Question

Let f(x)=-2+6 x+2 x². If h ≠ 0, then the difference quotient can be simplified as

f(x+h)-f(x)/h=A h+B x+C,
where A, B, and C are constants.

Answer 1

The simplified difference quotient for the function f(x) = -2 + 6x + 2x^2 is Ah + Bx + C, where A = 2, B = 4, and C = 6.

To find the difference quotient for the given function f(x) = -2 + 6x + 2x^2, compute (f(x+h) - f(x))/h:

Find f(x + h):

f(x + h) = -2 + 6(x + h) + 2(x + h)^2

Simplify f(x + h) - f(x):

f(x + h) - f(x) = 6h + 2(x + h)^2 - 2x^2

Divide by h:

(f(x + h) - f(x))/h = (6h + 2(x + h)^2 - 2x^2)/h

Simplify further:

= 6 + 2[(x + h)^2]/h - 2(x^2)/h

Expand [(x + h)^2]/h:

= 6 + 2[(x^2)/h + 2x + h] - 2(x^2)/h

Combine like terms:

= 4x + 2h + 6

So, the difference quotient is Ah + Bx + C, where A = 2, B = 4, and C = 6.