Calculus notation question

In summary, the conversation is discussing the notation used for writing an antiderivative and definite integral. The question is whether the designation of "t naught" in \int^{t}_{t_{0}} f(s) ds is commonly understood to mean an antiderivative with no arbitrary constant. The responses explain that the value of the integral is F(t) - F(t0), where t0 can be any value and F(t0) may or may not be zero. The conversation also touches on the use of integrating factor and how it relates to the notation for antiderivatives.
  • #1
1MileCrash
1,342
41
When one writes:

[itex]\int^{t}_{t_{0}} f(s) ds[/itex]
Do they generally mean "the antiderivative of f(t), and ignore the arbitrary constant/pick t naught so that it is 0?"
 
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  • #2
Antiderivative - yes, call it F(t).
The integral is F(t) - F(t0), t0 can be anything - not necessarily 0.
 
  • #3
mathman said:
Antiderivative - yes, call it F(t).
The integral is F(t) - F(t0), t0 can be anything - not necessarily 0.

Of course it can be anything, but I was asking if nothing else is said, then I could assume they mean an antiderivative with no arbitrary constant.
 
  • #4
1MileCrash said:
When one writes:

[itex]\int^{t}_{t_{0}} f(s) ds[/itex]



Do they generally mean "the antiderivative of f(t), and ignore the arbitrary constant/pick t naught so that it is 0?"
I'm not sure you're writing what you meant to. The above is the definite integral of f over the interval [t0, t].

If F is an antiderivative of f, then the value of the integral is F(t) - F(t0. t0 might or might not be zero, and F(t0) might or might not be zero.

If you're talking about this, however,
$$ \frac{d}{dt}\int^{t}_{t_{0}} f(s) ds$$
then that evaluates to f(t).
 
  • #5
Mark44 said:
If you're talking about this, however,
$$ \frac{d}{dt}\int^{t}_{t_{0}} f(s) ds$$
then that evaluates to f(t).
Actually, I'm talking about what I wrote.

I am aware that f(t0) may be 11, 42, grahams number, or batman riding a trex. I am asking if the designation of "tee naught" is commonly taken as an obvious intent to notate an antiderivative with no arbitrary constant. I don't know a better way to ask my question.
 
Last edited:
  • #6
http://en.wikipedia.org/wiki/Integrating_factor

Here is a very straight forward and common use of the notation. I'm asking if this is a routine and acceptable way to say "take the antiderivative and don't give me an arbitrary constant" since as far as I know, there is no other way to say that. I'm asking if I wrote that in a proof, people would know what I am talking about, but judging by the responses, the answer is no.
 
  • #7
The problem is a "floating pronoun". You ask if, in [tex]\int_{x_0}^x f(t) dt[/tex], "ignore the arbitrary constant/pick t naught so that it is 0?" What does "it" refer to? If F(t) is an anti-derivative of f(t), then the integral is F(x)- F(x_0) so, at [itex]x= x_0[/itex], the value of the function is 0. But you certainly cannot "pick t naught so that it is 0?" You cannot pick [itex]t_0[/itex], it is given in the integral.
 
  • #8
1MileCrash said:
http://en.wikipedia.org/wiki/Integrating_factor

Here is a very straight forward and common use of the notation. I'm asking if this is a routine and acceptable way to say "take the antiderivative and don't give me an arbitrary constant" since as far as I know, there is no other way to say that. I'm asking if I wrote that in a proof, people would know what I am talking about, but judging by the responses, the answer is no.

Near the top in the wiki article, they have this. (I made one change, from P(s) to p(s). You'll see why in a minute.)
$$ M(x) = e^{\int_{s_0}^x p(s)ds}$$

Let's assume that P(s) is an antiderivative of p(s).

Then the exponent on e is
$$ \left. P(s)\right|_{s_0}^x = P(x) - P(s_0)$$

So M(x) = eP(x) - P(s0) = eP(x)/eP(s0)

Since P(s0) is just a constant, we can write M(x) = KeP(x), where K = 1/eP(s0).

If you have an integrating factor, then a constant multiple of it will also work, so we can ignore the K.
 

1. What does the notation "df/dx" mean in calculus?

The notation "df/dx" is known as the derivative of the function f with respect to the variable x. It represents the rate of change of the function f at a particular point in terms of the variable x.

2. What does "∫" represent in calculus notation?

The symbol "∫" represents the integral in calculus, which is the reverse process of differentiation. It represents the sum of infinitely small values of a function over a specific interval.

3. What is the meaning of "lim" in calculus notation?

The notation "lim" stands for limit in calculus. It represents the value that a function approaches as its input approaches a certain value or as the value of the variable approaches infinity.

4. What does the notation "dx" mean in calculus?

The notation "dx" represents the differential of the variable x. It is used to indicate the variable with respect to which the function is being differentiated or integrated.

5. What is the purpose of using subscript notation in calculus?

Subscript notation is used in calculus to differentiate between multiple variables or functions that are being used in an equation. It helps to clearly identify which variable or function is being referred to in a particular context.

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