Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2007  OctNov  (P29709/02)  Q#4
Question
The equation of a curve is y = 2x − tan x, where x is in radians. Find the coordinates of the stationary points of the curve for which
Solution
We are required to find the coordinates of stationary points of the curve with equation;
A stationary point on the curve is the point where gradient of the curve is equal to zero;
We can find expression for gradient of the curve and equate it with ZERO to find the coordinates of stationary points.
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 of is;
Since we are looking for coordinates of stationary points of the curve and gradient at stationary point sis ZERO;
Using calculator;
Properties of 

Domain 

Range 

Periodicity 



Odd/Even 

Translation/ Symmetry 






We utilize the odd/even property of to find other solutions (roots) of :
Therefore;
Therefore, we have two solutions (roots) of the equation;


To find all the solutions (roots), we can utilize the periodic property of for both these values of but that will yield values beyond the desired interval of .
Hence, xcoordinates of the stationary points on the curve with equation are;


Two values of x indicate that there are two stationary points.
Corresponding values of y coordinate can be found by substituting values of x in equation of the curve.
For 
For 







Hence, coordinates of stationary points are;



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