## More On Vertical Tangent Lines

### October 24, 2010

This paper, More On Vertical Tangent Lines, proves a fact about vertical tangent lines. You need a solid understanding of the Mean Value Theorem as it is taught in undergraduate calculus courses. If you’re in the habit of reading proofs – in particular, the epsilon-delta proofs – then this should be manageable, given that you understand some of the more theoretical exercises about how to use the Mean Value Theorem.

To summarize the paper: Suppose f is continuous. In most calculus books, a function is said to have a vertical tangent line at x=a if and only if “f'(a)=+ or – infinity” (that is, [f(a+h)-f(a)]/h goes to infinity or to minus infinity as h goes to 0). If f is differentiable away from x=a and f ‘(x) approaches + or – infinity as x approaches a, then it follows that f has a vertical tangent at x=a (this is the theorem). For our favorite functions, this is the most likely way for a vertical tangent to occur. However, sometimes a vertical tangent can exist at x=a even if f'(x) does NOT approach + or – infinity as x goes to a.

The paper proves the theorem and provides a function f such that f has a vertical tangent at 0, but f'(x) does not approach + or – infinity as x approaches zero.

**UPDATE:** See also this nice graph by fellow grad student Bob Willenbring. He gave me some TeX tips for doing these, and I’ll have to figure out how it all works sometime soon. I’d like to include this in the document itself next draft.

[…] More On Vertical Tangent Lines – Intended for someone who at least has a solid understanding of undergraduate calculus, especially formal definitions and epsilon-delta proofs. Also essential is a solid understanding of the more theoretical applications of the Mean Value Theorem that are sometimes included in problem sets. This paper isn’t too over the top, but someone new to the material should maybe read slowly, as the Mean Value Theorem can tend to be deceptively subtle – there is always more to it than meets the eye. Posted by Matthew Bond Filed in Uncategorized Leave a Comment » […]

[…] 4, 2010 I revised More On Vertical Tangents, and included a nice graph of the example function made by Bob Willenbring. Thanks, […]