The diagram shows part of the curve of $y=\frac{1}{4}x^2-\frac{1}{2}\ln\left(\frac{x}{3}\right)$ a

Question

The diagram shows part of the curve of $y=\frac{1}{4}x^2-\frac{1}{2}\ln\left(\frac{x}{3}\right)$ and $P(3,2.25)$ is a point on the curve. By using calculus, determine, with full working, whether $y=\frac{4}{3}x-1$ is the equation of the tangent to the curve at $P$.[5] Differentiate $x\ln\left(\frac{x}{3}\right)-x$ with respect to $x$.[2] Using the result from part (b), find the exact area of the region enclosed by the curve $y=\frac{1}{4}x^2-\frac{1}{2}\ln\left(\frac{x}{3}\right)$, the $x$-axis, the line $x=1$ and the line $x=3$.[3]

Accepted Answer

(a) $y=\frac43x-1$ is not the tangent at $P$ (b) $\ln\left(\frac{x}{3}\right)$ (c) $\frac{19}{6}+\frac12\ln\left(\frac13\right)$

2025 SCGS PRELIMS P1 Q12

2025 SCGS PRELIMS P1 Q12

Solution:

Solution locked