Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2011  OctNov  (P29709/22)  Q#3
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Question
The diagram shows the curve y= x − 2 ln x and its minimum point M.
i. Find the xcoordinates of M.
ii. Use the trapezium rule with three intervals to estimate the value of
giving your answer correct to 2 decimal places.
iii. State, with a reason, whether the trapezium rule gives an underestimate or an overestimate of the true value of the integral in part (iii).
Solution
i.
We are required to find the xcoordinate of point M which is given as minimum point of the curve with equation;
A stationary point on the curve is the point where gradient of the curve is equal to zero;
Since point M is minimum point, therefore, it is stationary point of the curve and, hence, gradient of the curve at point M must ZERO.
We can find expression for gradient of the curve at point M and equate it with ZERO to find the x coordinate of point M.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
Therefore;
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation natural logarithmic function , for is;
Now we need expression for gradient of the curve at point M.
Gradient (slope) of the curve at a particular point can be found by substituting x coordinates of that point in the expression for gradient of the curve;
Since point M is a maximum point, the gradient of the curve at this point must be equal to ZERO.
Hence, xcoordinate of point M on the curve is 2.
ii.
We are required to apply Trapezium Rule to evaluate;
The trapezium rule with intervals states that;
If the graph is bending downwards over the whole interval from to , then trapezium rule will give an underestimate of the true area.
We are given that there are three intervals, .
We are also given that and .
Hence;




1 



2 



3 



4 


Therefore;
iii.
If the graph is bending upwards over the whole interval from to , then trapezium rule will give an overestimate of the true area (as shown in the diagram below).
It is evident that for the given graph trapezium rule will give an overestimate.
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