| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find stationary points of exponential curves |
| Difficulty | Moderate -0.3 This is a straightforward application of differentiation to find and classify a stationary point. Students need to differentiate exponentials (standard rules), set dy/dx = 0, solve a simple exponential equation, and apply the second derivative test. While it requires multiple steps, each is routine and the question follows a standard textbook pattern with no novel insight required. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Mark | Guidance |
| (i) Obtain derivative \(e^x - 8e^{2x}\) in any correct form | B1 | |
| Equate derivative to zero and simplify to an equation of the form \(e^u = a\), where \(a \neq 0\) | M1* | |
| Carry out method for calculating \(x\) with \(a > 0\) | M1(dep*) | |
| Obtain answer \(x = \ln 2\), or an exact equivalent (also accept \(0.693\) or \(0.69\)) | A1 | Accept statements of the form "\(u^a = a\), where \(u = e^x\)" for the first M1. |
| (ii) Carry out a method for determining the nature of the stationary point | M1 | |
| Show that the point is a minimum correctly, with no incorrect work seen | A1 | Max 2 marks |
| Content | Mark | Guidance |
|---------|------|----------|
| **(i)** Obtain derivative $e^x - 8e^{2x}$ in any correct form | B1 | |
| Equate derivative to zero and simplify to an equation of the form $e^u = a$, where $a \neq 0$ | M1* | |
| Carry out method for calculating $x$ with $a > 0$ | M1(dep*) | |
| Obtain answer $x = \ln 2$, or an exact equivalent (also accept $0.693$ or $0.69$) | A1 | Accept statements of the form "$u^a = a$, where $u = e^x$" for the first M1. | Max 4 marks |
| **(ii)** Carry out a method for determining the nature of the stationary point | M1 | |
| Show that the point is a minimum correctly, with no incorrect work seen | A1 | Max 2 marks |
---4 The curve $y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }$ has one stationary point.\\
(i) Find the $x$-coordinate of this point.\\
(ii) Determine whether the stationary point is a maximum or a minimum point.
\hfill \mbox{\textit{CAIE P3 2002 Q4 [6]}}