Simplify the difference quotient
[f(2 + h) − f(2)]/h ; if h ≠ 0.
f(x) = x2 − 3x
I keep getting the answer wrong. I must be messing up my steps...Please Help
f (2 + h ) = ( 2 + h ) ^ 2 - 3 * ( 2 + h )
f ( 2 ) = 2 ^ 2 - 3 * 2 = 4 - 6 = - 2
[ f ( 2 + h ) − f ( 2 ) ] / h =
[ ( 2 + h ) ^ 2 - 3 * ( 2 + h ) - ( - 2 ) ] =
[ 2 ^ 2 + 2 * 2 * h + h ^ 2 - 3 * 2 - 3 * h + 2 ] / h =
[ 4 + 4 h + h ^ 2 - 6 - 3 h + 2 ] / h =
( h ^ 2 + h ) / h = h * ( h + 1 ) / h = h + 1
[ f ( 2 + h ) − f ( 2 ) ] / h =
[ ( 2 + h ) ^ 2 - 3 * ( 2 + h ) - ( - 2 ) ] / h =
[ 2 ^ 2 + 2 * 2 * h + h ^ 2 - 3 * 2 - 3 * h + 2 ] / h =
[ 4 + 4 h + h ^ 2 - 6 - 3 h + 2 ] / h =
( h ^ 2 + h ) / h = h * ( h + 1 ) / h = h + 1
OH! thanks. I just messed up a tiny calculation haha. It's such a messy question..
To simplify the difference quotient for the function f(x) = x^2 - 3x, we need to follow a step-by-step process. Let's break it down with clear explanations for each step:
Step 1: Plug in the given values into the difference quotient formula.
[f(2 + h) − f(2)] / h
Step 2: Replace f(x) with the given function, f(x) = x^2 - 3x.
[(2 + h)^2 - 3(2 + h) - (2^2 - 3(2))] / h
Step 3: Expand the square term (2 + h)^2 and simplify.
[(4 + 4h + h^2) - 6 - 3h - (4 - 6)] / h
Step 4: Combine like terms within each expression.
[(h^2 + h + 4h - 6 - 4 + 6)] / h
Step 5: Further simplify by combining like terms.
[h^2 + 5h - 4] / h
Now, the difference quotient is simplified to (h^2 + 5h - 4) / h.
Make sure to double-check your calculations at each step to avoid any errors.