| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Chord gradient with h (algebraic) |
| Difficulty | Moderate -0.8 This is a straightforward first-principles differentiation question requiring students to find a gradient using the difference quotient, then take a limit as h→0. While it involves algebraic manipulation, it's a standard textbook exercise testing basic understanding of derivatives from first principles with no conceptual challenges or novel problem-solving required. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07g Differentiation from first principles: for small positive integer powers of x |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\{y(2+h)\} = \left[2 - (2+h)\right]\left[(2+h)+3\right]\) | M1 | Attempt to find \(y\) when \(x=2+h\). PI |
| Gradient \(= \frac{[2-(2+h)](5+h)+3-3}{2+h-2}\) | M1 | Use of gradient \(= \frac{y_2 - y_1}{x_2 - x_1}\) OE to obtain an expression in terms of \(h\) |
| \(= \frac{-h(3+h)}{h} = -3-h\) | A1 | CSO \(-3-h\) or \(-(3+h)\) or equally simplified |
| Answer | Marks | Guidance |
|---|---|---|
| (b) As \(h \to 0\), \(\{\)grad. of line in (a)\(\} \to\) grad. of curve at \((2,3)\) | E1 | '\(h \to 0\)' seen OE in words |
| \(\{\)Gradient of curve at point \((2,3)\} = -3\) | A1F | ft on \(c\)'s \(a\) value only if both M1s have been scored in part (a) and \(a+bh\) has been obtained convincingly for non zero \(a\) and \(b\). Final answer left as '\(\to -3\)' is A0 |
**(a)** $\{y(2+h)\} = \left[2 - (2+h)\right]\left[(2+h)+3\right]$ | M1 | Attempt to find $y$ when $x=2+h$. PI
Gradient $= \frac{[2-(2+h)](5+h)+3-3}{2+h-2}$ | M1 | Use of gradient $= \frac{y_2 - y_1}{x_2 - x_1}$ OE to obtain an expression in terms of $h$
$= \frac{-h(3+h)}{h} = -3-h$ | A1 | CSO $-3-h$ or $-(3+h)$ or equally simplified
**Total for (a): 3 marks**
**(b)** As $h \to 0$, $\{$grad. of line in (a)$\} \to$ grad. of curve at $(2,3)$ | E1 | '$h \to 0$' seen OE in words
$\{$Gradient of curve at point $(2,3)\} = -3$ | A1F | ft on $c$'s $a$ value only if both M1s have been scored in part (a) and $a+bh$ has been obtained convincingly for non zero $a$ and $b$. Final answer left as '$\to -3$' is A0
**Total for (b): 2 marks**
**Overall Total: 5 marks**
**Additional Notes:**
- (b) Differentiation to find $dy/dx = -3$ when $x=2$ scores E0A0F
- (b) Note: E0, A1F is possible
- (b) Marking the E1 with OE wording for '$\to$' eg 'tends to', 'approaches', 'goes towards'. Do NOT accept '='
- (b) Example: As $h \to 0$ gradient $\to -3-0=-3$ (E1A0F)...if c and had then written 'gradient is/ = $-3$' the A1F would have been scored
---A curve $C$ has equation $y = (2 - x)(1 + x) + 3$.
\begin{enumerate}[label=(\alph*)]
\item A line passes through the point $(2, 3)$ and the point on $C$ with $x$-coordinate $2 + h$.
Find the gradient of the line, giving your answer in its simplest form.
[3 marks]
\item Show how your answer to part (a) can be used to find the gradient of the curve $C$ at the point $(2, 3)$. State the value of this gradient.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2016 Q2 [5]}}