| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Topic | Differentiation from First Principles |
| Type | Expand f(a+h) algebraically |
| Difficulty | Moderate -0.3 Part (a) is straightforward algebraic substitution and expansion. Part (b) requires differentiation from first principles, which is a standard A-level technique but slightly more involved than using standard differentiation rules. The question is well-structured with clear steps, making it slightly easier than average but still requiring proper method and algebraic manipulation. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.07g Differentiation from first principles: for small positive integer powers of x |
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathbf{f}(x) = x^2 - 4x + 2$, find $\mathbf{f}(3 + h)$
Express your answer in the form $h^2 + bh + c$, where $b$ and $c \in \mathbb{Z}$. [2 marks]
\item The curve with equation $y = x^2 - 4x + 2$ passes through the point $P(3, -1)$ and the point $Q$ where $x = 3 + h$.
Using differentiation from first principles, find the gradient of the tangent to the curve at the point $P$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q8 [5]}}