2021 YIJC P1 Q1
(a)
The diagram shows part of the graph of $y=\ln x$ with $n$ rectangles of equal width, where $n$ is a positive integer.
(i) Show that the total area of the $n$ rectangles, $A$, is
$A=\frac{1}{n}\sum\limits_{r=1}^{n}{\left[ \ln \left( n+r \right) \right]-\ln n}$
[2]
(ii) Evaluate $\underset{n\to \infty }{\mathop{\lim }}\,A$, giving your answer correct to $4$ decimal places.
[2]
(b)
It is given that $\text{f}\left( x \right)=\frac{a}{{{x}^{2}}}+bx+c$, where $a$, $b$ and $c$ are constants. The curve with equation $y=\text{f}\left( x \right)$ has a minimum point with coordinates $\left( 1,2 \right)$ and $\int_{1}^{3}{\text{f}\left( x \right)}\,\text{d}x=\frac{20}{3}$. Find the equation of the curve.
[4]
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