The Tangent Line Problem

        The tangent line problem is widely considered to be the instigating idea behind the derivative. Computing the slope of a tangent line was a problem that the French mathematician Pierre de Fermat developed. Picking up on these ideas were Isaac Newton and Gottfried Liebniz, who then developed differential calculus.

What is a tangent line? For a general curve it is not easy to define what is meant by a tangent line; for example a tangent line might mean a line touches  the curve only once, but this does not work in all cases. Here are some examples of tangent lines:

Example (Secant and Tangent Lines) Let In this problem we will calculate a series of functions whose graphs are secant lines to the graph of and use them to infer the equation of the tangent line at the point

(a) Find the equation of the line that passes through the points and and sketch the graph of both and the secant line.

    Solution. The slope formula yields the slope of the line as follows:
    
         .

Using the formula with and we can easily find as follows:



Therefore the equation of the line is The secant line and the graph of are illustrated:



(b) Do the same as in part (a) for the points and

    Solution. The slope formula yields the slope of the line as follows:
    
         .

Using the formula with and we can easily find as follows:



Therefore the equation of the line is The secant line and the graph of are illustrated:



(c) In part (a), and in part (b), Fill in the table:



(d) From this table what would you say is the equation for the tangent line of the function at   Explain your conclusion.

    Solution. We can approximate the tangent line from this table because we see and as (if you are not sure of this, then work out some more rows in the table).  Therefore we infer the tangent line is

(e) Using the notation used in this example, label the points and distances in the following diagram.
    

    Solution.  The two points on the graph are and . Since , the horizontal distance between and is The vertical distance between and is ; the vertical distance between and is .

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Cite this as:
Tangent Line Problem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/tangent-line-problem.html