Title: InkMaths / Proof and numerical methods
On every note of this deck, unless a note says otherwise:
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Front: RECALL
State the shape of a proof by contradiction.
Back: Assume the opposite of what you want, reason correctly until you reach something impossible, then conclude the original statement. Open with the assumption in words and close by naming the contradiction you hit. Both sentences carry marks and both are routinely left out.

Front: RECALL
State whether the trapezium rule over- or under-estimates.
Back: The chords lie above a curve that bends upwards, so the rule overestimates there. They lie below a curve that bends downwards, so it underestimates. Sketch the curve with one trapezium on it and the answer is visible, which is what the examiner wants to see said.

Front: SPOT THE ERROR
Say what is wrong with this claim:
Newton-Raphson always converges to the root nearest your starting value.
Back: Near a stationary point f'(x) is tiny, the tangent is almost flat, and it meets the axis a long way off. The next value can land beyond a different root or outside the domain altogether. A starting value exactly at a stationary point fails outright, because you would be dividing by zero.

Front: RECALL
State what a change of sign does and does not prove.
Back: If f is continuous on the interval and f(a) and f(b) have opposite signs, there is at least one root between them. Continuity is the condition people skip. An asymptote flips the sign with no root there at all, and a pair of roots close together produces no sign change even though both exist.

Front: DEFINE
Define: proof by exhaustion.
Back: Splitting the statement into a finite set of cases that between them cover every possibility, then checking each one. It only proves anything if the cases genuinely leave nothing out, so say why they do.