Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2011  OctNov  (P29709/21)  Q#3
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Question
The diagram shows the part of the curve for . Find the xcoordinates of the points on this part of the curve at which the gradient is 4.
Solution
We are required to find the xcoordinate of the points on the curve where gradient is 4.
Therefore first we need to find the expression for gradient of the given curve.
Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that particular point.
Gradient (slope) of the curve at a particular point can be found by substituting xcoordinates of that point in the expression for gradient of the curve;
We are given;
Therefore;
For we use chain rule.
If we define , then derivative of is;
If we have and then derivative of is;
Let , then;
Rule for differentiation of is:
Rule for differentiation of is;
Therefore, we can equate expression of gradient of the curve with 4.
provided that
Now we have two options.




Using calculator, we can find; 



Properties of 

Domain 

Range 

Periodicity 



Odd/Even 

Translation/ Symmetry 






We utilize the odd/even property of to find another solution (root) of :
Hence;
Therefore, we have four solutions of of ;












However, only following values of are within the given interval .


Hence, these are the xcoordinates of the points on the curve where curve has gradient 4.
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