parametrized curve and tangent linePlot a tangent line to a spacecurve together.command templateWe need spacecurve and display: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<MAPLE comment: Here we see---in MAPLE notation---each individual curve while assigning it a string name as well so they can be displayed together below; afterwards one can replace the semicolons by colons to suppress the individual plots [as we did with the "with(plots)" listing of commands], but it is useful to first see that they are working first.Tangent line at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUkjbW5HRiQ2JVEiMUYnRjJGPEYyRjw= derived by hand, showing an equal parameter interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn on either side of the point of tangency and the corresponding interval on the tangent line: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Rotating this around, you see that indeed the tangent line is tangent to the curve. If your hand-derived equation for the tangent line is wrong, it should look wrong.
However, to be sure, use the next code to trial and error zoom in by setting the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn away from the value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8 here where the tangent is displayed in order to center the point of tangency in the plot window. [EDIT to use this template for a different value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn .] In this case first choosing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JVElMC4wNUYnRjJGPEYyLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y8 gets pretty close to merging of the two curve pixels, but removing the hashtag # from the second line shows 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 is about right without zooming in too far.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=For fun (if you are curious) you can right click on the original plot after rotating to a good view and change the Manipulator to Zoom In, and then left click on the point of tangency to zoom in. You may have to right click and change to Pan to move the point back to the center before zooming further.right margin context menu approach (too much work!)We need to load the VectorCalculus or Student[VectorCalculus] package for derivatives to work with a Maple vector-valued function. Then for a spacecurve vector-valued function output, we choose PlotBuilder, 3d parametric plot and narrow the parameter range. For a given value of the parameter, write the spacecurve representing the tangent line vector-valued position function for the tangent line. Use a narrower parameter interval with PlotBuilder, then copy and paste the tangent line onto the spacecurve. You can then rotate the plot around to verify that the tangent line looks right. You can also position the point of tangency roughly in the middle of the plot and with right click, Manipulator, Zoom in, we can zoom in to see the two curves begin to merge, but you need to switch to Manipulator, Pan to move the field of view to center it on the point of tangency. Here we use the Maple tangent line expression to illustrate. You can right click on each graph to choose color = red to see the curves a bit better.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 I already pasted into this spacecurve plot the tangent line from below.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 PlotBuilder applied to the tangent line vector valued function allows the 3d parametric plot and choice of color and interval of the 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 we just enter our hand calculated result this does not happen. This shows how a wrong entry looks: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It intersects the curve but is clearly not tangent to it.