Timeline for Find equations of the tangents to a parametric curve that pass through a given point
Current License: CC BY-SA 4.0
10 events
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| Jun 4, 2018 at 22:33 | comment | added | amd | The existence of a well-defined tangent line is indeed the essence of differentiability, but that wasn’t really the question. Regardless, the definition you give appeals well to intuition, but is too restrictive. It doesn’t allow a tangent to exist at an inflection point. | |
| Jun 4, 2018 at 18:53 | comment | added | nooneperfect | I missed the original main question in all this discussion. Anyways i was referring to the yellow highlighted question and Mr.Siong Thye Goh graph about the lines mentioned there. The main point i was making multiple tangents at a common point doesn't fit the tangent line definition. | |
| Jun 4, 2018 at 18:40 | comment | added | amd | It seems that all I can do for you at this point is to refer you back to my original comment. Neither you nor the O.P. are solving the problem that was actually posed, which is to find the tangents that pass through the given point, not the tangent at that point. The fact that it happens to lie in the curve is an irrelevancy that is tripping both of you up. | |
| Jun 4, 2018 at 18:18 | comment | added | nooneperfect | slideplayer.com/slide/231981 Hopefully this will help to clear definition. | |
| Jun 4, 2018 at 17:56 | comment | added | nooneperfect | I guess you are jumbling between touching and crossing,of course the second line is not tangent as you can clearly see it was crossing given curve. But the first line is touching exactly at (4,3). The point here im making is that for a given point there only exist one tangent line not the multiple tangent line at common point. More likely first line is a Secant Line as if we extended , but as in problem we derived tangent line so first line is the only line which will fit in the definition of tangent line | |
| Jun 4, 2018 at 17:46 | comment | added | amd | You’re misinterpreting what that page you’ve cited is telling you. That line in the illustration is certainly not tangent to the curve at the point that it crosses the curve but it is tangent to the curve elsewhere. The same situation exists in this problem. The second line is not tangent to the curve at $(4,3)$, but at some other point. The fact that it crosses the curve somewhere else is immaterial. In particular, the idea that a tangent line doesn’t cross the curve at the point of tangency only holds for locally convex curves, which is not the case for my earlier example. | |
| Jun 4, 2018 at 7:57 | comment | added | nooneperfect | Right, through (4,3) there is tangent at that point but slope form of line $(y-3)=-2(x-4)$ equation crosses curve not touching it. Also there is difference between secant ,tangent line and normal line definition. $(y-3)=-2(x-4)$ This line doesn't fit the definition of tangent line as you can see in graph. Im not so quiet sure about the origin you mentioned $y=x^3$ but i guess it will not be called as tangent if we take $x=0 and y=0$ line equation. tutorial.math.lamar.edu/Classes/CalcI/Tangents_Rates.aspx | |
| Jun 4, 2018 at 7:31 | comment | added | amd | You appear to be making the same error that the O.P. did: the original problem doesn’t say that $(4,3)$ is the point of tangency, only that the tangent lines must pass through this point. | |
| Jun 4, 2018 at 7:22 | comment | added | amd | So, would you say that $y=x^3$ has no tangent at the origin? | |
| Jun 4, 2018 at 6:38 | history | answered | nooneperfect | CC BY-SA 4.0 |