Based on how you've responded here, it seems like you simply don't have an appreciation for the material covered. The short answer is, in a first year math classroom you're given nice, tea party little questions for which you know solutions exist. For these situations, like say integrating f(x) = x from 0 to 1, you may as well just compute its anti-derivative and yes it seems kind of wasteful to compute Riemann Sums.
What if you can't compute a function's anti-derivative? Then when you can't compute a derivative explicitly, which happens all the time in "the real world" you do the next best thing: approximate to say, 5 decimal places. So using 'easy' examples to test you is part of the learning process...would you rather have learned the method with a really hard function that doesn't actually have an anti-derivative?
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Originally Posted by Krusenik
The applications were useless and the profs generally just always used the same application like amount of morphine in blood or something.
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Logistic equations, Half-life equations, etc. such as morphine in blood, are problems that Mathematical Biologists have to deal with all the time. When do you give that next shot of morphine? If you do it too soon, they'll OD on morphine...if you do it too late, they're in excruciating pain in the meantime.
Seems like an important example to me...and it's a beautiful example of how medical theory (ie. maintaining a certain blood concentration of morphine) melds beautifully with mathematics.
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Cobwebbing is useless to learn in first year calculus.
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Cobwebbing is similar, it's a trick used for approximating yes. Now I haven't learned this in a Math course because I'm not an applied Math buff (I do the theoretical, 'pure' maths such as Geometry and Topology, Set Theory, etc. They probably teach it in a 2nd year modelling course), but I have learned this method by myself to solve a problem I had in a 3rd year math course (Combinatorics, Math 3U03) regarding recurrences.
And so, this time it's a method for approximating solutions to
recurrence relations. Like I said, you may be able to explicitly solve a simple example like r(n+2) = r(n+1) + 2r(n) explicitly for a solution...
But in 'the real world' where you do population dynamics and have a situation like the one I'm going to link in a moment, it's not very easy to solve and you must approximate a solution.
http://www.public.iastate.ed u/~kmo...ed/cobweb.html
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They taught you EULER'S method...WTF??! forget that, euler's method is completely useless and totally inaccurate in calculating integration.
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Euler's method is for solving ODE's (Ordinary Differential Equations)...not Integrals.
http://en.wikipedia.org/wiki/Euler's_method
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Why bother learning this piece of trash method of approximation? Ask any other math majors, they'll tell you: they skip this euler's method because it is useless and inaccurate.
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Well I'm a math grad student...and the beauty part of being a math student is that you solve and prove everything for yourself. Other people's work/opinions don't matter until you read through the finer details and accept the work as valid for yourself. So I don't really need to ask other math majors :p
I had to learn Euler's method in Math 2C03, the 2nd year ODE course. Why?
Because not all differential equations are solvable. Sure, all of the nice ODEs you see in a first or second year
methods course are solvable, but what if you have a system of ODEs for which you haven't learned a 'trick' for solving them? (Or worse still, if it's been proven that a solution doesn't exist!)
Though there may or may not exist a way to sketch the solution directly (There are always
existence and uniqueness questions to ask when solving an ODE), using an approximation method helps shed a lot of insight into the problem. Again you have to remember that in the 'real world' math isn't elegant...the numbers don't 'work out nicely.'
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Euler's method??? COME ON.... and this other stupid method, sum method where you count the area literally by approximating unnder the graph? why...why......NO you cannot have a justification for this.
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Sure, I have justification for that too.
Riemann Sums are again, important in the real world. Be a sport and try to integrate f(x) = e^(x^2) from 0 to 1 for me...and you'll find no antiderivative F with F'(x) = f(x) exists in a closed form solution. You have to do something like sketch the curve and use Riemann Sums to approximate the answer, or expand it out as a Taylor Series and try to make educated guesses at the behaviour.
Now it's needless to say, f(x) = e^(x^2) is a standard model for some bacterial growth...which sounds a lot like the Life Sciences doesn't it?
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Some of the other applications that were used often were like maximizing the amount of fencing around a garden or something, that's great especailly when you spend a whole class covering things like that.
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I'm confused, particularly since optimization is something that's usually covered in Math 1A03 and Math 1N03 (engineering calc).
I'm just not sure what you expected. You hated the life sciences part of the course designed to have life sciences applications...I guess a bit of friendly advice would be, make sure you're in the right program.
