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I am reading a chapter titled APPLICATION OF DERIVATIVES from NCERT school book

In that chapter, it is mentioned that finding equations for tangent and normal is one of the application of derivative.

In this chapter, we will study applications of the derivative in various disciplines, e.g., in engineering, science, social science, and many other fields. For instance, we will learn how the derivative can be used (i) to determine rate of change of quantities, (ii) to find the equations of tangent and normal to a curve at a point, (iii) to find turning points on the graph of a function which in turn will help us to locate points at which largest or smallest value (locally) of a function occurs. We will also use derivative to find intervals on which a function is increasing or decreasing. Finally, we use the derivative to find approximate value of certain quantities.

My doubt is all about the tangent of a curve in general.

The equation of a curve $y = f(x)$ at $(x_0, y_0)$ is given by

$$y - y_0 = \dfrac{dy}{dx} (x - x_0)$$

The diagram provided by the book and my presumption on the (definition of) tangent coincides. The presumption is that the tangent line touches curve at single point only. It may be true for some curves. I cannot understand the fact that how can a tangent line touches curve at single point only provided the curve is non-convex. Most textbooks explains about tangents and derivatives using nice convex curves only.

Is it possible for any curve (assume continuous) to have tangent at any point where derivative exists? Or can a tangent of a curve at a point can also intersect the curve at other points? Or do tangents for curves exist at some points only and not at all points?

hanugm
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    Yes, a tangent can touch a curve at multiple points. For example, consider the tangents to the curve $y=x^3$. For a more extreme example, consider the tangent to $y=\cos x$ at the point $(0,1)$. For an even more extreme example, consider the tangents to the curve of a linear function. – Joe Sep 03 '21 at 23:54
  • @Joe Then, what about this: A tangent line is a straight line that touches a function at only one point. – hanugm Sep 03 '21 at 23:57
  • By the way, you might want to consider what the formal definition of "tangent line" is. In fact, the most elementary one is that a tangent line to a function $y=f(x)$ at $(x_0,y_0)$ is the line passing through $(x_0,y_0)$ with a slope of $f'(x_0)$! This definition is not quite as general as we would like (for instance, it does not account for vertical tangents), but more advanced definitions of tangent still lean heavily on the notion of the derivative. – Joe Sep 03 '21 at 23:57
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    @hanugm The link in your comment is nonsense. – Paul Frost Sep 04 '21 at 00:04
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    I am afraid that this is simply incorrect. Perhaps you can define tangents to a circle or a parabola in that way, but that definition is useless in a more general setting. Here is a reference that says a tangent line does not have to intersect the curve only once, taken from Michael Spivak's Calculus. – Joe Sep 04 '21 at 00:04
  • Refrain from posting the entire chapter as a PDF as it is hard to find what diagram you're referencing. You can just attach the image of the diagram. – ultralegend5385 Sep 04 '21 at 00:14
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    The word “touches” is doing an awful lot of work in that non-definition. That’s even worse than defining an asymptote as a line that approaches but never reaches the graph :). – Erick Wong Sep 04 '21 at 00:18
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    @hanugm Note that your link is careful to state that it's only a summary. That language is intended to give you an intuitive guide to, in particular, the relationship between secants and tangents. It's not intended as a rigorous definition of a tangent. – Robert Shore Sep 04 '21 at 00:41
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    A nice (though still informal) definition that avoids the "touches in one point" issue is: The tangent line to a curve at a point is the line that the curve looks like really-really-really-...-really close to that point. (In the context of, say, a graphing calculator: It's the line that the curve looks like when you zoom wwwwaaaayyyy in on the point.) – Blue Sep 04 '21 at 00:43
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    @Blue: I very much like that informal definition, because it motivates the following definition: the tangent line to a curve is the "best" linear approximation of the curve around a certain point. (If $f$ is differentiable at $a$, then a linear function $g$ is the best linear approximation around $a$ if $f(a+h)-g(a+h)$ is $o(h)$ as $h\to 0$.) This definition of tangent also helps to motivate the definition of the derivative. – Joe Sep 04 '21 at 00:54
  • @Blue: I like to think of $f(a+h)-g(a+h)$ not just as being tiny, but tiny relative to the size of the "zoom". That's why we require that $f(a+h)-g(a+h)$ be $o(h)$. – Joe Sep 04 '21 at 01:00
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    @Joe: "This definition of tangent also helps to motivate the definition of the derivative." Indeed! My informal definition of derivative is "The slope of the line that the curve looks like [etc, etc, etc]". :) ... A helpful feature of this definition is that you can say "If the curve doesn't 'eventually' look like a line, then it doesn't have a tangent (or derivative)", which explains the problem with, say, $y=|x|$: No matter how close you get to the origin, the graph still looks like a "V", not a line. (From there, discussion can turn to "one-sided" derivatives and all that.) – Blue Sep 04 '21 at 01:04
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    @Blue: That's a good example to bring up, because I have seen many students mistakenly think that the line $y=0$ is tangent to $y=|x|$ at the origin. Perhaps this is because they have been taught that the tangent line has to touch the curve once. With the definition you are proposing, it is obvious that the two-sided derivative cannot exist. – Joe Sep 04 '21 at 01:07
  • Does this answer your question? Problem with basic definition of a tangent line. – Joe Sep 04 '21 at 01:12
  • What does it mean to say that a curve is non convex??? – copper.hat Sep 04 '21 at 04:43
  • Apologies: earlier, I screenshotted the wrong page of Spivak. Here is a reference that says that the tangent line can touch the curve more than once. – Joe Sep 04 '21 at 22:21

1 Answers1

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The tangent to a curve can intersect the curve at infinitely many points. Let $y = \sin x$ and consider the tangent line $y=1$.

Robert Shore
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    In fact, the tangent to a curve can intersect the curve at an uncountably infinite number of points, but perhaps this is tangential to the question :-) – Joe Sep 04 '21 at 01:10