Math Is Fun Forum

It is indeed - it is also a generic term for superfluous things ("Etiquete? It's, you know, how you fold your cutlery at bed-time and all that dross"), not to mention a most excellent song by the Smashing Pumpkins.

Hey there,

I would really like to post some exercises at some point - do you think there's any that are particularly needed? Could it be useful to have a thread where people make suggestions of the types of exercises they would like to see?

Also, what level is this forum aimed at - any level of math?

I was thinking of doing some exercises on complex numbers, as well as the basics of calculus (limits, continuity, etc), matrices and polar coordinates. Would these be appropriate?

Let me make sure I understand you correctly - when you say the "differential operator", do you mean, for example, like the "d"s in

You can get a much nicer solution to (1) if you write h(x) as (x-2)[(x-1)^-2] and differentiate using the product rule:

So in this case, a = n = 3.

As for (2), the point of inflextion occurs where h''(x) = 0.

For the last question, since you know that x = 4 is a solution you know that the expression for f(x) has a factor of (x-4) in it. So it can be re-written as:

for some a, b, c, d. Find these and then solve

"The tangent of f at x = 3 is horizontal" is just telling you that f'(3) = 0. Given this, find f'(x), use the method I just mentioned to take out a factor of (x-3) and you should be left with a quadratic to solve to get f'(x) = 0.

Am at work, so will only answer question 2 for now. First, I direct you to my very crude drawing: .

Now, the north pole lies along the y-axis, and the distance that the rocket has to travel is x. Using basic trig:

The rest is easy. (and remember, cos(90-x) = sin(x)).

...but in that case, it would make sense for the question to ask for the distance traveled in some frame of reference (ie of the fly, of the ground etc) and it doesn't. Also, at 50 and 75 km/hr, relativistic effects are small to say the least.

That doesn't make sense to me... which quantities? 60,000,000 is already a real number...??? Is that all the question states? Is there anything else in the text that may clarify what it means?

Well, since I'm new around here and you asked so niceley, here are some semi-solutions (I don't like giving the whole game away, it spoils all the fun )

Anyway...

1. A) Turn the equations into their matrix form:

(1 -6 2 5)
(2 -9 -1 14)
(4 12 -3 19)

and use elementary row operations (e.g. start with Row2 -> Row2 - 2*Row1, etc) to reduce to echelon form. You should then have a last row like (0 0 <something> <something else>), which means you have a simple equation for z. use this value in the middle row to get y, and this value in the first row to get x and you've got your solution.

C) When you do the row operations, if one of the rows is some integer multiple of another, you have infinitely many solutions (since they are essentially the same equation, so you have three variables and two equations, leaving one free variable that can take any value it likes).
Use one of the rows with two variables to obtain an equation for one in terms of the other (e.g. y = 4z +2) and solve the other equation using this value. What you should end up with is two equations that give two of the variables in terms of the other (e.g. x = 2z +4, y = 5z - 7).

D) If you do some row operations, you should find that one of the rows becomes something like 0x + 0y + 0z = <something>.

2. A) Let f(x) = x[sup]3[/sup] - 3x[sup]2[/sup] + 2x +1. Now, if you can show that f(x) must cross the x-axis between x=-1 and x=0, it must have a solution (a point where f(x) = 0) between -1 and 0. Newton's method is just number crunching once you know what numbers to crunch.

B) z-w and zw are just as they look, remember when you work out z/w, it's always a good idea to multiply the numerator and denominator by the complex conjugate of the denominator, in this case -1-2i.

D) If one solution is (-1-5i), then another will be (-1+5i), the complex conjugate of the first. You then have two factors of the function, namely (z + 1 - 5i) and (z + 1 + 5i). Multiply them together and find what other term you need to use to get to your original answer, and you're home and dry.

If you need any more help, feel free to ask.

Dross