Question

Some one can help me to solve this question? find the slope of curve at a point by using `(f(x+h)-f(h))/h`

Answer 1

`"slope" = -9`

Explanation:

In this context, you have, with the essential limit idea added:

`"slope" = (f(x + h) - f(x))/(h) |_(color(red)(h to 0))`

`= (((x+h)^3 -12 (x+h)) -( x^3 -12 x))/(h) |_(h to 0)`

Expand the cubic term:

`=(((x^3 + 3 x^2 h + 3 xh^2 + h^3) -12 (x+h)) - x^3 +12 x)/(h) |_(h to 0)`

`=( 3 x^2 h + 3 x h^2 + h^3 - 12 h )/(h) |_(h to 0)`

`= 3 x^2 + 3 x h + h^2 -12 |_(h to 0)`

`= 3 x^2 -12 + h ( 3 x + h) |_(h to 0) = 3x^2 - 12`

It follows that the slope evaluated at `x = 1` in this way is:

`("slope" = (f(x + h) - f(x))/(h) |_(h to 0))\_(x = 1) = 3(1)^2 - 12 = -9`