Functions,
5 graphs,
Functions, graphs, and numbers
5.1 Graphs, slopes, and tangents It is quite possible that we first
heard about tangents, or tangent lines, when we were very young. We
may have been shown a circle as in Figure 5.11 and have been solemnly
told that some lines in the plane of the circle intersect the circle twice,
some others do not intersect the circle at all, and some others, the tan-
gents to the circle, intersect the circle just once. When graphs more
complicated than circles appear, no such simple story can adequately
describe tangents. For example, the line T of Figure 5.12 intersects the
graph twice and seems to be tangent to the graph at Pa, while the line
L intersects the graph only once and does not seem to be tangent to the
graph. To attack this rather delicate matter, we start with a given
function f defined over some interval and draw the graph G of y = f(x) as
in Figure 5.13. We next select an x within the domain of f and call it
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