| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find tangent line equation |
| Difficulty | Standard +0.3 This is a standard C3 differentiation question involving exponential functions. Parts (a)-(c) require routine techniques: identifying range, finding intercepts, and verifying a given tangent equation. Part (d) adds mild complexity by requiring a second tangent and solving simultaneous equations, but all steps follow predictable procedures with no novel insight needed. Slightly easier than the typical 0.0 benchmark. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06b Gradient of e^(kx): derivative and exponential model1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(f(x) > -2\) | B1 |
| (b) | \(x = 0, \quad y = e - 2 \quad \therefore P(0, e-2)\) | B1 |
| \(y = 0\): \(0 = e^{x+1} - 2\) | M1 | |
| \(3x + 1 = \ln 2\) | M1 | |
| \(x = \frac{1}{3}(\ln 2 - 1) \quad \therefore Q(\frac{1}{3}(\ln 2 - 1), 0)\) | A1 | |
| (c) | \(f'(x) = 3e^{3x+1}\) | M1 |
| At \(P\), grad \(= 3e\) | A1 | |
| \(y - (e-2) = 3e(x - 0)\) | M1 | |
| \(y = 3ex + e - 2\) | A1 | |
| (d) | At \(Q\), grad \(= 6\) | B1 |
| Tangent at \(Q\): \(y - 0 = 6(x - \frac{1}{3}(\ln 2 - 1))\) | M1 | |
| \(y = 6x - 2\ln 2 + 2\) | ||
| Intersect: \(3ex + e - 2 = 6x - 2\ln 2 + 2\) | ||
| \(x(3e - 6) = 4 - e - 2\ln 2\) | M1 | |
| \(x = \frac{4-e-2\ln 2}{3e-6} = -0.0485\) (3sf) | A1 | |
| (13) |
(a) | $f(x) > -2$ | B1 |
(b) | $x = 0, \quad y = e - 2 \quad \therefore P(0, e-2)$ | B1 |
| $y = 0$: $0 = e^{x+1} - 2$ | M1 |
| $3x + 1 = \ln 2$ | M1 |
| $x = \frac{1}{3}(\ln 2 - 1) \quad \therefore Q(\frac{1}{3}(\ln 2 - 1), 0)$ | A1 |
(c) | $f'(x) = 3e^{3x+1}$ | M1 |
| At $P$, grad $= 3e$ | A1 |
| $y - (e-2) = 3e(x - 0)$ | M1 |
| $y = 3ex + e - 2$ | A1 |
(d) | At $Q$, grad $= 6$ | B1 |
| Tangent at $Q$: $y - 0 = 6(x - \frac{1}{3}(\ln 2 - 1))$ | M1 |
| $y = 6x - 2\ln 2 + 2$ | |
| Intersect: $3ex + e - 2 = 6x - 2\ln 2 + 2$ | |
| $x(3e - 6) = 4 - e - 2\ln 2$ | M1 |
| $x = \frac{4-e-2\ln 2}{3e-6} = -0.0485$ (3sf) | A1 |
| | (13) |6.
$$\mathrm { f } ( x ) = \mathrm { e } ^ { 3 x + 1 } - 2 , \quad x \in \mathbb { R } .$$
\begin{enumerate}[label=(\alph*)]
\item State the range of f .
The curve $y = \mathrm { f } ( x )$ meets the $y$-axis at the point $P$ and the $x$-axis at the point $Q$.
\item Find the exact coordinates of $P$ and $Q$.
\item Show that the tangent to the curve at $P$ has the equation
$$y = 3 \mathrm { e } x + \mathrm { e } - 2 .$$
\item Find to 3 significant figures the $x$-coordinate of the point where the tangent to the curve at $P$ meets the tangent to the curve at $Q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q6 [13]}}