tag:blogger.com,1999:blog-5303307482158922565.post2647233541541552580..comments2024-03-22T13:39:55.941-07:00Comments on Math Mama Writes...: Sneaking Up On the Fundamental Theorem of CalculusSue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-5303307482158922565.post-43616012652191158732010-06-18T11:35:47.925-07:002010-06-18T11:35:47.925-07:00Sue, thank you for sharing the story. I especially...Sue, thank you for sharing the story. I especially appreciated the pictures.MariaDhttps://www.blogger.com/profile/00769513929584082597noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-20184454680642804692010-06-11T17:42:37.719-07:002010-06-11T17:42:37.719-07:00(And thanks, Hao. I don't usually prepare at a...(And thanks, Hao. I don't usually prepare at all. Having this conversation is helpful.)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-11170029997765205972010-06-11T17:42:04.180-07:002010-06-11T17:42:04.180-07:00I'm going to have to look that up before Monda...I'm going to have to look that up before Monday! I'm sure I know both notations, but I have no idea which one goes with which person.<br /><br />I've got to come up with some story about why we'd want to find the area out to t (x is in use already). Maybe we can afford a bit more weight and we want to see how long we can make the part.<br /><br />He's going to discover that it's the anti-derivative without me ever mentioning them. That's my hope.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-21817413746902467502010-06-11T16:41:24.897-07:002010-06-11T16:41:24.897-07:00You could always switch back and forth between New...You could always switch back and forth between Newton and Liebniz notation and include some history in the lesson. :)Haohttps://www.blogger.com/profile/02348974241652264510noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-27226518385769938812010-06-09T20:29:41.120-07:002010-06-09T20:29:41.120-07:00Aluminum is roughly 3 g/cm^3 (among friends), so t...Aluminum is roughly 3 g/cm^3 (among friends), so to have an areal density of 1 g/cm^2, the part would need to be around 1/3 of a centimeter thick. Steel is closer to 8 g/cm^3, so it would need to be around 1/8 cm thick.EMShttps://www.blogger.com/profile/17382583550255067568noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-80120338833317807442010-06-08T12:36:05.320-07:002010-06-08T12:36:05.320-07:00Dang it, I knew density would be the wrong word. B...Dang it, I knew density would be the wrong word. But what I'm asking is whether that's too few grams for the area (assume it's whatever thickness needed for a toy car, so can it be thin enough to make sense).<br /><br />I think putting anti-derivatives first points students to think anti-derivative too quickly in the area problem. (Especially given the notational overlap, where the integration symbol means anti-derivative for indefinite integrals, and does not mean anti-derivative for definite integrals - it means area then.)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-91645197428463340622010-06-08T10:44:31.295-07:002010-06-08T10:44:31.295-07:00"1 gram per square centimeter (is that a reas..."1 gram per square centimeter (is that a reasonable density?)"<br /><br />is not valid because density is mass per unit volume. 1 gram per cubic centimeter is the density of water, and metal is quite a bit denser than that (roughly an order of magnitude).<br /><br />Otherwise, what you are doing sounds very neat! I think it probably makes sense to think about antiderivatives (indefinite integrals) before getting to the whole riemann sum = definite integral bit, but it sounds like you are headed that way and tying together the ideas.Haohttps://www.blogger.com/profile/02348974241652264510noreply@blogger.com