Tangent and Secant Lines

Secant Line - m_sec- Average Rate of Change over a period of time.

Tangent Line - m_tan - Instantaneous Rate of Change at any given time during an equation.

Exp:

y = x²

Slope over your given period of time when x = 2 to x = 5

Slope (y₂-y₁)/(x₂-x₁) = (25-4)/(5-2) = 7

The equation of the secant line between 2 and 5 is:

(y-4) = 7(x -2) y=7x-10

To find the instantaneous rate of change easily just take the derivative of the equation.

Original y = xⁿ

Derivative (dy/dx) = (n)x^(n-1)

Original y=x²

Derivative (dy/dx)=2x

So when x= 3 the tangent line is (y-9) = 6(x-3)