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CAIE A-Level Mathematics AS 1.7.1 Gradient As Limit Questions

Compare gradients of chords approaching a point to estimate the tangent gradient, then interpret this limiting value as f′(x).

Syllabus
2028–2030
Course
Mathematics 9709
Level
AS

Exam points

  • Calculate a chord gradient from function values or tabulated data, including an expression in h.
  • Use gradients of chords approaching a point to estimate the curve gradient there as a limit.
  • Interpret f′(x) as the gradient of the curve at x, using the limiting chord-gradient evidence.

CAIE A-Level Mathematics AS 1.7.1 Gradient As Limit Questions question 1

[Maximum number: 3]

The equation of a curve is y=f(x), where f(x)=(2x−1)3x−2−2\mathrm{f}(x)=(2 x-1) \sqrt{3 x-2}-2. The following points lie on the curve. Non-exact values have been given correct to 5 decimal places.

A(2,4), B(2.0001, k), C(2.001,4.00625), D(2.01,4.06261), E(2.1,4.63566), F(3,11.22876)

Question (a)

(a)

Find the value of k. Give your answer correct to 5 decimal places.

The table shows the gradients of the chords A B, A C, A D and A F.

Table for Question (a) — CAIE A-Level Mathematics AS
[ 1 ]

Question (b)

(b)

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

[ 1 ]

Question (c)

(c)

Deduce the value of f′(2)f^{\prime}(2) using the values in the table.

[ 1 ]
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