A curve has equation $y=x^n\sqrt{a^2-x^2}$, where $a$ and $n$ are integer constants such

Question

A curve has equation $y=x^n\sqrt{a^2-x^2}$, where $a$ and $n$ are integer constants such that $a \ge 2$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$, $a$ and $n$, expressing your answer as a single fraction in a fully factorised form.[4] It is given that the curve has the same gradient value at both $x=1$ and $x=-1$. Find a condition that $n$ must satisfy.[3] Given that the curve has a gradient of $0$ at $x=\sqrt{2}$, find the values of $a$ and $n$. Justify your answers clearly.[4]

Accepted Answer

(i) $\dfrac{x^{n-1}\left(na^2-(n+1)x^2\right)}{\sqrt{a^2-x^2}}$ (ii) $n$ is odd (iii) $a=2$, $n=1$

NJC IP4 Techniques of Differentiation Tutorial Q3

NJC IP4 Techniques of Differentiation Tutorial Q3

Video Solution:

Solution:

Solution locked