simplify (f(x+h)-f(h))/h where f(x)=2x^2+x.
f(x)=2x^2+x
f(x+h) = 2(x+h)^2 + x+h
= 2x^2 + 4hx + 2h^2 + x + h
then (f(x+h)-f(h))/h
= (2x^2 + 4hx + 2h^2 + x + h - (2x^2+x) )/h
= (4hx + 2h^2 + h)/h
= h(4x + 2h + 1)/h
= 4x + 2h + 1 , h ≠ 0
In your next step you will be expected to take the limit of that
expression as h ---> 0
and leading up to derivatives. Fun times ahead! Enjoy the mathemagics.
To simplify the expression (f(x+h)-f(h))/h where f(x) = 2x^2+x, we need to substitute the given function into the expression and simplify.
1. Start by substituting f(x) = 2x^2 + x into the expression:
(2(x+h)^2 + (x+h) - (2h^2 + h)) / h
2. Expand and simplify the numerator:
(2(x^2 + 2hx + h^2) + (x + h) - (2h^2 + h)) / h
(2x^2 + 4hx + 2h^2 + x + h - 2h^2 - h) / h
(2x^2 + 4hx + x + h) / h
3. Divide each term in the numerator by h:
2x^2/h + 4hx/h + x/h + h/h
2x^2/h + 4x + x/h + 1
Thus, the simplified expression is:
2x^2/h + 4x + x/h + 1
To simplify the expression (f(x+h) - f(h))/h, we can start by substituting the given function f(x) = 2x^2 + x into the expression.
First, let's find f(x+h).
Replace x in f(x) with (x+h):
f(x+h) = 2(x+h)^2 + (x+h)
= 2(x^2 + 2xh + h^2) + x + h
= 2x^2 + 4xh + 2h^2 + x + h
Next, substitute f(x) and f(x+h) back into the original expression:
(f(x+h) - f(h))/h
= (2x^2 + 4xh + 2h^2 + x + h - (2h^2 + h))/h
= (2x^2 + 4xh + x + h)/h
Now, let's break down the expression further:
= 2x^2/h + 4xh/h + x/h + h/h
= 2x^2/h + 4x + x/h + 1
Therefore, the simplified expression is 2x^2/h + 4x + x/h + 1.