Find the difference quotient and simplify your answer. Step by step. f(x) = 3x2 − 6x, f(x + h) − f(x) h , h �‚ 0
f(x)=3x^2-6x
f(x+h)=3(x+h)^2 -6(x+h)
= 3x^2+6xh+3h^2-6x-6h
so f(x+h)-f(x)= 0x^2-0x+6xh-3h^2-6h
dividing that by h then
6x-3h-6
as h approaches zero, then the difference quotent= 6x-6
To find the difference quotient for a function f(x), we need to evaluate the expression [f(x + h) - f(x)] / h.
Given that f(x) = 3x^2 - 6x, let's start by substituting f(x + h) and f(x) into the formula:
[f(x + h) - f(x)] / h = [3(x + h)^2 - 6(x + h) - (3x^2 - 6x)] / h
Now, let's simplify this expression step by step.
1. Expand (x + h)^2 by using the formula (a + b)^2 = a^2 + 2ab + b^2:
[3(x^2 + 2hx + h^2) - 6(x + h) - (3x^2 - 6x)] / h
2. Distribute the terms inside the brackets:
[3x^2 + 6hx + 3h^2 - 6x - 6h - 3x^2 + 6x] / h
3. Combine like terms:
[6hx + 3h^2 - 6h] / h
4. Factor out h from the numerator:
h(6x + 3h - 6) / h
5. Cancel out the h terms:
6x + 3h - 6
So, the simplified difference quotient for f(x) = 3x^2 - 6x is 6x + 3h - 6.
To find the difference quotient, we need to calculate:
f(x + h) - f(x) / h
where f(x) = 3x^2 - 6x.
Step 1: Replace x in the given equation with x + h:
f(x + h) = 3(x + h)^2 - 6(x + h)
Step 2: Expand the equation:
f(x + h) = 3(x^2 + 2hx + h^2) - 6(x + h)
= 3x^2 + 6hx + 3h^2 - 6x - 6h
Step 3: Calculate the difference:
f(x + h) - f(x) = [3x^2 + 6hx + 3h^2 - 6x - 6h] - [3x^2 - 6x]
= 3x^2 + 6hx + 3h^2 - 6x - 6h - 3x^2 + 6x
= 6hx + 3h^2 - 6h
Step 4: Divide the difference by h:
(f(x + h) - f(x)) / h = (6hx + 3h^2 - 6h) / h
Step 5: Simplify the expression by canceling out common factors:
= (h(6x + 3h - 6)) / h
= 6x + 3h - 6
Therefore, the simplified difference quotient is 6x + 3h - 6.