Applied Calculus Math 215 - University of Hawaii

42 CHAPTER 2. THE DERIVATIVE
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Figure 2.5: The graph of sin x
Exercise 33. Use your graphing calculator, DfW, or any other means to
get the graph of f(x) =e x near x =1/2. You can also use one of the graphs
of the exponential functions from these notes. Use the graph to estimate
f ′ (1/2), the slope of the tangent line at this point.
2.1 Definition of the Derivative
In both of the previous examples we were able to suggest the tangent line
to the graph of a function at a point. In the first example we also discussed
the idea of the tangent line being close to the graph. We discussed the idea
numerically and in terms of a picture. The graph and the tangent line were
squeezed between two parabolas. In our upcoming definition of differentiability,
of the tangent line, and the derivative we will express ‘closeness’
analytically. We will use the absolute value. If you are not familiar with it,
then you may want to read about it in Section 5.2. But, for the moment it
suffices that you know that for any two real numbers a and b the absolute
value of their difference, i.e., |a−b|, is the distance between these two points.
Let us formalize the idea of an interior point.
Definition 2.1. Let D be a subset of the real line, and x0 an element in
D. We say that x0 is an interior point of D if D contains an open interval