| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | October |
| Marks | 7 |
| Topic | Tangents, normals and gradients |
| Type | Find tangent at given point (polynomial/algebraic) |
| Difficulty | Moderate -0.3 This is a straightforward calculus question requiring basic differentiation (power rule), sketching a cubic gradient function by finding stationary points, and finding a tangent line equation. All parts are routine A-level techniques with no problem-solving insight required, making it slightly easier than average but not trivial due to the multi-part nature and sketching component. |
| Spec | 1.07c Sketch gradient function: for given curve1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
A curve has equation $y = \frac{1}{4}x^4 - x^3 - 2x^2$.
\begin{enumerate}[label=(\roman*)]
\item Find $\frac{dy}{dx}$. [1]
\item Hence sketch the gradient function for the curve. [4]
\item Find the equation of the tangent to the curve $y = \frac{1}{4}x^4 - x^3 - 2x^2$ at $x = 4$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q7 [7]}}