# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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 Symmetry For The Win. Topic: Calculus. October 30th, 2006 Problem: Evaluate the improper integral . Solution: As the topic suggests, we will look for a symmetry to simplify the problem. Notice the identity (since we’re just integrating across the interval from different “directions.” Using this, we have . Adding the old integral to the new one, we have from the property of logs and the double-angle identity (here it is again!). But in fact this expression is simply (*) by the substitution . Taking into the account of the symmetry of from to , we get so . Plugging back into (*) we obtain so we get . QED. ——————– Comment: The function is not nice to actually integrate; it involves the polylogarithm function if you try here. This is an important example of how symmetry can help a ton in integration because there are so many functions that cannot be integrated with elementary functions but can be evaluated over an interval through different techniques. ——————– Practice Problem: Evaluate without actually finding the antiderivative. Posted in Calculus || No Comments » Out Of Place Trig. Topic: Geometry/Trigonometry. Level: AMC/AIME. October 29th, 2006 Problem: (2006-2007 Warm-Up 4) Two sides of a triangle are and and the angle between them is . If , what is the maximum area of the triangle? Solution: Well, given two sides and the angle between them, there is only one formula for the area of a triangle that comes to mind. . So plugging this in, we get the area to be . By the double-angle identity, this is equal to for . But the maximum of this is clearly and it is obtained when . QED. ——————– Comment: This problem shouldn’t have been too hard; finding that specific formula for the area of the triangle could have been derived by drawing an altitude. And the double-angle identity should always be known. Lastly, maximizing the sine function is a given. ——————– Practice Problem: (2006 Skyview) If and , what is ? Posted in AIME, AMC, Geometry, Trigonometry || 1 Comment » Bellevue Wins First At Skyview! October 28th, 2006 Posted in Announcements || 1 Comment » Square Sum Stuff. Topic: Polynomials/S&S. Level: AMC/AIME. October 27th, 2006 Problem: Evaluate the summation . Solution: Here’s a technique that will help you evaluate infinite series that are of the form polynomial over exponential. It’s based on the idea of finite differences: If is a polynomial with integer coefficients of degree then is a polynomial of degree (not hard to show; just think about it). So let . Then consider by simply multiplying each term by : . And now find the difference by subtracting the terms with equal denominators. We get . Notice that the numerator is now a polynomial of degree instead of . Repeating this, we have and . Notice that the latter part is just a geometric series which sums to so . QED. ——————– Comment: The method of finite differences is extremely useful and is basically a simplified version of calculus – in a very approximating sense. It’s a good thing to know, though, because then you have a better understanding of how polynomials work. ——————– Practice Problem: Let be a polynomial with integer coefficients. Using the method of finite differences, predict the degree of . . Posted in AIME, AMC, Polynomials, Sequences & Series || 2 Comments » Serious Substitution. Topic: Calculus. October 25th, 2006 Problem: Use the substitution to solve the differential equation . Solution: Well, let’s do what it says. From the chain rule, we have . Then we also have by the product rule and chain rule again . So we can make the substitutions and to obtain the differential equation . But we know the solution to this is so our final solution is . QED. ——————– Comment: This substitution is, of course, not really natural but was actually found after solving the ODE in another way. Fortunately, it simplifies the problem rather greatly and seems to be a useful technique to look out for. ——————– Practice Problem: Find another way to solve the differential equation. Posted in Calculus || 3 Comments »