# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#11

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Question

The diagram shows part of the curve  and the line x = 1.  The point A is the minimum point on the curve.

i.       Show that the x-coordinate of A satisfies the equation and find the  exact value of  at A.

ii.       Find, showing all necessary working, the volume obtained when the shaded  region is rotated through 360 about the x-axis.

Solution

i.

We are given that the curve is;

The point A is the minimum point on the curve.

A stationary value is the maximum or minimum value of a function.

Coordinates of stationary point on the curve can be found from the derivative of  equation of the curve by equating it with ZERO. This results in value of x-coordinate  of the stationary point  on the curve.

Therefore, we need to find the derivative of equation of the curve.

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:

To find the coordinates of the stationary (minimum) point of the curve;

We are required to find the exact value of at A.

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 x-coordinates of that point in the expression for gradient of the curve;

Therefore, first we need to find .

Second derivative is the derivative of the derivative. If we have derivative of the  curve   as , then expression for the second derivative of the curve is;

We have found above that;

Therefore;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:

To find the exact value of at point A we need to substitute x-coordinate of A in  above equation.

We have found above that x-coordinate of A satisfies the equation

Hence;

This is x-coordinate of point A.

Therefore;

ii.

Expression for the volume of the solid formed when the shaded region under the  curve is rotated completely about the x-axis is;

Therefore, for the given case;

We are given that;

Hence, for x=0 and x=1;

Rule for integration of  is:

Rule for integration of  is: