Question

The curve C has equation y= f(x) passes through the point (5,65).

Given that;

a.   Use integration to find f(x).

b.   Hence show that

c.   In the space provided on page 17, sketch C, showing the coordinates of the points where C crosses the x-axis.

Solution

a.
 

We are required to find f(x), when;

We are also given that the curve passes through the point (5,65).

Clearly it is the case of finding equation from its derivative.

We can find equation of the curve from its derivative through integration;

For the given case;

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. . 

We are also given that the curve passes through the point (5,65). 

Therefore, substituting given values of y and x.

Hence, above equation obtained from integration can now be written as;

b.
 

We have found in (a);

c.    

We are required to sketch curve C for which we have found equation in (a) as;

It is a cubic function.

ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities. 

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

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.

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.

The given cubic function has positive coefficient of .

Therefore, the function increases from left to right along x-axis. 

Next we need to find the y-crossing of the function which is also called y-intercept.

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line). 

We have found in (b) that given equation of the curve C can be written as; 

In this equation we substitute ; 

Hence, y-intercept of curve C has coordinates (0,0).

Now we find x-crossing of curve C which is also called x-intercept.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

In equation of curve C  we substitute ;

We have three options now.

Therefore, the curve C intersects the x-axis at 03 points with coordinates (0,0),  and (4,0).  

We can now sketch the graph of the curve C as shown below.