Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2017 | June | Q#2

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Question

A curve has the equation . The curve has a stationary point
at the point M where
.

a.  Find the coordinates of the other stationary point of the curve.

b.  Find the value of  at the point M, and hence determine, with a reason, whether M is a  minimum point or a maximum point.

c.  Sketch the curve.

Solution

a.
 

We are required to find the coordinates of stationary point on the curve with equation;

A stationary point  on the curve  is the point where gradient of the curve is equal to zero;

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, first we need to find the expression for the derivative of the given 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 is of  is:

Rule for differentiation is of  is:

To find the coordinates of the stationary point on the curve, we equate this derivative equal to  ZERO.

Now we have two options.

Two possible values of  imply that there are two stationary points on the curve one at each value  of .

We are already given that point M, where x=-2, is  a stationary point on the curve and hence x- coordinate of other stationary point on the curve is  .

 

b.    

We are required to find the second derivative.

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; 

Therefore;

Rule for differentiation is of  is:

Rule for differentiation is of  is:

Rule for differentiation is of  is:

Once we have the coordinates of the stationary point  of a curve, we can determine its  nature, whether minimum or maximum, by finding 2nd derivative of the curve.

We substitute  of the stationary point in the expression of 2nd derivative of the curve and  evaluate it;

If  or  then stationary point (or its value) is minimum.

If  or  then stationary point (or its value) is maximum.

We are given that point M, where x=-2, is a stationary point on the curve. Therefore, to determine  the nature of point M, we substitute x-coordinate of point M in the expression of second derivative of  the curve obtained above;

Since , the point M on the curve is minimum.

 

c.
 

We are required to sketch the curve with equation given as;

We are required to sketch a cubical polynomial given as;

ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities. If the  coefficient of  is such that  cubic graph is increasing from left to right at the extremities and  when  the cubic graph is inverted.

The coefficient of  in the given polynomial is negative. Therefore it will cause inverted cubic graph  increasing from right to left at the extremities.

ü Find the point where the graph crosses y-axis by finding the value of  when .

Therefore for ;

Hence, the cubic graph intercepts y-axis at , ie it passes through the origin (0,0).

ü Find the point(s) where the graph crosses the x-axis by finding the value of  when . If  there is repeated root the graph will touch the x-axis.

We already have x-coordinates of two stationary points on the curve at  and

We may like to find corresponding y-coordinate of stationary points on the curve for accurate  sketching of the curve by substituting corresponding value of x-coordinate in equation of the
curve.

When ;

When ;

We can sketch the graph with this information. However, we may like to do the following as well. 

ü Calculate the values of  for some value of . The is particularly useful in determining the  quadrant in which the graph might turn close to the y-axis.

ü Complete the sketch of the graph by joining the sections.

Sketch should show the main features of the graph and also, where possible, values where the  graph intersects coordinate axes.

desmos-graph (1).png

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