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 = ?
(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.
To find the final answer for (f(a+h) - f(a))/h, let's simplify the expression step by step:
1. Start with the expression (f(a+h) - f(a))/h.
2. Substitute f(a+h) and f(a) using the given function f(x) = x^3.
(f(a+h) - f(a))/h = ((a^3+3a^2h + 3ah^2 + h^3) - a^3)/h.
3. Simplify the numerator:
(a^3+3a^2h + 3ah^2 + h^3) - a^3 = 3a^2h + 3ah^2 + h^3.
4. Rewrite the fraction with the simplified numerator:
((3a^2h + 3ah^2 + h^3) / h.
5. Cancel out h from the numerator and denominator:
(3a^2 + 3ah + h^2).
Therefore, the final answer for (f(a+h) - f(a))/h is 3a^2 + 3ah + h^2.