Standard +0.8 This requires product rule differentiation of exponential and trigonometric functions, setting the derivative to zero, and solving a transcendental equation numerically. The algebraic manipulation and need for numerical methods elevates this above routine differentiation exercises, though it follows a standard stationary points framework.
4 A curve has equation \(y = \mathrm { e } ^ { - 3 x } \tan x\). Find the \(x\)-coordinates of the stationary points on the curve in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\). Give your answers correct to 3 decimal places.
Equate derivative to zero and obtain an equation of the form \(a\sin 2x = b\), or a quadratic in \(\tan x\), \(\sin^2 x\), or \(\cos^2 x\)
M1*
Carry out correct method for finding one angle
M1(dep*)
Obtain answer, e.g. 0.365
A1
Obtain second answer 1.206 and no others in the range (allow 1.21)
A1
[Ignore answers outside the given range.] [Treat answers in degrees, 20.9° and 69.1°, as a misread.]
[6]
| Use product or quotient rule | M1 |
| Obtain derivative in any correct form | A1 |
| Equate derivative to zero and obtain an equation of the form $a\sin 2x = b$, or a quadratic in $\tan x$, $\sin^2 x$, or $\cos^2 x$ | M1* |
| Carry out correct method for finding one angle | M1(dep*) |
| Obtain answer, e.g. 0.365 | A1 |
| Obtain second answer 1.206 and no others in the range (allow 1.21) | A1 |
| [Ignore answers outside the given range.] [Treat answers in degrees, 20.9° and 69.1°, as a misread.] | [6] |
4 A curve has equation $y = \mathrm { e } ^ { - 3 x } \tan x$. Find the $x$-coordinates of the stationary points on the curve in the interval $- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi$. Give your answers correct to 3 decimal places.
\hfill \mbox{\textit{CAIE P3 2009 Q4 [6]}}