Definition of The Tangent Line
Great!
The tangent line to
f(x) at A is plotted below. Move A and observe that the tangent line moves as A moves.
Type this code in to the input bar to keep track of the slope of the tangent lines--and therefore also the growth rates of f(x)--as we move A along f(x):
(x(A),slope(g))Quick Check: What's the slope of the tangent line to f when A is set to (0,0)?
Just to be clear, let's define exactly what a tangent line is:
Definition: The tangent line of a function at a point on the function is defined as the line through the point with slope equal to the limit of the secant lines through the point as
h goes to 0.
This language is meant to capture precisely what we have seen in the previous activities, but it definitely is a little clunky. To help you digest it, here's a slightly simpler but less precise alternate definition.
Simpler/Less Precise Alternate Definition: The tangent line of a function at a point is the line through the point with slope that most closely matches the function at that point.
Either way you define the tangent line, when you zoom way in on a tangent line to a function at a point, the function and the tangent line become indistinguishable from one another. No matter what function you look at, and no matter what tangent line you choose, this is always true. A better name for a tangent line might be a "matching line". Check it out in the applet below. The function is plotted in solid red, and the tangent line at (3,3) is plotted in dotted red. Zoom in on (3,3) and notice that the function and the tangent line become indistinguishable.It's pretty awesome that Geogebra can calculate tangent lines for us! There are methods to calculate tangent lines "by-hand" without a computer, and you can learn about those in a traditional algebra based calculus course. The short story is: pick any point on a function; use the differential quotient to find the slope of the tangent line; then use the "point/slope equation of a line" from algebra 1 to calculate the equation of the line. We won't bother with studying this method here however, and will instead rely on Geogebra.
The tangent line is a very important concept because at a zoomed in scale, it exactly matches
f(x). Therefore, the slope of the tangent line is used as a measurement of the growth rate of f(x) at A. If the tangent line has positive slope, the function is "increasing" (from left to right). If the tangent line has negative slope, the function is "decreasing".
Of critical importance are places where the functions are neither increasing or decreasing; at these places functions can be said to have "stopped" to take a break. Places where a function has a tangent line with slope equal to 0 are of critical importance to finding maximums and minimums of functions. We'll discuss this type of stuff more soon.
For now though we will continue the discussion of tangent lines.