The Constant Difference Theorem

    The following theorem says that two functions with equal derivatives on an open interval differ by a constant on that interval. This next theorem is important to integration theory.

Proposition (Constant Difference Theorem) Let and be functions that are continuous on and differentiable on If for all in then is constant on that is, where is a constant.

    Proof. Let Then for all in Thus by the Zero Derivative Theorem, for some constant and so as desired.
    

Example (Constant Difference Theorem) If possible, use the Constant Difference Theorem. Let Find a function with and

    Solution. Let where is some constant to be determined. Then and to determine we use to obtain



Therefore, is the function we desire.

Example (Constant Difference Theorem) If possible, use the Constant Difference Theorem. Show that and differ by a constant. Are the conditions of the constant difference theorem satisfied? Does

    Solution. We simplify



which is valid on any interval not containing Thus on any interval not containing the constant difference theorem applies. In fact, we have



when

Example (Constant Difference Theorem) If possible, use the Constant Difference Theorem. Let and Use and to demonstrate the constant difference theorem.

    Solution.  The functions and are polynomial functions so they are continuous and differentiable for all real numbers. Also,
    


for all real numbers. By the Constant Difference Theorem, we have for some real number

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Constant Difference Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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