The three curves (black, red, green) shown at right illustrate
three very different ways to travel from point $\,A\,$ to
point $\,B\,$. However, the average rate of change between $\,A\,$ and $\,B\,$ is precisely the same for all three curves. This is because average rate of change only uses information at the ends of the interval of interest—it is the slope of the line between the two points. How, then, can we capture what is happening between $\,A\,$ and $\,B\,$? Instantaneous rate of change to the rescue! 

Imagine walking along a curve, moving from left to right. At each point, the tangent line (when it exists) gives information about the ‘direction’ you're traveling at the instant you pass through that point. For example (at right):
Tangent Lines Don't Always Exist
Tangent lines don't always exist.
Mathematics Cannot Rely on Intuition
Most people have very good intuition about how to draw tangent lines (when they exist).
Here's the big problem with tangent lines: Calculus solves these problems using the concept of limits. Here's the idea, in a nutshell.


On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
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