CAIE A-Level Mathematics 1.7.1 Gradient as limit

CAIE A-Level Mathematics 1.7.1 Gradient as limit
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise estimating curve gradients from chords, tables and limiting behaviour, including interpreting f′(x) from nearby values.

How this is tested

  • calculate chord gradients to a stated accuracy from two points on a curve
  • explain that chord gradients approach the tangent gradient as h tends to 0
  • use table values to state an approximate value of f′(x) at a point

Question 3

[Maximum number: 2]

The equation of a curve is y=f(x), where f(x)=12x23(x−2)2\mathrm{f}(x)=\frac{1}{2} x^{\frac{2}{3}}(x-2)^{2}. The following points lie on the curve. Non-exact values of the y-coordinates are given correct to 6 decimal places.

A(8,72), B(8.001, k), C(8.01,72.300388), D(8.1,75.038882)

Question 3(b)

(a)

Find the gradient of the chord AD. Give your answer correct to 4 decimal places.

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Question 3(c)

(b)

State what the values in the table suggest about the value of f'(8).

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