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.
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)
`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.