Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2006  OctNov  (P29709/02)  Q#6
Question
The diagram shows the part of the curve for , and its minimum point M.
i. Find the coordinates of M.
ii. Use the trapezium rule with 2 intervals to estimate the value of
Giving your answer correct to 1 decimal place.
iii. State, with a reason, whether the trapezium rule gives an underestimate or an overestimate of the true value of the integral in part (ii).
Solution
i.
We are required to find the coordinates of point M which is minimum point of the curve;
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;
If and are functions of , and if , then;
If , then;
Let and ;
Rule for differentiation of natural exponential function , or ;
Rule for differentiation of 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 .
To find the ycoordinate of point M on the curve, we substitute value of xcoordinate in equation of the curve.
Hence;
Hence, ycoordinate of the point M is .
Therefore, coordinates of point M
ii.
We are required to apply Trapezium Rule to evaluate;
The trapezium rule with intervals states that;
We are given that there are two intervals, .
We are also given that and .
Hence;




1 



2 



3 


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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