Differentiation Application Example

 

Lets assume that the student asked what the solution to the below question.

 

Question: Write the equation of the curve, which is a third order polynomial, shown below. The two points, the curve crosses the x-axis are -2 and 2 and has a local minimum at .

 

 

Answer: The curve is a third order polynomial which crosses the x-axis at -2, 2 and another point to the right of . The general equation of a third order polynomial is

 

 

This equation has four unknowns, namely a, b, c and d and therefore four equations are needed to solve. The concepts you need to answer this question includes zeroes of a polynomial (Algebra Download), derivatives and stationary points (Differentiation Download).

 

You can solve this question by seeing the solutions to the given questions which use the same concepts.

 

 


 

Questions which will help you to solve your question.

 

Question 1: What is the value of y when in the third order polynomial sketched below?

 

Answer: The general equation of a third order polynomial is . When  . In other words  since when ,

                                                           (equation 1)

 

Hence .

 

Substituting in y gives:

 

 

What is the value of d in your question?

 


 

Question 2: If the third order polynomial y in question cuts x axis at points -1 and 2, further identify the curve y. (At what points the curve in your question cuts the x-axis?)

 

Answer: At the points where curve cuts the x-axis, y equals to zero; and the points (-1,0) and (2,0) are called the zeroes of the polynomial. A third order polynomial, at most has three zeroes.  and are two of the factors of the given third order polynomial. So when ,  and when , .  If we substitute this information in , we obtain:

That is:

                  ..............................................   (equation 2)

               ..............................................   (equation 3)

Multiplying equation 2 gives:

          ..............................................   (equation)

Adding equations  and 3 gives:

That is:

                              ..............................................   (A)

 

At what points the curve in your question cuts the x-axis?

 


 

Question 3: If the above third order polynomial has a local maximum at, identify the polynomial.

 

 Answer: Local maximum, minimum and point of horizontal reflection are stationary points. At stationary points of curve y, .

 

 

     when      

     ..............................................   (equation 4)

                  ..............................................   (equation 2)

 

Adding equations 2 and 4 gives:

 

That is:

     ..............................................   (B)

We already obtained the equation A in question 2.

                              ..............................................   (A)

Multiplying equation A by 0.8630 gives:

..............................................   ()

Adding equations and B gives:

Substituting b, in equation A gives:

Substituting the values of a and b in equation 2 gives:

                  ..............................................   (equation 2)

Finally, substituting a, b, c in   gives:

 

At what value of x, the curve in your question has a stationary (maximum, minimum or point of inflection)?