| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 9 |
| Topic | Differentiating Transcendental Functions |
| Type | Differentiate inverse trigonometric functions |
| Difficulty | Standard +0.3 Part (a) is a standard textbook derivation of the derivative of arcsin using implicit differentiation (3 marks of routine work). Part (b) requires computing a mean value integral involving splitting the integrand, recognizing one part as arcsin derivative and the other requiring substitution, then simplifying—this is methodical A-level integration practice with multiple steps but no novel insight. The 9 total marks reflect length rather than conceptual difficulty, making this slightly easier than average overall. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs4.08e Mean value of function: using integral4.08g Derivatives: inverse trig and hyperbolic functions |
Given that $y = \arcsin x$, $-1 \leqslant x < 1$,
\begin{enumerate}[label=(\alph*)]
\item show that $\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}$. [3]
\end{enumerate}
Given that $f(x) = \frac{3x + 2}{\sqrt{4 - x^2}}$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item show that the mean value of $f(x)$ over the interval $[0, \sqrt{2}]$, is
$$\frac{\pi\sqrt{2}}{4} + A\sqrt{2} - A,$$
where $A$ is a constant to be determined. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q7 [9]}}