Exercise 3: Forward vs. Central Difference on x^2 — A Special Case — Possible Solution
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GIVEN
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f(x) = x^2, x=1, h=0.5. True derivative: 2x = 2.

STEP 1: FORWARD DIFFERENCE
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(f(1.5) - f(1)) / 0.5 = (2.25 - 1) / 0.5 = 1.25 / 0.5 = 2.5

Error: |2.5 - 2| = 0.5

STEP 2: CENTRAL DIFFERENCE
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(f(1.5) - f(0.5)) / (2*0.5) = (2.25 - 0.25) / 1 = 2.0 / 1 = 2.0

Error: |2.0 - 2| = 0.0

RESULT
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Central difference gives EXACTLY 2.0 - zero error, even at a
fairly large step size of h=0.5, which is far larger than any of
the step sizes needed to get comparable accuracy in this chapter's
own x^3 example.

DOES THIS MATCH THE x^3 PATTERN? NO - AND HERE'S WHY
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This is NOT the same pattern as x^3, where central difference was
dramatically more accurate than forward difference but still had a
small, nonzero error. For x^2 specifically, central difference is
EXACTLY correct, for any h at all, not just very accurate.

This happens because central difference's own error term depends on
the THIRD derivative of the function (this is a genuine
mathematical fact about how the central difference formula's error
behaves, tied to how quickly the function's own curvature is
changing). For f(x)=x^2, the third derivative is 0 everywhere (the
second derivative of x^2 is the constant 2, and the derivative of a
constant is 0) - so central difference's error term vanishes
completely for any quadratic function, regardless of h. x^3's own
third derivative is a nonzero constant (6), which is exactly why
central difference still had a small residual error there.

WHY THIS WORKS AS AN ANSWER
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Both differences are computed explicitly with real numbers rather
than assumed to follow the same trend as the earlier x^3 example,
and the surprising exact-zero-error result is explained using a
specific, correct mathematical reason (central difference's error
term depending on the third derivative, which is zero for any
quadratic) rather than just noting the numbers happened to come out
differently.