| Exam Board | AQA |
|---|---|
| Module | Paper 3 (Paper 3) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Standard +0.8 This is a multi-part question requiring second derivative analysis to find concavity (involving product/chain rule with exponentials), numerical integration via trapezium rule, understanding of why trapezium rule underestimates on concave functions, and a proof involving bounds. While each individual technique is A-level standard, the combination of calculus concepts, numerical methods, and proof reasoning with error bounds makes this moderately challenging—above average but not exceptionally difficult. |
| Spec | 1.07f Convexity/concavity: points of inflection1.07p Points of inflection: using second derivative1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a) | Finds 2nd derivative and sets up an | |
| inequality | AO3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains correct first derivative | AO1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| from ‘their’ first derivative | AO1.1b | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| (could use set notation) | AO2.2a | A1 |
| (b) | Uses trapezium rule | AO1.1a |
| Answer | Marks | Guidance |
|---|---|---|
| Trapezium rule entries all correct | AO1.1b | A1 |
| Finds correct value | AO1.1b | A1 |
| (c) | References area being completely |
| Answer | Marks | Guidance |
|---|---|---|
| So… | AO2.4 | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| awarded) | AO2.4 | E1 |
| (d) | Uses suitable rectangle to obtain | |
| over-estimate | AO3.1a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| above the curve | AO2.4 | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| contains no slips. | AO2.1 | R1 |
| Total | 12 | |
| Q | Marking Instructions | AO |
Question 7:
--- 7(a) ---
7(a) | Finds 2nd derivative and sets up an
inequality | AO3.1a | M1 | dy
=−2xe−x2
dx
d2y
=−2e−x2 +4x2e−x2
dx2
−2e−x2 +4x2e−x2 <0
4x2 −2<0
2 2
− < x<
2 2
Obtains correct first derivative | AO1.1b | A1
Obtains second derivative correct
from ‘their’ first derivative | AO1.1b | A1F
Deduces correct final inequality
(could use set notation) | AO2.2a | A1
(b) | Uses trapezium rule | AO1.1a | M1 | 0.5 0.1
∫ e−x2 dx≈ (e−0.01+e−0.25
2
0.1
+2(e−0.04 +e−0.09 +e−0.16))
0.3611
Trapezium rule entries all correct | AO1.1b | A1
Finds correct value | AO1.1b | A1
(c) | References area being completely
within concave section
So… | AO2.4 | E1 |
2 2
[ 0 . 1 , 0 . 5 ]≈⊂ − ,
2 2
∴ area is completely within
concave section
Hence trapezia lie below curve
and give an under-estimate for
the area
Trapezia all fall completely
underneath the curve therefore
underestimate (only award this
mark if previous E1 has been
awarded) | AO2.4 | E1
(d) | Uses suitable rectangle to obtain
over-estimate | AO3.1a | B1 | Using a rectangle with the left
hand edge the same height as
the curve will produce an over-
estimate
Area of rectangle =
0.4×e−0.12
= 0.396...
∴ 0.36< A<0.40
So A = 0.4 to 1 dp
Explains that this rectangle lies
above the curve | AO2.4 | E1
Constructs rigorous mathematical
argument about accuracy, which
leads to required result
Only award if they have a
completely correct solution, which
is clear, easy to follow and
contains no slips. | AO2.1 | R1
Total | 12
Q | Marking Instructions | AO | Marks | Typical SolutionThe diagram shows part of the graph of $y = e^{-x^2}$
\includegraphics{figure_7}
The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $x$ for which the graph is concave. [4 marks]
\item The finite region bounded by the $x$-axis and the lines $x = 0.1$ and $x = 0.5$ is shaded.
\includegraphics{figure_7b}
Use the trapezium rule, with 4 strips, to find an estimate for $\int_{0.1}^{0.5} e^{-x^2} dx$
Give your estimate to four decimal places. [3 marks]
\item Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate. [2 marks]
\item By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 Q7 [12]}}