Timeline for Examples where it's useful to know that a mathematical object belongs to some family of objects
Current License: CC BY-SA 3.0
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| Sep 2, 2011 at 12:39 | comment | added | KConrad | Moreover, the same issue arises in differential calculus. To find the max/min points for a function, we want to see where the tangent lines to its graph have slope 0 (necessary condition). In order to find the places where the slope is 0, we use differential calculus to find the slope of the tangent line everywhere along the graph first. Only then do we set the result equal to 0. Thus we are considering a horizontal tangent line as lying in the family of all tangent lines in order to locate the horizontal tangent lines. | |
| Sep 2, 2011 at 12:18 | comment | added | KConrad | I think this is a really great example since it is at an undergraduate level and is an aspect that is hard to get students in calculus to appreciate. I would make this situation even more concrete: in order to find the area under y = sin x over the interval [0,pi], the simplest solution is to consider the more general problem of finding the (signed) area over [0,t] for all t and then we can use calculus, specializing to t = pi at the end. | |
| Sep 1, 2011 at 6:06 | history | answered | anonymous | CC BY-SA 3.0 |