| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of product |
| Difficulty | Standard +0.3 This question requires applying the product rule twice and then computing mean values via integration. While it involves multiple steps (differentiation, integration, evaluation), each step is straightforward application of standard techniques. The product rule application is routine, and finding mean values by integration is a standard A-level further maths procedure. The calculations are somewhat lengthy but conceptually straightforward, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08i Integration by parts4.08e Mean value of function: using integral |
| Answer | Marks |
|---|---|
| (i) \(\text{MV of } y_1 \text{ over } 0 \leq x \leq \pi/2 = 2/\pi [x^3 \sin x]_0^{\pi/2}\) | M1 |
| \(= ... = \pi/2\) (AG, CWO) | A1 |
| (ii) \(\text{MV of } y_2 \text{ over } 0 \leq x \leq \pi/2 = 2/\pi [2x \sin x + x^2 \cos x]_0^{\pi/2} = 2\) | M1A1 |
(i) $\text{MV of } y_1 \text{ over } 0 \leq x \leq \pi/2 = 2/\pi [x^3 \sin x]_0^{\pi/2}$ | M1 |
$= ... = \pi/2$ (AG, CWO) | A1 |
(ii) $\text{MV of } y_2 \text{ over } 0 \leq x \leq \pi/2 = 2/\pi [2x \sin x + x^2 \cos x]_0^{\pi/2} = 2$ | M1A1 |1 Given that
$$y = x ^ { 2 } \sin x$$
(i) show that the mean value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ with respect to $x$ over the interval $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$ is $\frac { 1 } { 2 } \pi$,\\
(ii) find the mean value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ with respect to $x$ over the interval $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$.
\hfill \mbox{\textit{CAIE FP1 2009 Q1 [4]}}