EggMath: Embryo Calculus
Tangents and Slopes
At each point on a graph there are many straight
lines passing through that point. Most of them
cut right through the graph, but one special line
just brushes the graph. This one line is said to
be tangent to the graph at that point.
Click twice on the graph to draw a line that
intersects the graph at the chosen points. (This
is called a secant line for the graph.) As
you bring the two points together, the secant line
gets closer and closer to a tangent line,
which cuts the graph only at one point.
One way to measure the direction of a line is
to measure the angle it makes with the horizontal.
It's often more useful to use the tangent
of this angle, which is defined using a little
right triangle with its hypotenuse lying on the
line, as in the diagram. The tangent is the ratio
of the vertical to the horizontal side, and this
is called the slope of the line.
The diagram shows the slope m of a
tangent line to the graph of an exponential