Which are tangents

Question

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We are asked to see which are tangents and which aren't. I think B3, bottom left and bottom middle are not tangents

Newb · Accepted Answer · 2013-10-10T05:52:11.160

2

A tangent to a curve is a straight line that just touches the curve, such that the slope of the tangent is exactly that as the slope of the curve. By that definition, all except for B5 display tangents.

EDIT: The picture is unclear, so it is difficult for me to tell in the cases of B3 and B4. A larger picture would help, or you could judge this yourself using the definition I provided above.

edited Oct 10 '13 at 05:52

answered Oct 10 '13 at 04:45

Newb

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What do you mean by "just touches" ? – DBFdalwayse Oct 10 '13 at 05:00
Are you sure about that? – J. W. Perry Oct 10 '13 at 05:22
@J.W.Perry The cusp at B5 is non-differentiable, so it doesn't have a defined tangent. Do you think otherwise? – Newb Oct 10 '13 at 05:51
@DBF There's a point $(x1,y1)$ on the curve which is also on the tangent line. – Newb Oct 10 '13 at 05:53
@DBF I forgot to mention: at that point (x1,y1) the tangent and the curve must have the same gradient. You can verify this by differentiating the curve at that point. – Newb Oct 10 '13 at 06:10

J. W. Perry · Answer 2 · 2013-10-10T16:29:21.070

B1 and B2 are clearly expressing the unique tangent lines at $x=0$ in your image.

There are infinitely many lines "touching" at $x=0$ on B5, but no unique defined tangent line.

B6 is clearly expressing the unique tangent line at $x=0$.

There are definitely no tangent lines being expressed in B3.

In B4 one is definitely not the tangent line, but the other does appear to be the tangent line at $x=0$ (I cannot see the names of the two lines but I think it says $c_2$).

We define the tangent line to a curve $f(x)$ at a point $P(x_0,f(x_0))$ as being the unique line through $P$ with slope $$\lim_{h \rightarrow 0} \frac{f(x_0+h)-f(x_0)}{h}.$$ If this slope does not exist, then there is no tangent line at $P$.

You might want to clarify that a tangent is necessarily unique. It seems implied, but a lot of people don't understand this very basic concept (during senior year of high school, I would draw graphs like B3 in an attempt to prove my calculus teacher wrong). — Ryan, Oct 10 '13 at 05:59

Which are tangents

2 Answers2