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CAIE A-Level Mathematics 1.7.1 Gradient as limit

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

Syllabus
2028–2030
Course
Mathematics 9709
Level
AS

Exam points

  • 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

1.7.1—Gradient as limit 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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