Question

find the function whose first difference is 9x^2+11x+5​

Answer 1

Answer:

The function of first difference is f(x) = $3x^{2} + x^{2} +x + C$

Step-by-step explanation:

Given: The given function f(x) =$9x^{2} + 11x +5$

To find: The first difference of the given function

Solution

Let f(x) be the required function then ∆f(x) = $9x^{2} + 11x +5$

Now we have to change ∆f(x) in factorial notation.

f(x) = $9x^{2} + 11x +5$ = $Ax^{2} + Bx^{1} + C$

    = Ax(x-1) + Bx +C

Putting  x = 0 then we get

C = 5

If we sub x = 1 then we get,

$9x^{2} + 11x +5$  = B +C

25 = B +C

25 - 20 =B

B = 20

On comparing  we get the value of A = 9

∆f(x) = $9x^{2} + 11x^{1} +5$

Integrating we get

f(x) = $\frac{9x^{3} }{3} + \frac{20x^{2} }{2} + 5x +C_{1}$

Where $C_{1}$ is the constant of integration.

f(x) = $\frac{9x^{3} }{3} + \frac{20x^{2} }{2} + 5x +C_{1}$

    = 3x(x-1) (x-2) + 10x (x-1) + 5x + $C_{1}$

f(x) = $3x^{2} + x^{2} +x + C$

Final answer:

The function of first difference is f(x) = $3x^{2} + x^{2} +x + C$

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Answer 2

Answer:

The function whose first difference is $9x^2+11x+5$ is $f(x) = 3x^{3} + x2 + x + C$.

Step-by-step explanation:

Given:

First difference is $9x^2+11x+5$.

To Find:

The function whose first difference is $9x^2+11x+5$.

Solution:

Let f(x) be the required function then $\delta f(x) = 9x2 + 11x + 5$.

First, we change $\delta f(x)$ in factorial notation.

Let $\delta f(x) = 9x^{2} + 11x + 5=Ax^{(2)} +Bx^{(1)} +C$

             $= Ax(x -1) + Bx + C$  ------ equation no.01.

Putting x = 0 we get $C = 5.$

Putting x = 1 we get $9 + 11 + 5 = B + C$

                                 $B+C=25$

                                $B+5=25\\B=20$        

                                                           

On comparing like term in equation no.01. we get A = 9.

On putting in equation no.01. we get.

$\delta f(x) = 9x^{2} + 20x + 5$

On Integrating, we get.

$f(x) = \frac{9x^{(3)} }{3} + \frac{20x^{(2)} }{2} + 5x+C_{1}$

                       where C1 is constant of Integration.

       $= 3x(x -1) (x- 2) + 10x(x -1) + 5x + C1$

        $f(x) = 3x^{3} + x2 + x + C$

    Thus,the function whose first difference is $9x^2+11x+5$ is $f(x) = 3x^{3} + x2 + x + C$.

#SPJ2