Given f(x) = 4x 2 + 2x - 1, the simplified form of the difference quotient, [f(x+h) - f(x)]/h is, what?
in the limit as h>zero, it is 8x+2.
To find the difference quotient, we need to evaluate the expression [f(x+h) - f(x)]/h.
Given f(x) = 4x^2 + 2x - 1, let's first find f(x+h).
Substitute (x+h) into the function:
f(x+h) = 4(x+h)^2 + 2(x+h) - 1
Next, let's find f(x) by substituting x into the function:
f(x) = 4x^2 + 2x - 1
Now we can substitute these expressions back into the difference quotient formula:
[f(x+h) - f(x)]/h = [4(x+h)^2 + 2(x+h) - 1 - (4x^2 + 2x - 1)]/h
Simplify the expression inside the brackets:
[f(x+h) - f(x)]/h = [(4x^2 + 8xh + 4h^2 + 2x + 2h - 1) - (4x^2 + 2x - 1)]/h
Combine like terms within the brackets:
[f(x+h) - f(x)]/h = [(8xh + 4h^2 + 2h)]/h
Now, factor out an h from the numerator:
[f(x+h) - f(x)]/h = [h(8x + 4h + 2)]/h
Cancel out the common factors of h:
[f(x+h) - f(x)]/h = 8x + 4h + 2
So, the simplified form of the difference quotient is 8x + 4h + 2.