| Date | May 2015 | Marks available | 2 | Reference code | 15M.1.hl.TZ1.4 |
| Level | HL only | Paper | 1 | Time zone | TZ1 |
| Command term | Expand | Question number | 4 | Adapted from | N/A |
Question
Expand \({(x + h)^3}\).
[2]
a.
Hence find the derivative of \(f(x) = {x^3}\) from first principles.
[3]
b.
Markscheme
\({(x + h)^3} = {x^3} + 3{x^2}h + 3x{h^2} + {h^3}\) (M1)A1
[2 marks]
a.
\(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{{{(x + h)}^3} - {x^3}}}{h}\) (M1)
\( = \mathop {\lim }\limits_{h \to 0} \frac{{{x^3} + 3{x^2}h + 3x{h^2} + {h^3} - {x^3}}}{h}\)
\( = \mathop {\lim }\limits_{h \to 0} (3{x^2} + 3xh + {h^2})\) A1
\( = 3{x^2}\) A1
Note: Do not award final A1 on FT if \( = 3{x^2}\) is not obtained
Note: Final A1 can only be obtained if previous A1 is given
[3 marks]
Total [5 marks]
b.
Examiners report
Well done although some did not use the binomial expansion.
a.
Fine by those who knew what first principles meant, not by the others.
b.
Syllabus sections
Topic 1 - Core: Algebra » 1.3 » The binomial theorem: expansion of \({\left( {a + b} \right)^n}\), \(n \in N\) .
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