We use a new dummy variable 't' to denote that.

Then in the last line where they say g'(x) = f(x), it makes sense to use the same dummy variable, because they're saying that the value of g' and the value of f are the same for any given x.

Perhaps it helps to think of choosing an x_0; then you let t run from a to x_0 and you get a number that's g(x_0). Then you pick another x_1, and let t range over a to x_1, and so forth. That's why you need a separate dummy variable.

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- #3

HallsofIvy

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[tex]\int_a^x f(t)dt[/tex]

is a function of

[tex]\int_a^x f(t)dt= F(x)- F(a)[/tex]

a function of x.

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- #5

HallsofIvy

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Pretty much, yes.

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That's... despicable. For understanding's sake it's probably best to just focus on what the theorem is trying to say without getting caught up in the details of how it goes about saying it. Given a continuous, differentiable function, the rate of change of that function's area function is the function itself. Done.

And that makes sense because as the function gets higher and higher, so too does the area bound between it and the x-axis. A small change in the function when f = 9089078907 will produce a huge change in the total area underneath it.

Sorry, just venting my thoughts. Thanks for the replies.

And that makes sense because as the function gets higher and higher, so too does the area bound between it and the x-axis. A small change in the function when f = 9089078907 will produce a huge change in the total area underneath it.

Sorry, just venting my thoughts. Thanks for the replies.

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- #7

HallsofIvy

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what's despicable?That's... despicable.

For understanding's sake it's probably best to just focus on what the theorem is trying to say without getting caught up in the details of how it goes about saying it. Given a continuous, differentiable function, the rate of change of that function's area function is the function itself. Done.

And that makes sense because as the function gets higher and higher, so too does the area bound between it and the x-axis. A small change in the function when f = 9089078907 will produce a huge change in the total area underneath it.

Sorry, just venting my thoughts. Thanks for the replies.

- #8

Fredrik

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Hey, i was just going to say exactly that.That's why when people are being picky and pedantic, they say thatf is the function; and f(x) is the value of the function at the point x.

(But I don't consider it pedantic. I think the other terminology is really sloppy and causes a lot of unnecessary confusion).

JamesGold, I even refuse to use phrases like "f(x) is continous". f(x) is a number in the range of the function f, and a number can't be continuous. This is how I would state the theorem in the image you posted:

If f is continous on [a,b], then the function g defined by [tex]g(x)=\int_a^x f(t)dt[/tex] for all x≥a, is continuous on [a,b], differentiable on (a,b), and satisfies g'(x)=f(x) for all x in (a,b).

Note that we could also have said that g is defined by [tex]g(y)=\int_a^y f(x)dx[/tex] for all y≥a. The statement defines the same function no matter what variable symbol we use.

- #9

lavinia

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I'm confused.

What's the difference between f(x) and f(t)?

they are the same only when t=x.

The derivative of the integral at x is f(x).

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