Difference Quotient of a Function

Question

Find the difference quotient for the following function

f(x) = 7x^2-3x-2

Answer 1

f(x) = 7x^2 -3x -2

f'(x) = 14x -3 = the derivative of f(x)

the difference quotient, when h=0 also equals 14x -3

f(x+h) = 7(x+h)^2 -3(x+h) -2 = 7x^2 + 14xh + 7h^2 - 3x -3h -2 -7x^2 +3x +2

f(x+h) - f(x) = 14xh -3h + 7h^2

[f(x+h) -f(x)]/h = (14xh -3h +7h^2)/h = 14x -3 +7h which would equal 14x-3 if h=0

Answer 2

The difference quotient is really just the same as the slope formula where m = (y₂ - y₁)/(x₂ - x₁) but it is written in terms of the point (x, y) which written in function nomenclature is (x, f(x)), and the point that is "h" more than "x" or ((x + h), f(x + h)). Putting these together, we get:

or, simplified:

In this case:

f(x + h) = 7(x + h)² - 3(x + h) - 2

f(x + h) = 7(x² + 2xh + h²) - 3(x + h) - 2

f(x + h) = 7x² + 14xh + 7h² - 3x - 3h - 2

So:

f(x + h) - f(x) = (7x² + 14xh + 7h² - 3x - 3h - 2) - (7x² - 3x - 2)

f(x + h) - f(x) = 7x² + 14xh + 7h² - 3x - 3h - 2 - 7x² + 3x + 2

f(x + h) - f(x) = 14xh + 7h² - 3h

f(x + h) - f(x) = h(14x + 7h - 3)

And:

[f(x + h) - f(x)]/h = h(14x + 7h - 3)/h

[f(x + h) - f(x)]/h = 14x + 7h - 3

Answer 3