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 ofchange, which is the instantaneous rate of change at
x=