Do you agree or disagree with the statement "the marginal rate of change of the function f(x)= x5  at the point (2,32) is 80”? Please show working to support answer.

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This is related to derivatives.

`y = x^5 ` ----(1)

Let us say for for marginal change in `deltax` the change in y is `deltay `

`y+deltay = (x+deltax)^5` ----(2)

 

(2)-(1)

`deltay = (x+deltax)^5-x^5`

 

Rate of change means `deltay/(deltax)`

`(deltay)/deltax = [(x+deltax)^5-x^5]/(deltax)`

 

When `(deltay)/(deltax) ` be a small marginal change;

`lim_(xrarr0)(deltay)/deltax = lim_(xrarr0)[(x+deltax)^5-x^5]/(deltax)`

 

`lim_(xrarr0)(deltay)/(deltax)` referred as the derivative of a function.

Therefore rate of marginal change can be expressed as the derivative.

 

`(dy)/dx = 5x^4`

Rate of change at x = 2 and y = 32;

`((dy)/dx)_(x=2) = 5*2^4 = 80`

 

So the statement is correct and we should agree with that.

 

 

 

 

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