Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2019  FebMar  (P29709/22)  Q#5
Question
The curve with equation
and the point P on the curve has ycoordinate 10.
i. Show that the xcoordinate of P satisfies the equation
ii. Use the iterative formula
to find the xcoordinate of P correct to 4 significant figures. Give the result
of each iteration to 6 significant figures.
iii. Find the gradient of the curve at P, giving the answer correct to 3 significant figures.
Solution
i.
We are given that the curve and the point P on the curve has ycoordinate 10.
We are required to find the xcoordinate of the point P.
We can find the corresponding xcoordinate of point P by substituting ycoordinate of the same point in equation of the curve.
Taking logarithm of both sides;
and are inverse functions. The composite function is an identity function, with domain the positive real numbers. Therefore;
ii.
Iteration method can be used to find the root of the given equation using sequence defined by;
If the sequence given by the inductive definition , with some initial value , converges to a limit , then is the root of the equation .
Therefore, if , then is a root of .
We are given that as initial value.



1 


2 


3 


4 


5 


6 


7 


It is evident that .
Hence, is a root of .
The root given correct to 4 significant figures is 2.316.
iii.
We are required to find gradient of the curve at the point .
We have demonstrated in (iii) that, .
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;
Therefore, first we need to find expression for gradient of the curve.
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is:
We are given equation of the curve;
We use Quotient Rule to differentiate.
If and are functions of , and if , then;
If , then;
Let and , then;
We differentiate and one by one.
First we differentiate .
Rule for differentiation of natural exponential function is;
Next, differentiate .
Rule for differentiation of is:
Rule for differentiation of is:
Rule for differentiation of is:
Hence;
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