| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Numerical gradient deduction |
| Difficulty | Easy -1.2 This is a straightforward application of reading gradient values from a table to estimate the derivative at a point, requiring only basic understanding that gradient of a chord approximates the derivative. It's simpler than average A-level questions as it involves minimal calculation and is primarily about interpreting given numerical data rather than performing differentiation. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07b Gradient as rate of change: dy/dx notation |
| Chord | \(A E\) | \(B E\) | \(C E\) | \(D E\) | ||
| 4 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Gradient of \(CE = 2.5\) | 1 B1 | |
| Gradient of \(DE = 2.1\) | 1 B1 | |
| \(f'(2) = 2\) | 2 B1 | Accept reasonable conclusion following their gradient |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f'(2) = 2\) | 1 B1 | Accept reasonable conclusion following their gradient |
## Question 1(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Gradient of $CE = 2.5$ | 1 B1 | |
| Gradient of $DE = 2.1$ | 1 B1 | |
| $f'(2) = 2$ | 2 B1 | Accept reasonable conclusion following their gradient |
## Question 1(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(2) = 2$ | 1 B1 | Accept reasonable conclusion following their gradient |1 The following points
$$A ( 0,1 ) , \quad B ( 1,6 ) , \quad C ( 1.5,7.75 ) , \quad D ( 1.9,8.79 ) \quad \text { and } \quad E ( 2,9 )$$
lie on the curve $y = \mathrm { f } ( x )$. The table below shows the gradients of the chords $A E$ and $B E$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
Chord & $A E$ & $B E$ & $C E$ & $D E$ \\
\hline
\begin{tabular}{ c }
Gradient of \\
chord \\
\end{tabular} & 4 & 3 & & \\
\hline
\end{tabular}
\end{center}
(a) Complete the table to show the gradients of $C E$ and $D E$.\\
(b) State what the values in the table indicate about the value of $\mathrm { f } ^ { \prime } ( 2 )$.\\
\hfill \mbox{\textit{CAIE P1 2020 Q1 [3]}}