Question

Question #93377

Answer 1

The phrase "alternative form" is not universal in mathematics. I will assume that you are referring to the Khan Academy terminology.

Explanation:

We need to find `lim_(xrarr2) (g(x) - g(2))/(x-2)`

`lim_(xrarr2) (g(x) - g(2))/(x-2) = lim_(xrarr2) ([x^3-6x] - [(2)^3-6(2)])/(x-2)`

`= lim_(xrarr2) (x^3-6x+4)/(x-2)`

Note that, if we try to evaluate the limit by substitution, we get the indeterminate form `0/0`.

Since `2` is a zero of the polynomial numerator, we can be sure that `x-2` is a factor. So we can simplify the ratio and try again.

Use polynomial division or trial and error to get

`x^3-6x+4 = (x-2)(x^2+2x-2)`.

Continuing:

`= = lim_(xrarr2) ((x-2)(x^2+2x-2))/(x-2)`

`= = lim_(xrarr2) (x^2+2x-2)`

`= 4+4-2 = 6`

The slope of the line tangent to the graph of `g(x) = x^3-6x` at the point where `x=2` is `6`.

Terminology

A textbook discussion (or other presentation) of the derivative (the slope of the tangent line) has some choices to make. The choices are limited, but not all instructors make the same choices.

For instance, we can define the slope of the line tangent to the graph of `y = f(x)` at the point where `x=a` by either

`lim_(xrarra)(f(x)-f(a))/(x-a)` `" "` OR `" "` `lim_(hrarr0)(f(a+h)-f(a))/h`

(OR by defining the derivative at `x` first and then later discussing tangent lines).

Whichever choice is made, that becomes the official definition fro that textbook (or or class or whatever presentation) and the other is the "alternative" of "equivalent" or "Theorem 2.1" or "Equation 3.4" or whatever.