The Slope of theTangent Line

Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity and acceleration problem 3. The minimum and maximum problem 4. The area problem Each problem involves the notion of a limit, and calculus can be introduced with any of the four problems. The Tangent Line Problem

First attempt at finding the equation of the line tangent to a curve at a given point

As we have seen graphically, most functions are locally linear. When zoomed in at a point, the function looks like a line, actually, it looks like the line tangent. At the point of tangency, the function and the tangent line share: same value (same point) and the same slope

We have also seen graphically, that if we find use various secant lines, we can approximate (and eventually, find exactly) the slope of the tangent line. So, lets say we want to find the slope of the function at point A. The plan would be: 1) Find the slope of the secant line connecting point A with point B (another point on the function). 2) “Move” point B closer to A and find the slope of the new secant line. 3) Repeat taking notice each time of the results in order to “guess” the slope of the curve. First attempt at finding the equation of the line tangent to a curve at a given point

Finally,

What if we wanted to find the slope of the function at another point?