Tangent Lines and Normal Vectors
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If a function f(x) is differentiable at x0, then f(x) will have a tangent line at (x0) with the equation l(x) = f(x0) + f '(x0)(x - x0).
The normal vector at x0, (-f '(x0), 1), is perpendicular to the tangent line.
In this demonstration you can input a function f(x) in the control panel. The graph of f(x) appears in green in a separate window. Next, use the tapedeck in the control panel to choose a point x0 in the domain. The tangent line to the graph at x0 is shown in cyan.
This demo shows the unit normal vector (in white) to the tangent line for a point x0 in the domain.
Exercises1. Find equations for the tangent line for each of the following sets of conditions, and check using one of the demos:
2. Why is the tangent line not defined for f(x) = |x| at x = 0?
3. Describe the tangent lines and normal vectors for any linear function.
4. Note that since the normal vector in this demo always has a vertical component of 1, it can never be parallel to the x-axis. Is this a problem? Why or why not?
- f(x) = x2 at x = 1
- f(x) = sin(x) at x = π
- f(x) = |x| at x = 1