Secant Lines Approximate a Tangent Line
David W. Lyons
lyons@lvc.edu
Click in the picture box below to begin.
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Motivating Problem
An equation for a function y = f(x)
expresses a relationship between two variable quantities x and
y. A picture of the graph gives a visual representation of that
relationship. To analyze a function or its graph, it can be useful to
replace the difficulties of a complicated equation and a complicated
graph with a simpler equation and a simpler graph. One might hope that
the simpler object would be easier to understand, yet also be a
reasonable approximation to the more complicated curve. But how do you
find such a simplification?
Observation
A tangent line to a curve can be thought of as a close
approximation to the curve in the vicinity of the point of tangency.
Equations and graphs of lines are easy to use. It will turn out that
approximating a curve using its tangent line (near the point of
tangency) is a very fruitful strategy. This leads to a geometric
problem.
Problem (refinement of the motivating problem above)
Given a curve which is a piece of the graph of a function, and given
a point on the curve, how can you find the line tangent to the curve at
the given point?
Illustration of a Solution
The picture in the box above illustrates a method for solving the
problem of finding a tangent line.
The blue curve is part of the graph of a function f, and point
P is the point at which we wish to find a tangent line.
The idea is to choose another point Q on the graph. It is then a
simple matter to connect P and Q with a secant
line. When Q is close to P, the secant line should be
close to the tangent line. By allowing Q to approach P
and "taking a limit," we obtain the desired tangent line. The procedure
of "taking a limit" is at the heart of calculus. You can get a visual
idea of the process by moving the point Q closer and closer to
the point P.
The green numbers "x coord" and "y coord" give the location for
Q. The blue numbers "dx" and "dy" give the horizontal and
vertical distances from P to Q, repsectively. The magenta
number "dy/dx" is the slope of the secant line through P and
Q.
By moving the mouse (click or click and drag), you can reposition the
vertical green line. By moving the vertical green line closer to the
red line, the point Q moves closer to the basepoint P, and
the slope dy/dx of the secant line becomes a better approximation of the
slope of the line tangent to the blue curve at P.