Question

How do you use the limit definition to find the slope of the tangent line to the graph `F(x) = ((4 (x^2)) – 3x + 7)` at (2, 17)?

Answer 1

The details will depend on which limit definition for the slope of the tangent line you are working with.

Explanation:

The slope of the line tangent to the graph of `f(x)` at the point `(a,f(a))` can be defined as

`lim_(xrarra) (f(x) - f(a))/(x-a)` or by `lim_(hrarr0)(f(a+h)-f(a))/h`.

(Note that different variable names may be used as well.)

For `f(x) = 4x^2-3x+7` and `a=2`, we get

`lim_(xrarr2) (f(x) - f(2))/(x-2) = lim_(xrarr2)([4x^2-3x+7]-[4(2)^2-3(2)+7])/(x-2)`

`= lim_(xrarr2) (4x^2-3x-10)/(x-2)`

If we try to evaluate by subsitution, we get the indeterminate form `0/0`.

This tells us that `2` is a zero of the polynomial numerator. Therefore `x-2` is a factor of the numerator. We'll factor and simplify, then try again to evaluate the limit.

`(4x^2-3x-10)/(x-2) = ((4x+5)(x-2))/(x-2) = 4x+5` `" "` (for `x != 2`)

So, `lim_(xrarr2) (4x^2-3x-10)/(x-2) = lim_(xrarr2)(4x+5) = 13`

The slope of the tangent line in question is `13`.