Difference quotient problem

Question

Suppose f(x) = x^2 + x − 1, simplify the following:

(a) f(x + h) =

(b) f(x + h) − f(x) =

(c) Compute the difference quotient f(x + h) − f(x) h . Simplify until the h in the denominator is canceled.

Answer 1

To evaluate f(x+h), we have to replace each instance of x by x+h, that means

(a)f(x+h) = (x+h)^2 + (x+h) -1 = x^2 + h^2 + 2xh + x + h - 1

Now we can compute the difference as

(b)f(x+h) - f(x) = [ x^2 + h^2 + 2xh + x + h - 1] - x^2 - x + 1 = h^2 + 2xh + h = h(h+2x+1)

So now we can compute the difference quotient

(c)[f(x+h) - f(x)]/h = [ h(h+2x+1)]/h = h+2x+1