(f(a+h) - f(a))/h = ((a^3+3a^2h + 3ah^2 + h^3) - a^3)/h
Using the distributive property and combining like terms in the numerator, we have:
(f(a+h) - f(a))/h = ((a^3 + 3a^2h + 3ah^2 + h^3) - a^3)/h
= (3a^2h + 3ah^2 + h^3)/h
Now, we can cancel out the h in the numerator and denominator:
(f(a+h) - f(a))/h = 3a^2 + 3ah + h^2
Therefore, the final answer is (f(a+h) - f(a))/h = 3a^2 + 3ah + h^2.
for any function y = f(x), the difference quotient if defined as (f(a+h) - f(a))/h. our function if f(x) = x^3. therefore, f(a) = a^3. similarly, f(a+h) = (a+h)^3 = (a+h)(a+h)(a+h) = a^3(3)a^2h + 3ah^2 + h^3. Substituting these into the difference quotient, we have (f(a+h) - f(a)/h = ((a+h)^3 = a^3))/h = ((a^3+3a^2h + 3ah^2 + h^3) - a^3)/h. subtracting a^3 and canceling h from the numerator and denominator gives us the final answer (f(a+h)-f(a))/h = ?
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