Is FTOC valid only in interval that the function is continuous? ∂ x ∫ 0 x f ( x ′ ) d x ′ = f ( x ) − f ( 0 )

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Is FTOC valid only in interval that the function is continuous?      ∂    x        ∫    0    x    f  (      x    ′    )    d      x    ′    =  f  (  x  )  −  f  (  0  )

Sydnee Villegas · Accepted Answer

Before worrying about what kind of hypotheses you need for the Fundamental Theorem of Calculus you need to understand better what it means. You seem to be mixing up two forms of the Fundamental Theorem of Calculus. The first says that      d          d      x            ∫    0    x    f  (  t  )    d  t  =  f  (  x  )  .Intuitively, that says that if you know how to calculate definite integrals (using Riemann sums to find areas) you have a way to find a function whose derivative is the function   f.The second form says that      ∫    0    x    f  (  t  )    d  t  =  F  (  x  )  −  F  (  0  )whenever   F is a function whose derivative is   f. Intuitively, that tells you that if you can somehow guess a function   F whose derivative is   f (an indefinite integral) you can find areas without fussing with sums.The second form says that      ∫    0    x    f  (  t  )    d  t  =  F  (  x  )  −  F  (  0  )whenever   F is a function whose derivative is   f. Intuitively, that tells you that if you can somehow guess a function   F whose derivative is   f (an indefinite integral) you can find areas without fussing with sums.The equation you wrote (with the proper notation for the derivative)       d          d      x            ∫    0    x    f  (  t  )    d  t  =  f  (  x  )  −  f  (  0  )is just wrong. You can see that if you try it out for the function   f whose value is always   2. Then      d          d      x            ∫    0    x    f  (  t  )    d  t  =      d          d      x        2  x  =  2while  f  (  x  )  −  f  (  0  )  =  2  −  2  =  0.

Is FTOC valid only in interval that the function is continuous? <mi mathvariant="normal">

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