Solve the difference quotient for the given function
f(x) = x^3, (f(a+h)-f(a))/h
i forgot how to do this problem
Divide [(a+h)^3 - a^3] by h.
Begin by multiplying out the (a+h)^3 term.
(a+h)^3 = a^3 + 3a^2h + 3ah^2 + h^3.
After two more steps,
(f(a+h)-f(a))/h = 3a^2 + 3ah + h^2
As h-> 0, it approaches 3 a^2, the derivative at x=a.
You will need
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
so (f(x+h) - f(x) )/h
= (x^3 + 3x^2h + 3xh^2 + h^3 - x^3)/h
= 3x^2 +3xh + h^2
To solve the difference quotient for the given function f(x) = x^3, we need to find the expression (f(a+h) - f(a))/h.
Step 1: Substitute f(x) = x^3 into the formula (f(a+h) - f(a))/h.
(f(a+h) - f(a))/h = ((a+h)^3 - a^3)/h
Step 2: Expand the numerator using the binomial expansion formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
(f(a+h) - f(a))/h = ((a^3 + 3a^2h + 3ah^2 + h^3) - a^3)/h
Step 3: Simplify the expression by canceling out like terms.
(f(a+h) - f(a))/h = (3a^2h + 3ah^2 + h^3)/h
Step 4: Divide each term by h to simplify further.
(f(a+h) - f(a))/h = 3a^2 + 3ah + h^2
Therefore, the difference quotient for the given function f(x) = x^3 is 3a^2 + 3ah + h^2.