given the function f(x) = 2x^2-3x-1 simplify f((x+h)-f(x))/h
f(x) = 2x^2-3x-1
f(x+h) = 2(x+h)^2 - 3(x+h) - 1
= 2(x^2 + 2xh + h^2) - 3x - 3h - 1
= 2x^2 + 4xh + 2h^2 - 3x - 3h - 1
then : f((x+h)-f(x))/h
= ( 2x^2 + 4xh + 2h^2 - 3x - 3h - 1 - (2x^2-3x-1) )
= ( 4xh + 2h^2 - 3h)/h
= 4x - 3, h ≠ 0
( 4xh + 2h^2 - 3h)/h = 4x-3+2h
Well, well, well, it looks like we have a math problem to solve! Don't worry, I'm here to add a little humor to make it more bearable.
Let's break this down one step at a time, shall we?
First, let's find f(x+h):
f(x+h) = 2(x+h)^2 - 3(x+h) - 1
Now, let's find f(x):
f(x) = 2x^2 - 3x - 1
Alright, now we can substitute these expressions into the given formula:
[f(x+h) - f(x)]/h = [(2(x+h)^2 - 3(x+h) - 1) - (2x^2 - 3x - 1)]/h
Now, we can expand the expressions and simplify:
= [(2(x^2 + 2xh + h^2) - 3x - 3h - 1) - (2x^2 - 3x - 1)]/h
Time to simplify further:
= (2x^2 + 4xh + 2h^2 - 3x - 3h - 1 - 2x^2 + 3x + 1)/h
Some terms start canceling out:
= (4xh + 2h^2 - 3h)/h
Now let's factor out that lovely h from the numerator:
= h(4x + 2h - 3)/h
Finally, h cancels out:
= 4x + 2h - 3
And there you have it! The simplified expression is 4x + 2h - 3.
Remember, math doesn't always have to be boring. Anytime you need a laugh or a joke, I'm here to brighten your day!
To simplify the expression f((x+h)-f(x))/h, let's break it down step by step:
First, let's find f(x+h):
f(x+h) = 2(x+h)^2 - 3(x+h) - 1
Expanding this using the binomial expansion, we get:
f(x+h) = 2(x^2 + 2xh + h^2) - 3(x+h) - 1
= 2x^2 + 4xh + 2h^2 - 3x - 3h - 1
Next, let's find f(x):
f(x) = 2x^2 - 3x - 1
Substituting these values into the expression, we have:
f((x+h)-f(x))/h = (2x^2 + 4xh + 2h^2 - 3x - 3h - 1 - (2x^2 - 3x - 1))/h
Simplifying further:
f((x+h)-f(x))/h = (2x^2 + 4xh + 2h^2 - 3x - 3h - 1 - 2x^2 + 3x + 1)/h
= (4xh + 2h^2 - 3h)/h
= 4x + 2h - 3
So, the simplified expression is 4x + 2h - 3.