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What interesting applications are there for theorems or other results studied in first-year calculus courses? A good example for such an application would be using a calculus theorem to prove a result in group theory. On the other hand, the importance of calculus in applied mathematics or in physics is well known, therefore is not a good example. |
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20
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You're looking for something fun for a calculus course? If a rectangle $R$ is tiled by rectangles, each of which has a side with integer length, then $R$ has a side with integer length. This is from
and one of those fourteen proofs goes by a double integral. |
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16
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Following on from the Galois theory example of Johannes, one straightforward way to produce an explicit polynomial with non-soluble Galois group over ${\mathbb Q}$ is to use an irreducible quintic with exactly three real roots, which necessarily has Galois group $S_5$. To check that an explicit polynomial (such as $x^5-4x+2$ if I am not mistaken, I am typing from memory) has this latter property reduces to standard calculus arguments such as "differentiate, find turning points, estimate values, use intermediate value theorem". I always find this calculus interlude at the end of half a semester of algebra quite amusing. |
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16
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The interesting application in Spivak's Calculus is the proof of the irrationality of pi. I guess this is the proof due to Niven. |
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14
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An interesting application of calculus is the elementary polynomial case of Mason's ABC theorem. This yields, for instance, a completely trivial proof of the polynomial case of FLT (Fermat's Last Theorem). That this works so effectively for polynomials (functions) vs. numbers is due to the fact that for functions we have available the derivative, which implies that we can exploit Wronskians as a measure of algebraic independence. Such Wronskian estimates serve as fundamental tools in diophantine approximation. See my post [1] for further details and references. [1] sci.math.research, 1996/07/17 |
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14
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The mean-value theorem (of differential calculus) can be used to prove that Liouville numbers are transcendental. The proof is quite simple, taking only a couple of lines. See Theorem 191 of Hardy and Wright's "An Introduction to the Theory of Numbers" on Google books. I believe, historically, that these were the first known examples of transcendental numbers. |
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14
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The intermediate value theorem is a basic ingredient in a Galois theory-based proof of the fundamental theorem of algebra. It is used as "Every real polynomial of odd degree has a real zero". |
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6
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The irrationality of $e$ !! First use the Taylor expansion of $e^x$ to show that $|e-S_n|<\frac{3}{(n+1)!}$ where $S_n= 1+\frac{1}{1!}+\frac{1}{2!}+\cdots + \frac{1}{n!}$. Then, deduce the irrationally of $e$. |
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6
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An example that I like is the proof that $e^{A+B}=e^A e^B$ for commuting matrices $A,B$. Since the matrix exponential is defined by the usual exponential series, we have to prove that $\sum \frac{(A+B)^n}{n!}=\sum\frac{A^n}{n!}\sum\frac{B^n}{n!}$ This follows, without actually computing the two sides, by observing that it is the same computation as for real numbers $A,B$ (because $A$ and $B$ commute). And for real numbers we know the result is correct by first-year calculus. |
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5
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Going in a completely different direction: a surprising application of calculus is the use of the Leibniz and chain rules to differentiate data types to create new types that represent structures with 'holes' in them. See here for an elementary exposition. (This is closely related to the differentiation of generating functions and combinatorial species.) |
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3
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I think you could probably show a smart group of first year calculus students how to get an exact formula for the Fibonnaci numbers using generating functions, which basically just boils down to knowing partial fraction decomposition and a few standard power series. You could then point them to Wilf's book if this makes them curious about generating functions in combinatorics. |
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2
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The notion of a formal derivative of a polynomial over some ring comes from the ordinary derivative of a polynomial over the real and complex numbers. Furthermore, results true over the real numbers, such as that $(fg)'=f'g+g'f$ and $(f \circ g)' = (f' \circ g) g'$, continue to hold over arbitrary rings. However, these results are much easier to prove over the real numbers using analytic techniques, and one might legitimately argue that mathematicians were only led to the corresponding formal results by the inspiration of the results in calculus. Furthermore, using something along the lines of the Lefschetz principle, one can probably derive the identities for formal derivatives from the corresponding facts for derivatives of polynomials over the complex numbers. |
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2
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A few years ago I gave a departmental colloquium talk, aimed at beginning M.A. students, on "An application of calculus to ring theory." A slightly facetious little abstract can be found here. The example establishing the main results—very well known to workers in commutative ring theory—was the ring of germs at $0$ of class $C^{\infty}$ functions on $\mathbb{R}$. A bit of calculus is needed in verifying the requisite properties. |
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2
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See Robert M. Young's lovely book titled "Excursions In Calculus" for some examples. |
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2
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A cool example: The intermediate value theorem may be used to prove the following theorem about continued fractions: Let $\alpha>1$, and suppose that $$ \left|\alpha-\frac{p}{q}\right|<\frac{1}{2q^2}. $$ Then, $\dfrac{p}{q}$ is one of the convergents (truncated continued fractions) of $\alpha$. |
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1
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Do approximations of $\pi$ count? If so, see http://math.stackexchange.com/questions/1956/is-there-an-integral-that-proves-pi-333-106 and the references there. |
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1
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Shanks' simplest cubic $x^3-ax^2-(a+3)x-1$ has discriminant $D=(a^2+3a+9)^2$ and hence 3 real roots. A calculus way to see this is to rewrite it as inverting $f(x)=a$ where $$f(x)=\frac{x^3-3x-1}{x^2+x}=x-1 -\frac{1}{x}-\frac{1}{x+1}$$ whose graph has three monotone branches which clearly intersects $y=a$ at three real points for any real $a$. In particular, we can pick $a$ to be any of the (real) roots which means we can iterate the construction and get a cubic tower of totally real fields. Also (though this is not relevant to the question), it's nice to see $f$ is a trace $$f(x)=x+\rho(x)+\rho^2(x),$$ where $\rho(x)=-1/(x+1)$ is of order 3 in $ \in PSL_2(Z)$ which show that if $\alpha$ is a root of the cubic , then so is $\rho(\alpha)$. |
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1
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A nice application of calculus that leads to a surprising and far reaching result, first obtained by the great Gauss himself, is the computation of the Arithmetic-Geometric Mean of two numbers $a > b > 0$. A comparatively short way to this end is presented on the first pages of J. and P. Borweins "Pi and the AGM". |
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0
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What I like(d) most is defining an analytic function that describes some number theoretic phenomena. One thing I remember is from Winfried Kohnen's postech lecture http://www.mathi.uni-heidelberg.de/~winfried/siegel2.pdf , see pages 1-3 for more details. He starts with the standard inner product on $\mathbb{R}^m$ viewed as a quadratic form $$Q(x):=x^t x.$$ We are interested in the number $r_Q(t)$ of tuples of squares of inetegers that add up to a natural number $t$, i.e. $$ r_Q(t):= # \left{ g \in \mathbb{Z}^m : Q(g)=(g_1)^2+ \dots + (g_4)^2=t \right} .$$ They can be computed via this power series $$ \theta_Q(z) = 1+ \sum_{t\geq 1} r_Q(t)\ \exp(2\pi i tz) $$ that is in fact $\theta_Q$ is a modular form of weight 2 w.r.t. $\Gamma_0$. Therefore (ok here is some kind of black box for the students), its Fourier coefficients can be given by $$r_Q(t)= 8 \left( \sigma_1(t)-4\cdot \sigma_1\left(\frac{t}{4}\right) \right)$$ where $\sigma_k(t)$ denotes the divisor function $$\sigma_k(t):=\sum_{d|t} d^k.$$ While writing this I was wondering whether the prime number theorem and elegant proofs of the fundamental theorem of algebra are too well known. p.s. sorry for messing up the formulas again. |
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-3
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I've always enjoyed dabbling in some of the ridiculous third dimensional object types for which one can create general SA/V formulas with a general knowledge of the integration involving solids of revolution, categorization of polynomials based on the number of POIs and PPOIs, a little bit of abstract thought, and a truckload of time to burn in a curriculum designed for the unfortunately all too common "American't" mathematical mindset. Cheers, Stephen |
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