Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663A/01) | Year 2014 | January | Q#9
Question
A curve with equation y=f(x) passes through the point (3,6). Given that
a. use integration to find f(x). Give your answer as a polynomial in its simplest form.
b. Show that , where p is a positive constant. State the value of p.
c. Sketch the graph of y = f(x), showing the coordinates of any points where the curve touches or crosses the coordinate axes.
Solution
a.
We are given;
We are given coordinates of a point on the curve (3,6).
We are required to find the equation of y in terms of x ie f(x).
We can find equation of the curve from its derivative through integration;
Therefore,
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
If a point lies on the curve
, we can find out value of
. We substitute values of
and
in the equation obtained from integration of the derivative of the curve i.e.
.
Therefore, substituting the coordinates of point (3,6) in above equation;
Therefore, equation of the curve C is;
b.
We have found in (b) that;
We are given that;
We can now compare the given and the found equations.
This yields that;
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
Hence, p=3 and we can write the given expression as;
c.
We are required to sketch;
As demonstrated in (b), we can write it as;
Substituting p=3 as found in (b);
It is evident that it is a cubic equation.
We can now sketch the curve as follows.
ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities.
It is evident that with positive coefficient of will shape the curve at extremities like increasing from left to right.
ü Find the point where the graph crosses y-axis by finding the value of when
.
We can find the coordinates of y-intercept from the given equation of the curve.
Hence, the curve crosses y-axis at point .
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 can find the coordinates of x-intercepts from the given equation of the curve.
Now we have two options.
|
|
|
|
|
|
|
|
|
Hence, the curve crosses x-axis at two points and
.
ü Calculate the values of for some value of
. This 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.
ü Sketch should show the main features of the graph and also, where possible, values where the graph intersects coordinate axes.
Comments