October 2021 Paper 3 Q4

OCR MEICurrent spec3 marksDifferentiation

4 The diagram shows points A and B on the curve \(y = \left(\dfrac{x}{4}\right)^{-x}\). The \(x\)-coordinate of A is 1 and the \(x\)-coordinate of B is 1.1.

Curve y = (x/4)^(−x) rising from (0, 1) to a maximum and then falling; points A and B marked close together on the rising part of the curve, joined by a chord
(a) Find the gradient of chord AB. Give your answer correct to 2 decimal places. [2]
(b) Give the \(x\)-coordinate of a point C on the curve such that the gradient of chord AC is a better approximation to the gradient of the tangent to the curve at A. [1]
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