| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative of simple polynomial (integer powers) |
| Difficulty | Moderate -0.8 This is a straightforward AS-level differentiation question requiring basic power rule application and understanding of the derivative as a limit. Part (a) is routine differentiation, part (b) is chord gradient using the difference quotient formula, and part (c) tests conceptual understanding that the tangent gradient is the limit as h→0. All techniques are standard with no problem-solving insight required. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07b Gradient as rate of change: dy/dx notation |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Attempts \(\frac{dy}{dx}\) at \(x=2\) | M1 | Allow \(3x^2 - 2 \to \ldots x\) and substitutes \(x=2\) |
| \(\frac{dy}{dx} = 6x \Rightarrow\) gradient at \(P\) is \(12\) | A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Gradient \(PQ = \dfrac{3(2+h)^2-2-10}{(2+h)-2}\) | B1 | Correct expression for gradient of chord, seen or implied |
| \(= \dfrac{3(2+h)^2-12}{(2+h)-2} = \dfrac{12h+3h^2}{h}\) | M1 | Attempts \(\frac{\delta y}{\delta x}\), condoning slips; denominator must be \(h\) |
| \(= 12+3h\) | A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| As \(h\to 0\), \(12+3h \to 12\); gradient of chord tends to gradient of tangent to the curve | B1 | Must state \(h\to 0\) and \(12+3h\to 12\) AND state chord gradient tends to gradient of curve |
# Question 5:
## Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Attempts $\frac{dy}{dx}$ at $x=2$ | M1 | Allow $3x^2 - 2 \to \ldots x$ and substitutes $x=2$ |
| $\frac{dy}{dx} = 6x \Rightarrow$ gradient at $P$ is $12$ | A1 | cso |
## Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Gradient $PQ = \dfrac{3(2+h)^2-2-10}{(2+h)-2}$ | B1 | Correct expression for gradient of chord, seen or implied |
| $= \dfrac{3(2+h)^2-12}{(2+h)-2} = \dfrac{12h+3h^2}{h}$ | M1 | Attempts $\frac{\delta y}{\delta x}$, condoning slips; denominator must be $h$ |
| $= 12+3h$ | A1 | cso |
## Part (c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| As $h\to 0$, $12+3h \to 12$; gradient of chord tends to gradient of tangent to the curve | B1 | Must state $h\to 0$ and $12+3h\to 12$ AND state chord gradient tends to gradient of curve |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-10_680_684_255_694}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows part of the curve with equation $y = 3 x ^ { 2 } - 2$\\
The point $P ( 2,10 )$ lies on the curve.
\begin{enumerate}[label=(\alph*)]
\item Find the gradient of the tangent to the curve at $P$.
The point $Q$ with $x$ coordinate $2 + h$ also lies on the curve.
\item Find the gradient of the line $P Q$, giving your answer in terms of $h$ in simplest form.
\item Explain briefly the relationship between part (b) and the answer to part (a).
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 2021 Q5 [6]}}