Do I have this idea of antiderivatives correct?So is all it's saying that if there are two functions that have the same derivatives for every single x in the interval, then f ( x ) = g ( x ) + α, means that the second function is just the exact same as f(x) except raised or lowered?

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Do I have this idea of antiderivatives correct?So is all it&#039;s saying that if there are two functions that have the same derivatives for every single x in the interval, then   f  (  x  )  =  g  (  x  )  +  α, means that the second function is just the exact same as f(x) except raised or lowered?

Dominique Holmes · Accepted Answer

Step 1Yes,   ∫  f  (  x  )     d  x is a family of functions that only differ by a constant, which is typically denoted by c. This constant c can be viewed as a shift vertically of the function.Step 2The idea of a family of antiderivatives is easily shown by, say   f  (  x  )  =      x    2    +  1 and   g  (  x  )  =      x    2    −  5. While   f  ≠  g, it is true that       f    ′    =      g    ′  . Likewise   ∫  2  x     d  x  =      x    2    +  c, which contains f and g.

Jordyn Calhoun · Accepted Answer

Explanation:Yes, because the difference   f  −  g must then have zero derivative everywhere and therefore be constant. This means the graph of f is the graph of g shifted vertically.

Do I have this idea of antiderivatives correct? So is all it's saying that if there are two functio

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