Question

Explain why the slope of the tangent line can be interpreted as an instantaneous rate of change.
The average rate of change over the interval​ [a, x] is
StartFraction f left parenthesis x right parenthesis minus f left parenthesis a right parenthesis Over x minus a EndFraction ..
StartFraction f left parenthesis x right parenthesis minus f left parenthesis a right parenthesis Over x minus a EndFraction .f(x)−f(a)x−a.
StartFraction f left parenthesis x right parenthesis minus f left parenthesis a right parenthesis Over x plus a EndFraction .f(x)−f(a)x+a.
StartFraction f left parenthesis a right parenthesis minus f left parenthesis x right parenthesis Over x minus a EndFraction .f(a)−f(x)x−a.
StartFraction f left parenthesis x right parenthesis plus f left parenthesis a right parenthesis Over x minus a EndFraction .f(x)+f(a)x−a.
The limit
ModifyingBelow lim With x right arrow minus a StartFraction f left parenthesis x right parenthesis minus f left parenthesis a right parenthesis Over x minus a EndFraction
is the slope of the
​line; it is also the limit of average rates of​change, which is the instantaneous rate of change at
x=

Answer 1

Explanation:

It looks like you're trying to make the coherent statement ...

  The average rate of change over the interval [a, x] is ...

     $\dfrac{f(x)-f(a)}{x-a}$

  The limit ...

     $\lim\limits_{x \to a}{\dfrac{f(x)-f(a)}{x-a}}$

  is the slope of the line. It is also the limit of the average rate of change, which is the instantaneous rate of change at x=a.