| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Expand f(a+h) algebraically |
| Difficulty | Moderate -0.8 This is a straightforward application of differentiation from first principles requiring only algebraic expansion of (2+h)³ and simplification. The binomial expansion is standard, and finding f'(2) by taking the limit as h→0 is immediate once the h terms are factored out. No problem-solving insight needed—purely mechanical execution of a well-practiced technique. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x |
| Answer | Marks | Guidance |
|---|---|---|
| \((2 + h)^3 = 8 + ah + bh^2 + h^3\); \((2 + h)^3 = 8 + 12h + 6h^2 + h^3\); \(f(2+h) - f(2) = 13h + 6h^2 + h^3\) | M1, A1A1, m1A1F | 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Divide by \(h\) and let \(h \to 0\); \(f'(2) = p = 13\) | M1, A1F | 2 marks |
**Part (a)**
| $(2 + h)^3 = 8 + ah + bh^2 + h^3$; $(2 + h)^3 = 8 + 12h + 6h^2 + h^3$; $f(2+h) - f(2) = 13h + 6h^2 + h^3$ | M1, A1A1, m1A1F | 5 marks | A1 for each of $a, b$; PI; Ft one coeff. wrong; NMS B1F |
**Part (b)**
| Divide by $h$ and let $h \to 0$; $f'(2) = p = 13$ | M1, A1F | 2 marks | ft wrong value of $p$ |
**Total: 7 marks**
---4 The function f is defined for all real values of $x$ by
$$\mathrm { f } ( x ) = x ^ { 3 } + x$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( 2 + h ) - \mathrm { f } ( 2 )$ in the form
$$p h + q h ^ { 2 } + r h ^ { 3 }$$
where $p , q$ and $r$ are integers.
\item Use your answer to part (a) to find the value of $\mathrm { f } ^ { \prime } ( 2 )$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q4 [7]}}