These Ten-Year-Series (TYS) worked solutions with video explanations for 2021 A Level H2 Mathematics Paper 2 Question 2 are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
2021 A Level H2 Math Paper 2 Question 2
The diagram shows a sketch of the curve $y=\text{f}\left( x \right)$. The region under the curve between $x=1$ and $x=5$, shown shaded in the diagram, is $A$. This region is split into $5$ vertical strips of equal width, $h$.
(a)
State the value of $h$ and show, using a sketch, that $\sum\limits_{n=0}^{4}{\left( \text{f}\left( 1+nh \right) \right)}\,h$ is less than the area of $A$.
[2]
(a) State the value of $h$ and show, using a sketch, that $\sum\limits_{n=0}^{4}{\left( \text{f}\left( 1+nh \right) \right)}\,h$ is less than the area of $A$.
[2]
(b)
Find a similar expression that is greater than the area of $A$.
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(b) Find a similar expression that is greater than the area of $A$.
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You are now given that $\text{f}\left( x \right)=\frac{1}{20}{{x}^{2}}+1$.
(c)
Use the expression given in part (a) and your expression from part (b) to find lower and upper bounds for the area of $A$.
[2]
(c) Use the expression given in part (a) and your expression from part (b) to find lower and upper bounds for the area of $A$.
[2]
(d)
Sketch the graph of a function $y=\text{g}\left( x \right)$, between $x=1$ and $x=5$, for which the area between the curve, the $x$-axis and the lines $x=1$ and $x=5$ is less than $\sum\limits_{n=0}^{4}{\left( \text{g}\left( 1+nh \right) \right)}\,h$.
[1]
(d) Sketch the graph of a function $y=\text{g}\left( x \right)$, between $x=1$ and $x=5$, for which the area between the curve, the $x$-axis and the lines $x=1$ and $x=5$ is less than $\sum\limits_{n=0}^{4}{\left( \text{g}\left( 1+nh \right) \right)}\,h$.
[1]
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