Slopes of Secants and Tangents

Slope

Slope is defined as rise over run or \(\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\). It measures how quickly the dependent variable changes with respect to the independent variable.

Rates of Change

Average rate of change: the rate of change (slope) between two points (an interval). This is the secant line.
Instantaneous rate of change: the rate of change at a specific instant. This is the tangent line.

The rate of change can be calculated for both functions and relations.

Slope of Secant to Slope of Tangent

When finding the tangent at point P, we can make a point Q on the function. As \(Q \to P\), the slope of the secant \(\to\) the slope of the tangent.

Sandwich Theorem

Calculate what the slope of \(\overline{PQ}\) as Q approaches P from the negative side and as Q approaches P from the positive side. Use what the secant lines are approaching towards to find the tangent at P.