THE MEAN VALUE THEOREM

Let's review some basic concepts of Geometry:

The Mean Value Theorem is an important idea, not just in Calculus, but in every facet of mathematics. In basic terms, the Mean Value Theorem states:

Given ANY interval on a smooth curve, there will exist a point at which the slope of the tangent line will equal the slope of the segment connecting the endpoints of the interval

Let's look at the graph of the function

The points (-1,1) and (3,9) are connected on the above graph. The slope of the segment calculates out to [9 - 1] / [3 - (-1)] = 2

Line up a straightedge with this line [the secant line] and move it toward the graph until you hit a point on the curve. If you've pulled down the edge in a parallel manner, you will hit the point where the tangent line will be parallel to the secant line.

What are the approximate coordinates of this point?

Determine two other points on this graph - calculate the slope and use a straightedge to determine where the tangent line is parallel.

QUESTIONS TO CONSIDER

1. On the graph above, plot points (-2,1) and (6,9). Determine the approximate coordinates of the point at which the tangent line is parallel.
   
   
2. Will the Mean Value Theorem work if you connected (-4,4) and (4,4)? Explain

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