Wikipedia:Reference desk/Archives/Mathematics/2010 February 21

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February 21

How to find the domain and range of this function?

A function f(x) has domain {x e R | x ≥ -4} and range {y e R | y < -1}. Determine the domain and range of this function:

y = -2f ( -x + 5 ) + 1

Apparently, the answer is

domain {x e R | x ≤ 9}, range { y e R | y > 3}

but I'm not sure how to arrive at this answer. I don't even know where to begin! If anyone could help, I would greatly appreciate it. Could someone tell me what steps would be necessary to get to the answer? -- —Preceding unsigned comment added by 74.12.20.185 (talk) 04:46, 21 February 2010 (UTC)

Perhaps the confusion comes from the use of x and y in two separate equations. (This is perfectly valid, but it can been disorienting.) Let's restate the second half of the problem as:

Define the function g(t) = -2f(-t+5) + 1. What are the domain and range of g(t)?

The domain of g(t) are the permissible values for t. You know the domain of f(x), so you know the permissible values for the argument of the function f, and in this case the argument is -t + 5. So, what values can t be if -t + 5 ≥ -4? The range of g(t) are the possible values that the function can obtain. You know the range of the function f(x), which is the same as the range of f(-x + 5) [Does that make sense? It is still just the function f.] so what values can -2f(-x + 5) + 1 obtain if f(-x + 5) < -1?

Relevant articles are range, domain, function composition, inequalities. 58.147.58.28 (talk) 05:16, 21 February 2010 (UTC)

Integration

Suppose we wanted to calculate the work done in going around a small rectange in the xy plane with dimensions Δx and Δy, with (Δx,Δy)-->0 (the two top sides are parallel to the x-axis). So we start by looking at the work done in travelling through the paths parallel to the x-axis. This is equal to $\int _{x,y}^{x+\Delta x,y}\!F_{x}(x,y)\,dx$ + $\int _{x+\Delta x,y+\Delta y}^{x,y+\Delta y}\!F_{x}(x,y)\,dx.$ Because Δx is said to be small, F_x is assumed to be roughly constant over the interval x to x+Δx. Thus, we get W=F_x(x,y)Δx - F_x(x,y+Δy)Δx, which we say is NOT equal to zero. Thus, the we are allowing for the fact that y is different between the two paths but are ignoring the variation of x along each of the paths. In other words, we are saying that the variation of F_x isn't significant in comparison to Δy. Why would this be true? 173.179.59.66 (talk) 05:34, 21 February 2010 (UTC)

Basically it's like using just the first term of a Taylor series, which is ok when the interval is small. Suppose you want to include a second-order correction: you end up with terms like $F_{xx}(\Delta x)^{2}$ and so forth, where since $\Delta x$ is so small, its square is swamped out by the first-order term. Try writing it out that way and seeing what the limit looks like as $\Delta x$ and $\Delta y$ approach zero. 75.62.109.146 (talk) 07:01, 21 February 2010 (UTC)

Hmm, I seem to be having trouble seeing how to write the integral as a Taylor series. Can you please show me? Thanks! —Preceding unsigned comment added by 173.179.59.66 (talk) 08:08, 21 February 2010 (UTC)

I don't mean literally write a Taylor series, I just mean it's the same idea. The Taylor series says f(x+h)=f(x)+hf'(x)+(h²/2)f''(x)+... . The integral you showed is a first-order approximation, so try writing out a few of the second-order correction terms and seeing what they look like. The point is that when h is small, h² is very small. 75.62.109.146 (talk) 21:41, 21 February 2010 (UTC)

You can though. $\int _{x,y}^{x+\Delta x,y}\!F_{x}(x,y)\,dx$ = F(x+Δx,y) - F(x,y). The Taylor expansion of F(x+Δx,y) is F + ΔxF_x + Δx²F_xx/2! +..., so we get F(x+Δx,y) - F(x,y) = ΔxF_x + Δx²F_xx/2! +...

For the second integral we get F(x,y+Δy) - F(x+Δx,y+Δy) = -ΔxF_x(x,y+Δy) - Δx²F_xx(x,y+Δy)/2! -...

The sum of those two is Δx(F_x(x,y) - F_x(x,y+Δy)) + Δx²(F_xx(x,y) - F_xx(x,y+Δy))/2! +...

Then we use a Taylor series over y to get -Δx(ΔyF_xy + Δy²F_xyy/2! +...) - Δx²(ΔyF_xxy + Δy²F_xxyy/2! +...)/2! -...

All the subsequent terms are dominated by -ΔxΔyF_xy as Δx and Δy get small. Treating F_x as constant over the x interval is equivalent to ignoring the terms with Δx², which is good. But treating F_x as constant over the y interval is equivalent to ignoring the terms with Δy, which is bad because that's everything. Rckrone (talk) 22:30, 21 February 2010 (UTC)

Schlafli symbol

Should the Schlafli symbol {6/2} be interpreted as a polygon compound or as a doubly-wound triangle? 4 T C 07:03, 21 February 2010 (UTC)

It's a hexagram. That Schlafli symbol tells you to take 6 evenly spaced vertices and connect every second one. Rckrone (talk) 07:27, 21 February 2010 (UTC)

are there any free & meritorious statistics lessons/courses to be found online?

Through various quirks of scheduling, a bit of self-study, and odd luck I managed to complete a BS in Geology and an MS in Environmental Policy without ever once taking a course in statistics, not even in high school. At this point in my life it is a source of inward embarrassment and occasionally a professional impediment. As such, I'd like to fill this gap in my knowledge on my own time. I would be grateful if someone could point me towards a reputable online source for statistics instruction from basically scratch? 218.25.32.210 (talk) 08:29, 21 February 2010 (UTC)

http://ocw.mit.edu/OcwWeb/Mathematics/index.htm seems to have some. 75.62.109.146 (talk) 21:38, 21 February 2010 (UTC)

name of method for generating a circular path in x, y plane

Hi!
Does anyone know the name for the method of generating a circular path using the two equations
(1) x += y / R
(2) y -= x / R ?

I remember reading about this years ago in wikipedia and now the name of the method escapes me.
It's got to be something simple, but looking under the list of List_of_numerical_analysis_topics I can't find it again... It's based on starting from a known point (x, y) anywhere on the circle of radius R, and moving to another spot on the circle by adding a velocity vector V = ( y/R, -x/R). Actually, the velocity vector V's magnitude |V| can be any fractional component. It's a nifty way to generate sine and cosine values.

Just to clarify, R is the radius of the circle plotted. Fex, a circle of radius R = 10 would go like this...

Step	x	y
0	10	0
1	10	-0.1
2	9.9	-1.09

Thanks! --InverseSubstance (talk) 22:45, 21 February 2010 (UTC)

CORDIC Dmcq (talk) 23:22, 21 February 2010 (UTC)

You might also be interested in Midpoint circle algorithm if doing circles Dmcq (talk) 23:26, 21 February 2010 (UTC)