Question about constants appearing in antiderivativesSay we have two functions f(x) and g(x) where d f ( x ) d x = d g ( x ) d x . This surely means that we can write f ( x ) = g ( x ) + C. Now let's say we are graphing these two functions and we want to edit one of them, say f(x) in such a way that the graph of f(x) lies perfectly on the graph of g(x). This would be possible to do by adding C to f(x) however only if C is a rational constant; if it is an irrational constant then we would need to use infinitely many digits of C to accomplish our goal since if we are using a finite number of digits of C it would be possible to "zoom in" the graph and see that the two functions don't overlap. This is only the context from which the question I want to ask emerged.What makes this constant irrational? Is the rationality/irrationality of the constant somehow dependent on the nature of the derivative of f(x) and g(x)?

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Question about constants appearing in antiderivativesSay we have two functions f(x) and g(x) where             d      f      (      x      )              d      x        =            d      g      (      x      )              d      x      . This surely means that we can write   f  (  x  )  =  g  (  x  )  +  C. Now let&#039;s say we are graphing these two functions and we want to edit one of them, say f(x) in such a way that the graph of f(x) lies perfectly on the graph of g(x). This would be possible to do by adding C to f(x) however only if C is a rational constant; if it is an irrational constant then we would need to use infinitely many digits of C to accomplish our goal since if we are using a finite number of digits of C it would be possible to &quot;zoom in&quot; the graph and see that the two functions don&#039;t overlap. This is only the context from which the question I want to ask emerged.What makes this constant irrational? Is the rationality/irrationality of the constant somehow dependent on the nature of the derivative of f(x) and g(x)?

empatteMattmkezo · Accepted Answer

Step 1Let&#039;s think about the simplest possible case.Let&#039;s suppose that       f    ′    (  x  )  =      g    ′    (  x  ) for all x but further, suppose that these are both equal to 0. Then both f and g are constant functions.So   f  (  x  )  =  A and   g  (  x  )  =  B for some A and B, and we see that   f  (  x  )  =  g  (  x  )  +  C for   C  =  A  −  B.Step 2You can imagine that you discover f by integration, and I discover g the same way. One possibility is that you find   A  =  0, and that I find   B  =  1. In this case,   C  =  −  1 is a rational.But another possibility is that you come up with   A  =  π and I come up with   B  =  2, in which case   C  =  π  −  2, which is irrational.There&#039;s nothing in the nature of the derivative - the constant function 0 - that says anything about what the values A and B can be (indeed, any real number is a possible value).

Question about constants appearing in antiderivatives Say we have two functions f(x) and g(x) where

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