tag:blogger.com,1999:blog-5303307482158922565.post1454501132016550283..comments2021-04-05T17:28:31.416-07:00Comments on Math Mama Writes...: Calculus: Anti-derivatives and Area Under a CurveSue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-5303307482158922565.post-69278177222816679422012-12-21T06:24:49.207-08:002012-12-21T06:24:49.207-08:00We can avoid using any symbol for anti=derivative ...We can avoid using any symbol for anti=derivative at first, by offering students functions labeled f'(x), ansking asking them to find f(x). I think that's my preference. And doing this project has made me notice that the definite integral is an odd new function, in that it needs two inputs. Our area function avoids this by keeping the left side constant, so I like the F(x) notation for this, which clarifies that it's just another function.<br /><br />I think you're right about starting off more gradually. Maybe I should make those modifications now, before I forget.<br /><br />I had done a section like the textbooks do, on anti-derivatives, so they saw how polynomials work, and probably sine, cosine, and e^x. I think we had done simple chain rule problems too, like find y when y'=sin(2x).<br /><br />I'm reading a very interesting <a href="http://www.lightandmatter.com/calc/" rel="nofollow">open source textbook</a> right now by Crowell, and he makes the connection between rate of change and area clear in the first chapter of the book. He's starting out with discrete change. It's an intriguingly different approach.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-67959373418058953192012-12-20T13:39:59.058-08:002012-12-20T13:39:59.058-08:00I like to start out by using F(x) as the symbol fo...I like to start out by using F(x) as the symbol for the antiderivative of f(x), and the integral sign for the area. Then the fundamental theorem is really exciting and nonobvious! I certainly remember in my first calculus class, seeing that theorem and thinking "So the integral of the function is the integral? What is this pointless thing?" and then the problems we were given for it all involved d/dt of the integral from a to t of f(x) dx, and with all those letters flying around the whole thing didn't make any sense. <br /><br />Your project looks fantastic -- I love the way you start, introducing the notation and the concept and estimation without any heavy machinery involved. Then moving on to linear functions -- maybe a bit too big a leap? I might use y = 3 first, and then y = 2x, and then their sum, with the bonus of introducing another fact about integrals. <br /><br />Then they get to the Fundamental Theorem in such a way that it seems almost natural, like they're discovering it for themselves!<br /><br />I'm not sure how much they know about antiderivatives before this project -- I'd be curious to know a little more about where in your course this lands.Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-30663671434092178992012-11-18T14:47:36.848-08:002012-11-18T14:47:36.848-08:00This makes so much sense. The area function is ve...This makes so much sense. The area function is very concrete and a much better point of departure for introducing integration. <br /><br />I also agree that the unfamiliar notation and terminology is disconcerting. It can help to "demystify" it if you give students some helpful ways to associate meaning with those terms so they "own" them.<br /><br />Your "area function" first approach lends itself very naturally to that, since you can talk about the integration symbol being just a fancy form of "S" for summation (and approximating areas obviously involves summing up many smaller areas with widths of dx).<br /><br />And the word "integrate" also has helpful etymological associations, since the word literally means "to render something whole" and you are literally putting many different puzzle pieces together to compute the area of the whole when you integrate.Mary O'Keeffehttps://www.blogger.com/profile/14662977706706048151noreply@blogger.com