Paper collated from year 2015

Content Pure Chapters 1-13

Marks 100

Time 2 hours

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14. Differentiate 𝑓(𝑥) = 8𝑥3 + 5 from first principles. (4)

Mark scheme

2.

4.

5(a)

(a)

6

7

8a

8(a)

9

10

11

12

𝑓′(𝑥) = limℎ→0

𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ

𝑓′(𝑥) = limℎ→0

8(𝑥+ℎ)3+5 −(8𝑥3+5)

ℎ

𝑓′(𝑥) = limℎ→0

8(𝑥3+3𝑥2ℎ+3𝑥ℎ2+ℎ3)+5 −8𝑥3−5)

ℎ

𝑓′(𝑥) = limℎ→0

(24𝑥2ℎ+24𝑥ℎ2+8ℎ3)

ℎ

𝑓′(𝑥) = limℎ→0

ℎ(24𝑥2+24𝑥ℎ+8ℎ2)

ℎ

𝑓′(𝑥) = limℎ→0

24𝑥2 + 24𝑥ℎ + 8ℎ2

As ℎ → 0, 𝑓′(𝑥) → 24𝑥2 (1)

13

14 States the formula for differentiation from

first principles.

Correctly applies the

formula to the specific

function and expands

and simplifies.

Factorises the ‘h’ out of

the numerator and

divides to simplify.

(1)

(1)

(1)