Recent content by drewfstr314

I'm trying to pull some old Olympiad questions for some students, but I can't get a handle on this one. I'd really like to include it, though. In answering general knowledge questions (framed so that each question is answered yes or no), the teacher's probability of being correct is A and a...

f(x) = \cos x (\sin x + \sqrt{\sin^2 x + \sin^2 \alpha}) Well, to find the inverse, switch x and y, and solve for y = f^-1(x): x = \cos y (\sin y + \sqrt{\sin^2 y + \sin^2 \alpha}) I think the first step is to get rid of the square root by isolating it: x\sec y - \sin y =...

My advice would be to break it down and look at each part individually. Another strategy that probably won't work here is that the range of f(x) is the same as the domain of f^-1(x) [f-inverse]. Since domain is easier to find, that might work.

Our textbook defines an exponential function as f(x) = ab^x. However, a question was brought up about a function, g(x) = 5^sqrt(x). Is g an exponential function? It looks like an exponential graph for x>0, but is not continuous on R. Thanks in advance!

If one is solving a modular equation: 4k \equiv 1 \: (\text{mod } n) with n even, known, for k, then one needs to find the inverse of 4 modulo n: 4x - 1 = nc 4x - nc = 1 But this only has solutions iif (4,n) = 2 (n is even, but not a multiple of 4), which doesn't divide 1, so...

Assume A is positive. By the Additive Identity Law,we have A - A = 0 From the Definition of Subtraction, we have A + (-A) = 0 Multiplying by a positive B, we have B(A + (-A)) = 0B Distributing, AB + (-A)(B) = 0B By the Converse of the Zero Product Property, AB + (-A)(B) = 0...

I have seen double sums, but have come across a problem involving sums over primes. However, this sum is inside a second sum, and is taken over all primes that divide the outside index, like this: \sum_{k=1}^{n} \sum_{p | k} \frac 1p for p prime. Is there any way to manipulate this...

This is only true in the sense of limits. While it is not true to say that \frac{x}{\infty} = 0, it is correct to say: \lim_{n \to \infty} \frac{x}{n} = 0 .

Your comment about one solution threw me off. We had gone over the problem (graphically, of course) in class, and there were two solutions. I then noticed that I had typed the wrong equation, and simply copied it from my first post. I switched the signs: \sqrt{7-x} = \frac{x^2}{2} + 12x - 10...

The original equation, -\sqrt{7-x} = -\frac{x^2}{2} + 12x - 10 has two solutions, at x1≈0.996 and x2≈-25.242, but the quartic has two extra solutions: at x3≈0.607 and at x4≈-24.361. These last two are extraneous solutions, but x1 and x2 are also solutions of the quartic.

On a math test, one of the questions was to solve -\sqrt{7-x}=-\frac{x^2}{2}+12x-10. I solved graphically with a calculator, but later tried to solve algebraically, when I had more time. The equation is equivalent (with extraneous solutions) to x^4 + 48x^3 +536x^2 -956x + 372=0. This quartic has...

I guess what I was looking for was something like this: f(x) = x^3 - 4x + 2 \Rightarrow f'(x) = 3x^2 - 4 and the solutions of f'(x)=0 are x = \pm \frac{2\sqrt3}{3} Based on this, is there any way to find the zeros of f(x)? Thanks!

Is there any similarity between the zeros of a function and the zeros of its derivative? That is, if A = set of all x such that f(x) = 0 B = set of all x such that f'(x) = 0 then is there any pattern to finding A if B is known (or vice versa)? Thanks!

Is there any way to simplify \sum_{n=1}^{\eta} \left(\sum_{p|n} \frac{1}{p} + \frac{1}{\prod_{p|n}p} \right) for a known η, and where "p|n" is a prime that divides n, i.e. p is a factor of n? Thanks

Is there a general algebraic way to write the quotient of two definite integrals as one? I mean, what would be \frac{\int_a^b f(s) ds}{\int_c^d g(t) dt} Is it analogous to the product of integrals creating a double integral? Thanks in advance!