- #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.
In summary: 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.
- #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. :tongue2:
- #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.
1. What is the definition of an integral in calculus?
An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value or accumulation of a quantity over a certain interval.
2. How is the integral symbol 'F read in calculus?
The integral symbol 'F is read as "the integral of" or "the antiderivative of" in calculus. It is often used in integration problems to represent the solution or answer.
3. What is the difference between a definite and indefinite integral?
A definite integral has specific limits or boundaries, while an indefinite integral does not. A definite integral gives a numerical value, while an indefinite integral gives a general function or equation.
4. How is the integral related to the derivative?
The integral and derivative are inverse operations of each other. The derivative of a function gives its rate of change, while the integral of a function gives its total value or accumulation.
5. What are some practical applications of integrals in real life?
Integrals are used in various fields such as physics, engineering, economics, and statistics. They can be used to calculate areas, volumes, and total amounts in real-life situations, such as finding the area under a velocity-time graph or calculating the total cost of production for a company.
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