Tangent Lines and Normal Vectors
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) Text If a function f(x) is differentiable at x_{0}, then f(x) will have a tangent line at (x_{0}) with the equation l(x) = f(x_{0}) + f '(x_{0})(x  x_{0}).
The normal vector at x_{0}, (f '(x_{0}), 1), is perpendicular to the tangent line.
Demos
Tangent Lines
 
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 x_{0} in the domain. The tangent line to the graph at x_{0} is shown in cyan.

Normal Vectors
 
This demo shows the unit normal vector (in white) to the tangent line for a point x_{0} in the domain.

Exercises 1. Find equations for the tangent line for each of the following sets of conditions, and check using one of the demos:
 f(x) = x^{2} at x = 1
 f(x) = sin(x) at x = π
 f(x) = x at x = 1
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 xaxis. Is this a problem? Why or why not?
