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GraphStatement

That is the statement and the graph given in my book. I get why they have the intervals that they do. It's because for the specific solution of the differential equation, the graph is that line which does not make a circle. And it has a vertical tangent line in its solution in the interval (-3, 7).

However, the part that I don't get is, why can't the graph of a differential equation have a vertical tangent line anywhere? It seems reasonable to me that dy/dx can be 0 sometimes. Why can't it be?

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    $\begingroup$ What does $y(x)$ having a vertical tangent at $x=a$ mean for $y^\prime(a)$? What is the slope of a vertical line? $\endgroup$ Commented May 19, 2017 at 6:35

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$\frac{dy}{dx}$ can certainly be zero sometimes, but that's a horizontal tangent line, not a vertical one. A vertical tangent line occurs when $dx=0$, and division by zero is undefined.

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  • $\begingroup$ Oh...got it! Thank you so much. $\endgroup$ Commented May 19, 2017 at 6:36

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