← Differentiation methods

4 Reading a graph: differentiability, f′ = 0, and limits

Every paper opens Question Three with a sketched graph and no equation. This is worth Achieved-Merit marks for reading, not calculating.

Reading differentiability off a sketch

  • A sharp corner — two different tangent gradients meet at the same point, so no single gradient exists there.
  • A jump or a hole (any discontinuity) — the curve is not even continuous, so it cannot be differentiable either.
  • A vertical tangent — the gradient would be infinite, which is not a real value.
  • Everywhere else the curve looks smooth and unbroken, it is differentiable.

Reading f′(x) = 0 and a limit off a sketch

  • f′(x) = 0 at every point where the tangent is horizontal — read this directly off the sketch, no algebra needed.
  • A horizontal tangent could be a maximum, a minimum, or a (rarer) stationary inflection — the flat tangent alone does not say which.
  • For lim(x→a) f(x): check the curve’s height approaching x = a from the left and from the right.
  • Same height from both sides → that height is the limit. Different heights (a jump) → the limit does not exist, even if f(a) is defined.
✓

A flat stretch is part of the answer to f′(x) = 0

A horizontal line segment has a gradient of zero at every point along it, so f′(x) = 0 on the whole stretch, written as an interval such as 3 < x < 5. Leave out an end where the graph jumps or turns a corner, because there is no single gradient there. A horizontal arrow running off the edge (x < −6, say) counts too. ✕

"Not differentiable" and "not continuous" are different claims

A function can be perfectly continuous (no jump, no hole — you could draw it without lifting your pen) and still fail to be differentiable, if it has a corner or a cusp. y = |x| is the standard example: continuous everywhere, but the tangent gradient jumps from −1 to +1 at x = 0, so it is not differentiable there. This exact distinction — "continuous but not differentiable" — is the paper’s own phrasing in three of the last four years; do not treat the two words as meaning the same thing. Worked example Reading a sketch A sketch of y = f(x) shows: a smooth curve for x < −1, a sharp corner at x = −1 where the curve then rises to a smooth rounded peak at x = 1, then a sudden jump upward at x = 3, continuing smoothly after that. State: (i) where f is not differentiable, (ii) where f′(x) = 0, (iii) lim(x→3) f(x). (i) Not differentiable at x = −1 (a corner — two different one-sided gradients meet) and at x = 3 (a jump — no derivative exists at a discontinuity). (ii) f′(x) = 0 at x = 1 only — the smooth rounded peak is the one point with a horizontal tangent. The corner at x = −1 has no single tangent, and the jump at x = 3 has none either. (iii) At x = 3 the curve steps up, so the left-hand and right-hand limits differ. ✓ Not differentiable at x = −1 and x = 3. f′(x) = 0 only at x = 1. lim(x→3) f(x) does not exist.

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