| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| 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 guided, multi-step differentiation from first principles question where part (a) scaffolds the algebra needed for part (b). The expansion of (5+h)³ is straightforward binomial expansion, and part (b) follows a standard template: substitute x=5+h, find the difference quotient, and take the limit as h→0. While it requires careful algebraic manipulation, the question provides significant structure and all steps are routine for Further Maths students who have practiced this technique. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((5+h)^3 = 125 + 75h + 15h^2 + h^3\) | B1 | Accept unsimplified coefficients |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y(5+h) = 100 + 65h + 14h^2 + h^3\) | B1F | PI; ft numerical error in (a) |
| Use of correct formula for gradient | M1 | |
| Gradient is \(65 + 14h + h^2\) | A2,1F | A1 if one numerical error made; ft numerical error already penalised |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| As \(h \to 0\) this \(\to 65\) | E2,1F | E1 for '\(h = 0\)'; ft wrong values for \(p, q, r\) |
| Total | 2 |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(5+h)^3 = 125 + 75h + 15h^2 + h^3$ | B1 | Accept unsimplified coefficients |
| **Total** | **1** | |
### Part (b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y(5+h) = 100 + 65h + 14h^2 + h^3$ | B1F | PI; ft numerical error in (a) |
| Use of correct formula for gradient | M1 | |
| Gradient is $65 + 14h + h^2$ | A2,1F | A1 if one numerical error made; ft numerical error already penalised |
| **Total** | **4** | |
### Part (b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| As $h \to 0$ this $\to 65$ | E2,1F | E1 for '$h = 0$'; ft wrong values for $p, q, r$ |
| **Total** | **2** | |
---6
\begin{enumerate}[label=(\alph*)]
\item Expand $( 5 + h ) ^ { 3 }$.
\item A curve has equation $y = x ^ { 3 } - x ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item Find the gradient of the line passing through the point $( 5,100 )$ and the point on the curve for which $x = 5 + h$. Give your answer in the form
$$p + q h + r h ^ { 2 }$$
where $p , q$ and $r$ are integers.
\item Show how the answer to part (b)(i) can be used to find the gradient of the curve at the point $( 5,100 )$. State the value of this gradient.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q6 [7]}}