Applied Calculus Math 215 - University of Hawaii

40 CHAPTER 2. THE DERIVATIVE
out, only one line can be placed in between two parabolas as in the picture,
and this line is the tangent line to the graph of f(x)at(x0,f(x0)). The slope
of the line l(x) ise, so that the derivative of f(x) =e x at x =1isf ′ (1) = e.
We will talk more about this geometric interpretation in Section 2.5.
Exercise 31. Use DfW (or any other accurate tool) to graph f(x) =lnx,
l(x)=x−1, p(x) =2(x−1) 2 + x − 1andq(x)=−2(x − 1) 2 + x − 1onthe
interval [.5, 1.5].
Second Example
Before we discuss the second example, let us think more about the tangent
line. What is its geometric interpretation? Which line looks most like the
graph of a function f near a point x. Sometimes (though not always) you
can take a ruler and hold it against the graph. The edge of the ruler on the
side of the graph gives you the tangent line. You find a line l which has the
same value at x as f (f(x) =l(x)), and the line does not cross the graph of
f (near x the graph of f is on one side of the line). This rather practical
recipe for finding the tangent line of a differentiable function works for all
functions in these notes at almost all points, see Remark 18 on page 164. It
works in the previous example as well as in the one we are about to discuss.
For x ∈ (−1, 1) we define the function
(2.1)
f(x) =y= 1−x 2 .
We like to use practical reasoning and a little bit of analytic geometry to
show that
f ′ (x) = −x
√
1−x2 .
(2.2)
The function describes the upper hemisphere of a circle of radius 1 centered
at the origin of the Cartesian coordinate system. To see this, square
the equation and write it in the form x 2 + y 2 = 1, which is the equation of
the circle. Thus we are saying that the slope of the tangent line to the circle
at a point (x, √ 1 − x 2 ) in the upper hemisphere is −x/ √ 1 − x 2 .Thecircle
and the tangent line are shown in Figure 2.4.
What is the slope of the tangent line to the circle at a point (x, y)?
Your intuition is correct if you say that it is perpendicular to the radial line
through the point (0, 0), the origin of the Cartesian plane, and the point