Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2003  MayJun  (P29709/02)  Q#6
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Question
The equation of a curve is
i. Show, by differentiation, that the gradient of the curve is always negative.
ii. Use the trapezium rule with 2 intervals to estimate the value of
giving your answer correct to 2 significant figures.
iii.
The diagram shows a sketch of the curve for . 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 show that the gradient of the curve with following equation is always negative.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
Therefore, we need to differentiate the given equation of the curve for the gradient.
We apply chain rule to solve this derivative.
If we define , then derivative of is;
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is;
Another way of applying chain rule is as follows.
If we have and then derivative of is;
We have;
Let , therefore;
First we find ;
Rule for differentiation of is:
Next we find .
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is;
Now we can find .
Since , therefore;
It is evident that and both
being squares will always be positive. There is a negative sign in , therefore, derivative will always be negative and since the derivative of a curve is its gradient, the gradient of the curve will be always negative.
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;




0 



1 



2 


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.