Simplify the difference quotient
[f(2 + h) − f(2)]/h if h ≠ 0.
f(x) = x^2 − 3x
I keep getting the answer wrong...please help.
[f(2 + h) − f(2)]/h if h ≠ 0
if f(x) = x^2 - 3x
f(2+h) = 4+4h+h^2 - 3(2+h)
= h^2 + h - 2
f(2) = 4-6 = -2
[f(2 + h) − f(2)]/h if h ≠ 0.
= (h^2 + h - 2 - (-2))/h
= (h^2 + h)/h
= h + 1
To simplify the difference quotient, you need to find f(2 + h) and f(2), and then subtract f(2) from f(2 + h) and divide the result by h. Let's go through the steps:
1. Start by finding f(2 + h):
Substitute (2 + h) into the function f(x):
f(2 + h) = (2 + h)^2 - 3(2 + h)
Simplify inside the parentheses: (2 + h)^2 = (2 + h)(2 + h) = 4 + 2h + 2h + h^2 = 4 + 4h + h^2
f(2 + h) = 4 + 4h + h^2 - 6 - 3h
Combine like terms: f(2 + h) = h^2 + h - 2
2. Next, find f(2):
Substitute 2 into the function f(x):
f(2) = 2^2 - 3(2)
Simplify: f(2) = 4 - 6
f(2) = -2
3. Now, subtract f(2) from f(2 + h):
f(2 + h) - f(2) = (h^2 + h - 2) - (-2)
Simplify: f(2 + h) - f(2) = h^2 + h - 2 + 2
f(2 + h) - f(2) = h^2 + h
4. Finally, divide the above result by h:
[f(2 + h) - f(2)]/h = (h^2 + h)/h
Cancel out the h in the numerator and denominator: [f(2 + h) - f(2)]/h = h + 1
So, the simplified difference quotient is h + 1.
Make sure to double-check your work and calculations to avoid any errors.