Question

Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. f(x)

Answer 1

`(f(x + \triangle x) - f(x))/(\triangle x) = 3x^2+ 3x \cdot \triangle x + (\triangle x)^2`

Explanation:

Given

`f(x) = x^3`

Required

Evaluate

`(f(x + \triangle x) - f(x))/(\triangle x)`

`(f(x + \triangle x) - f(x))/(\triangle x)` becomes

`(f(x + \triangle x) - f(x))/(\triangle x) = ((x + \triangle x)^3 - x^3)/(\triangle x)`

Expand

`(f(x + \triangle x) - f(x))/(\triangle x) = (x^3 + 3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 - x^3)/(\triangle x)`

Collect like terms

`(f(x + \triangle x) - f(x))/(\triangle x) = (x^3 - x^3+ 3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 )/(\triangle x)`

`(f(x + \triangle x) - f(x))/(\triangle x) = (3x^2 \cdot \triangle x+ 3x \cdot (\triangle x)^2 + (\triangle x)^3 )/(\triangle x)`

Factorize

`(f(x + \triangle x) - f(x))/(\triangle x) = (\triangle x(3x^2+ 3x \cdot \triangle x + (\triangle x)^2) )/(\triangle x)`

`(f(x + \triangle x) - f(x))/(\triangle x) = 3x^2+ 3x \cdot \triangle x + (\triangle x)^2`