When is a tangent line vertical?
Understand the Problem
The question is asking under what conditions the slope of a tangent line becomes undefined, which typically occurs when the derivative of a function approaches infinity. This usually happens at points where the function has a vertical tangent.
Answer
The slope of a tangent line is undefined at points where the derivative approaches infinity.
Answer for screen readers
The slope of a tangent line becomes undefined at points where the derivative of the function approaches infinity, typically at vertical tangents, discontinuities, or vertical asymptotes.
Steps to Solve
- Determine where the derivative is undefined
The first step is to find where the derivative of the function does not exist. This usually occurs at points of vertical tangents. To find these points, we take the derivative of the function and set it equal to infinity, or check the limits at those points.
- Identify vertical tangents
If the derivative approaches infinity as you approach a certain value of $x$, then the tangent line at that point is vertical. You can find these values by analyzing the function behavior around discontinuities or points where the derivative changes dramatically.
- Check for vertical asymptotes or discontinuities
Examine if the function has any vertical asymptotes or is undefined at certain points which also indicates potential vertical tangents. For instance, if a function is represented as $f(x) = \frac{1}{x}$, it is undefined at $x = 0$, showing a vertical line there.
The slope of a tangent line becomes undefined at points where the derivative of the function approaches infinity, typically at vertical tangents, discontinuities, or vertical asymptotes.
More Information
Vertical tangents signify rapid changes in the function's values; they can often occur with functions defined in piecewise manners or rational functions approaching specific limits.
Tips
Mistakes often include assuming that the slope is undefined only when a function is not defined. Students should be cautious to check derivative behavior and limits instead of solely relying on function observations.
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