Additional comments related to material from the class. If anyone wants to convert this to a blog, let me know. These additional remarks are for your enjoyment, and will not be on homeworks or exams. These are just meant to suggest additional topics worth considering, and I am happy to discuss any of these further.
We started with one of my favorite problems, given \(S = a_1 + \cdots + a_n\) with each \(a_i\) a positive integer and the goal to maximize the product of the \(a_i\). We quickly see the optimal is when each \(a_i\) is 2 or 3, and since \(2\ast 2 \ast 2 < 3 \ast 3\) we want \(3\)'s over \(2\)'s. We converted to a real problem and assumed there were \(n\) summands, each a real number. We got a function defined on the integers to maximize, replaced it with a function defined on the reals so calculus would be applicable. We then curve sketched and saw the function was increasing to its maximum and decreasing past it, so the optimal integer soln was either to the left or right of the optimal real soln (here optimal soln is referring to the number of summands). It's unusual to be this fortunate.
We had to maximize \(a_1 \ast \cdots \ast a_n\) given \(a_1 + \cdots + a_n = S\) and each \(a_i > 0\). We can do this with Lagrange multipliers, or since each \(a_i\) is in \([1, S]\) we can appeal to the \(n=2\) case because a real continuous function on a compact set attains its max and min. What is nice is that this existence result from real analysis improves to being constructive; if we were at the optimal point and all coordinates were not equal, we could simply replace two of them with the average and improve the product.
A nice application of this problem is that for disk storage (see radix economy), base 3 has advantages over base 2, though base 2 has the very fast binary search. Another nice example of base 3 occurs with the Cantor set.
We talked a lot about how to use logarithms to make an analysis easier, or to exponentiate. For example, \((S/x)^x = \exp(x \log(S/x))\).
We ended with a warning that you may have been fooled into believing you learned certain derivatives when you hadn't. For example:
\(f(x) = x^n\) has derivative \(n x^{n-1}\). This follows from the definition of the derivative and the binomial theorem to expand \((x+h)^n\) when \(n\) is a positive integer.
\(f(x) = x^{p/q}\) has derivative \(\frac{p}{q} x^{p/q-1}\). This follows by setting \(g(x) = f(x)^q = x^p\) and then differentiating, which gives \(g'(x) = q f(x)^{q-1} f'(x) = p x^{p-1}\), and then substituting and solving for \(f'(x)\). We cannot get it the same was as the derivative of \(x^n\), as that would require knowing the binomial theorem for non-integral exponents.
\(f(x) = x^{\sqrt{2}\) has derivative \(\sqrt{2} x^{\sqrt{2}-1}\). This follows from using the exponential function and the chain rule: \(x^{\sqrt{2}} = \exp(\sqrt{2} \log x)).
Thus, \(x^r\) does have derivative \(r x^{r-1}\), but the proof for general \(r\) goes through the exponential function.
Twenty-eighth day lecture: http://youtu.be/UfL8o_IetYQ (May 12, 2014: Review by Partitioning)
Bonus clip on Tangent Lines: http://youtu.be/lEJ06epMLEQ
It is possible to get so caught up in reductions and compactifications that the resulting equation hides all meaning. A terrific example is the great physicist Richard Feynman's reduction of all of physics to one equation, U = 0, where U represents the unworldliness of the universe. Suffice it to say, reducing all of physics to this one equation does not make it easier to solve physics problems / understand physics (though, of course, sometimes good notation does assist us in looking at things the right way).
Sixth day lecture: http://www.youtube.com/watch?v=71_8kHSAE4w (February 21, 2014: Limits, Defn Partial Derivatives)
Wednesday, February 12. We continued our list of applications of the Pythagorean Theorem. We saw how it leads to the law of cosines, which leads to our angle formula relating the angle in terms of the dot product. We then talked about determinants, which will be really useful when we get to the multidimensional change of variable formula. The cross product will be very useful in dealing with the geometry of various functions, and occurs all the time in physics and engineering, ranging from Maxwell's equations for electromagnetism to the Navier-Stokes equation for fluid flow.