While limit is what makes all of the calculus work, people usually think of calculus as starting with the derivative. The first problem in calculus is finding the slope of a line tangent to a graph at a point and then writing the equation of that tangent line.
Local Linearity is the graphical manifestation of differentiability. If you zoom-in of the graph of a function (at a point where we will soon say the function is differentiable), the graph eventually looks like a line: the graph appears to be straight, and its slope is the number we will call its derivative.
To do this we need to zoom-in numerically. Zooming-in numerically is accomplished by finding the slope of a secant line, a line that intersects the graph twice, and then finding the limit of that slope as the two points come closer together.
This week’s posts start with local linearity and tangent lines. They lead to the difference quotient and the equation of the tangent line.
Local Linearity II Working up to difference quotient. The next post explains this in more detail.
Tangent Lines approaching difference quotients on calculator by graphing tan line.
Next week: Difference Quotients.
Teaching Calculus 