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Prove that differentiation inverts the parity of a function. Hint: use (1.3.1.5) and see separately the cases of an even and an odd function.
(1.3.1.5) |
In order to solve correctly this exercise, the following relation: should be used for even and for odd parity. It should not be proven, because it follows directly from the definition. It means that as a consequence of inverting the sign of one of the increments the sign of the derivative is also inverted.
The proof is given here for completeness:In case of an even function it follows from the definition that: but from part 1 :
In case of an odd function it follows from the definition that: but from part 1 :
Part 1 should not be proven.
The solution of this exercise can be done in a different way (e.g. graphically), but it should be correct.
If only one of the cases (even or odd function) is solved correctly the score should be 6 points.
If both are correct - 10 points. Otherwise the score should be zero.