First find the secant line. The slope of the secant line through [tex](-1,1)[/tex] (when [tex]x=-1[/tex]) and [tex](2,4)[/tex] (when [tex]x=2[/tex]) is the average rate of change of [tex]y=x^2[/tex] over the interval [tex][-1,2][/tex]:
[tex]\text{slope}_{\text{secant}}=\dfrac{2^2-(-1)^2}{2-(-1)}=\dfrac33=1[/tex]
The tangent line to [tex]y=x^2[/tex] will have a slope determined by the derivative:
[tex]y=x^2\implies y'=2x[/tex]
Both the secant and tangent will have the same slope when [tex]2x=1[/tex], or when [tex]x=\dfrac12[/tex].
