Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2002  OctNov  (P29709/02)  Q#7
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Question
The equation of a curve is
i. Show that
ii. Find the coordinates of the points on the curve where the tangent is parallel to the xaxis.
Solution
i.
We are given;
We are required to find .
To find from an implicit equation, differentiate each term with respect to , using the chain rule to differentiate any function
We differentiate each term of the equation, on by one, with respect to x applying following rules.
Rule for differentiation of
If
Rule for differentiation of
Now we can combines derivatives of all terms of the equation as;
ii.
We are given that tangent to the curve at a point is parallel to xaxis.
If two lines are parallel to each other, then their slopes
Since slope of xaxis is ZERO, therefore, slope of tangent to the curve is also ZERO.
The slope of a curve
Therefore, slope of the curve at the point where tangent meets the curve is equal to the slope of the tangent. Hence;
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve
As demonstrated in in (i), gradient of the curve is given by;
As found above;
We can substitute this
With xcoordinate of a point at hand, we can find the ycoordinate of the point by substituting value of xcoordinate of the point any of the two equations.
We substitute








Therefore, at two points with coordinates
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