Lesson A tricky Calculus problem on derivative and anti-derivative

A tricky Calculus problem on derivative and anti-derivative

Problem 1

We want to find f(x) as a polynomial  f(x) =  +  + . . . + .

Taking derivative decreases the degree of a polynomial by one unit.
Taking antiderivative increases the degree of a polynomial by one unit.


Since the sum   f ' (x) + int f (x) dx   is a polynomial of degree 4,
                ----------------------

it means that the sough polynomial f(x) is of degree 3:

    f(x) = ax^3 + bx^2 + cx + d.


Then 

    f ' (x)     =                         + 2bx + c,

    int f(x) dx =   +  +  + dx  + E.



So, in the sum   f ' (x) + int f(x) dx
                 ----------------------

    (a)  coefficient at    is  , which gives an equation

              = 1;   hence  a = 4.



    (b)  coefficient at    is  0, which gives an equation

              = 0;   hence  b = 0.



    (c)  coefficient at    is  13, which gives an equation

              = 13,  or   = 13  --->  = 13 - 12 = 1  --->  c = 2.



    (d)  coefficient at    is  0, which gives an equation

             2b + d = 0,  which implies  2*0 + d = 0;  hence,  d = 0.



    +------------------------------------------------------------+
    |    At this point, the problem is just solved to the end.   |
    |            a = 4;  b = 0;  c = 2;  d = 0.                  |
    +------------------------------------------------------------+



The sough polynomial is  f(x) = 4x^3 + 2x.     ANSWER



CHECK.  The derivative is f ' (x) = .

        The anti-derivative is  F(x) =   = .

        The sum f ' (x) + F(x) =  +  = .   ! correct !