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Sun, I just wanted to add a comment since Roman didn't even respond to the method you tried to use. Your idea was to guess that the solution is a power function (xa). You produced the equation x^2((a^2-a)(x^(a-2))+4x(ax^(a-1))+2x^a=0 The key idea is that this equation...

Hi Sun, You can see why this limit fails to exist if you consider approaching the origin (0,0) along different trajectories. First consider approaching the origin along the linear trajectory y=x. Under this constraint, substituting in y=x, the function becomes 3x3/(x4+x2) What happens...

Hi Corrine, The notation in your question is ambiguous. It's not clear whether you mean cos(2x)-cos(6x)=0 or cos2x-cos6x=0 These equation are very different and have different solutions. I'm guessing that you meant the second equation. George's approach works to solve the first equation,...

Hi Jill, You can make some progress if you use the double angle formulas for sine and cosine and the sum formula for cosine. The double angle formula for sine says the first term on the left-hand side is sin(2x)=2sin(x)cos(X) The sum formula for cosine says cos(a+b)=cos(a)cos(b)-sin(a)sin(b) You...

Hi Sun, There's a shorter and a longer answer to this question. The short answer is that this function f(x,y) is continuous because of the following facts: Every polynomial function is continuous. The sine and cosine functions are continuous. The composition of two continuous functions...

Hi again Sun, It looks like what you're bad at is understanding how to set up the limits of the integral. You need to think about how to describe the region in space defined by the boundary conditions -- the region enclosed by the given plane 3x+2y+z=6 and the coordinate axes. The...

It depends on what you mean by "elementary." There's a quick proof if you use the two facts that (1) the angles bisectors of a triangle intersect at the incenter, the center of the inscribed circle, and (2) tangents drawn from a point external to a circle are congruent.  (To...

Hi Sun, You're on the right path. To calculate this flux integral, you should first realize that parametrizing the surface Q would be hard, so it's better to try to use the Divergence Theorem and instead compute the integral of the divergence of F not over Q but over the volume enclosed by...

Hi Caity, Unfortunately neither Tamara nor Jeremy gave you a correct answer, much less a correct explanation. There are two ways you can solve this inequality: 1) You can interpret the problem as asking when f(x) is less than or equal to 0, where f(x) is the function f(x)=(x+7)/(x-1)...

Roman is right that the answer is very close to (1000000000000000000001)20 + 1/2 = (1021+1)20 + 1/2 In particular, he's right that the error is very small, much less than 1/2. But you can be even more precise using the Arithmetic Mean-Geometric Mean-Harmonic Mean (AM-GM-HM)...

Hi Marisol, I'll try to help get you started, but I don't want to do the entire assignment for you. The Fundamental Theorem of Algebra says that every polynomial of degree n (meaning the highest power of x is n) has exactly n complex roots, if you count repeated ones separately. What...

Hi Joseph, You may be interested to know that there is a general formula for condensing the sum of a sine and a cosine wave into a single sine or cosine wave. I'll show you how to condense such a sum into a single sine wave, but the derivation for a single cosine wave is exactly the same. Goal:...

Hi Kayla, Kevin and John are right that there are no solutions to this system. But using substitution to see that is a little weird here, even though the problem asked you to use substitution. Here's a faster way to see the system has no solutions: The first equation just tells you x-y=0...

Hi Lar, Mykola is right that you can get the answer to this problem using the distance formula -- but simply plugging-and-chugging numbers in a formula doesn't teach you anything. Why does the distance formula work? Where in the world does that weird expression come from? To see why the...

At room temperature (i.e. everyday temperatures), definitely not! You can't pour glass into a container and have it conform to the shape of the container; that's the essential characteristic of a liquid. But if you were to heat glass up to a very high temperature (around 1500 degrees C), it would become...

Robert's answer didn't really help, I imagine; all he did is list the rows of Pascal's Triangle for you. But how is Pascal's Triangle related to your problem, the problem of expanding (a+b)6? It turns out that the numbers in the nth row of Pascal's Triangle tell you the coefficients...

Hi Alezzo, The statement 12=-1x +1 is called an equation; it tells you that the number on the left-hand side (12) is equal to the number on the right-hand side (-1x+1). "Solving" this equation just means finding the values of x that make the equation true. What in the world...

Hi Faith, As I mentioned in my comment above, the notation "f^2" is ambiguous. I'm going to do the problem two ways, depending on what "f^2" means. But since you said in the title of your post that the problem is a composition problem, I'm assuming the black dot means composition...

Hi Olga, You're right that the equations you gave define a function T from R3 -- this is the domain, consisting of ordered triples (x1, x2, x3) -- to R3, the codomain, consisting of ordered triples (w1, w2, w3). (In mathematical shorthand, this is written T:R3 -> R3.) You don't need...

Robert's answer is right, but he uses a clever trick to rewrite the form of the function. You can also analyze the problem by taking the derivative. By the quotient rule, we get f'(x)=(x2+a)(2x)-(x2)(2x) all divided by (x2+a)2 But the numerator simplifies to 2x3+2ax-2x3, which...