For the function f(x) = x^3, determine ((f(a+h) - f(a))/h
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The function f(x) = x^3.
To determine f(a+h) substitute x with a+h.
f(a+h) = (a+h)^3
= a^3 +3*a^2*h +3*a*h^2+ h^3
f(a) = a^3
`(f(a+h) - f(a))/h`
= `(a^3+3*a^2*h +3*a*h^2+ h^3 - a^3)/h`
= `(3*a^2*h +3*a*h^2+ h^3)/h`
= `3*a^2 +3*a*h+ h^2`
The simplified form of `(f(a+h) - f(a))/h` for `f(x) = x^3` is `3*a^2 +3*a*h+ h^2`
Which equals `3a^2` since` ` h = 0.
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Given `f(x)=x^3 ` .
To determine `(f(a+h)-f(a))/h ` :
put x=(a+h) in f(x) to obtain
`f(a+h)=(a+h)^3 = a^3 + h^3 + 3*a^2*h + 3*a*h^2 `
Therefore, `(f(a+h)-f(a))/h = ((a^3 + h^3 + 3*a^2*h + 3*a*h^2) - (a^3))/h`
`= (h^3 + 3*a^2*h + 3*a*h^2)/h`
`= h^2 + 3*a^2 + 3*a*h `
and when `h->0` , `(f(a+h) - f(a))/h -> f'(x) = 3*a^2 `
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