| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Complete numerical table |
| Difficulty | Easy -1.8 This question involves only direct substitution into a given function and calculating chord gradients using the formula (y₂-y₁)/(x₂-x₁). Part (a) requires substituting x=2.0001 into the function, part (b) is a simple gradient calculation, and part (c) asks students to observe that as the chord gets shorter, the gradient approaches the derivative—a conceptual understanding tested through pattern recognition rather than actual differentiation from first principles. Despite the topic label, no limit definition or algebraic manipulation is required; it's purely numerical and observational work suitable for early calculus students. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07b Gradient as rate of change: dy/dx notation |
| Chord | \(AB\) | \(AC\) | \(AD\) | \(AE\) | \(AF\) |
| Gradient of chord | 6.2501 | 6.2511 | 6.2608 | 7.2288 |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) | [k] = 4.00063 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4(b) | [Gradient AE] = 6.3566 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4(c) | Suggests that f'2 6.25 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(a) ---
4(a) | [k] = 4.00063 | B1 | CAO
1
--- 4(b) ---
4(b) | [Gradient AE] = 6.3566 | B1 | CAO
1
--- 4(c) ---
4(c) | Suggests that f'2 6.25 | B1 | CAO
1
Question | Answer | Marks | GuidanceThe equation of a curve is $y = \text{f}(x)$, where f$(x) = (2x - 1)\sqrt{3x - 2} - 2$. The following points lie on the curve. Non-exact values have been given correct to 5 decimal places.
$A(2, 4)$, $B(2.0001, k)$, $C(2.001, 4.00625)$, $D(2.01, 4.06261)$, $E(2.1, 4.63566)$, $F(3, 11.22876)$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$. Give your answer correct to 5 decimal places. [1]
\end{enumerate}
The table shows the gradients of the chords $AB$, $AC$, $AD$ and $AF$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Chord & $AB$ & $AC$ & $AD$ & $AE$ & $AF$ \\
\hline
Gradient of chord & 6.2501 & 6.2511 & 6.2608 & & 7.2288 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the gradient of the chord $AE$. Give your answer correct to 4 decimal places. [1]
\item Deduce the value of f$'(2)$ using the values in the table. [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q4 [3]}}