- #1
Line
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In Calculas I've learned that 'F means the integral of a function. SO what does ''F mean, th integral or an integral of a function?
In Calculas I've learned that 'F means the integral of a function.
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Discussion Overview
The discussion revolves around the notation used in calculus, specifically the meaning of symbols such as 'F and ''F in relation to integrals and derivatives of functions. Participants explore the implications of these notations and their relationships to concepts like derivatives and integrals.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant states that 'F represents the integral of a function, questioning whether ''F signifies the integral or an integral of a function.
- Another participant suggests that ''F could be analogous to shorthand notation for derivatives, providing a list of derivative and integral notations.
- Questions arise about the definitions of second derivatives and double integrals, with one participant implying a lack of understanding in calculus.
- A participant explains that the second derivative is the derivative of a derivative, while a double integral is similarly defined.
- Claims are made about the relationships between functions and their derivatives, with specific examples provided, such as f = x³ leading to ''f = x³ + C.
- Another participant argues that for a general solution, the entire form must be included, challenging the sufficiency of just x³ or x³ + C.
- Discussion includes a personal anecdote about test-taking strategies related to writing +C for antiderivatives.
- One participant introduces alternative notations for derivatives and integrals, such as f(n) for the nth derivative and f(-n) for the nth antiderivative, expressing unfamiliarity with 'f and ''f.
Areas of Agreement / Disagreement
Participants express differing views on the sufficiency of certain forms for general solutions and the clarity of notation used in calculus. There is no consensus on the meaning of ''F or the implications of the various notations discussed.
Contextual Notes
Some participants reference specific functions and their derivatives or integrals, but there are unresolved assumptions regarding the definitions and applications of the notations discussed. The conversation reflects varying levels of familiarity with calculus concepts.
- #2
berkeman
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I hadn't seen that notation before. I suppose it could be the analog of the shorthand notation for derivative?
f = function
f' = derivative of f
f'' = 2nd derivative of f
'f = integral of f
''f = double integral of f
f = function
f' = derivative of f
f'' = 2nd derivative of f
'f = integral of f
''f = double integral of f
- #3
Line
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What's a 2nd derivative and double integral?
- #4
d_leet
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Line said:
You didn't learn much in calculus did you? A 2nd derivative is pretty much what it sounds like the derivative of a derivative, and sort of similar for a double integral.
- #5
Gelsamel Epsilon
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f = x³
f' = 3x²
f'' = 6x
f'' is just the derivative of f'.
Also, f, is the ''f (double intergral) of f'' so if f = 6x then ''f = x³. I hope that wasn't confusing.
f' = 3x²
f'' = 6x
f'' is just the derivative of f'.
Also, f, is the ''f (double intergral) of f'' so if f = 6x then ''f = x³. I hope that wasn't confusing.
- #6
Hootenanny
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Gelsamel Epsilon said:
''f = x3 +C
- #7
Gelsamel Epsilon
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If you're asked for the general solution x³ is sufficient. 
- #8
HallsofIvy
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Actually, if f= 6x, then 'f= 3x2+ C so "f= x3+ Cx+ D.Hootenanny said:''f = x3 +C
I'm very puzzled by
It is precisely when you are asked for the "general solution" that you must have the entire form! x3 does satisfy the conditions but is not the "general solution". Neither is x3+ C. The general function "f such that f= 6x is "f(x)= x3+ Cx+ D.Gelsamel Epsilon said:
- #9
JonF
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When I was in calc and I took test the first thing I always did was write +C down the most the right side of the page.
- #10
Gelsamel Epsilon
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Ha, sorry HallsofIvy I confused myself, I meant if you're asked for "an" (rather then "the") antiderivative then that satisfies.
- #11
HallsofIvy
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It is not uncommon to use the notation f(n) to mean the nth (notice the parentheses) rather than f', f", since it is clearer if n is large. Similarly, a common notation for "nth anti-derivative" is f(-n). I had never seen 'f, "f before.
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