Quick Math Problem?

Question

Beginning of Calculus:

f(x) = x^3 - 225x

Instructions: Determine an appropiate viewing rectangle on a Calculator. In other words I'm looking for domain and range. My problem is, I've been away from Math for a bit, thus I'm rusty on some of the skills. When it comes to looking for the domain and range (x/y min and max), would I want to use the intercepts to do so? How else would one go about stating the appropiate domain and range, or "viewing rectangle" for this guy. Thanks a lot.

Camaren

Answer 1

Since it factorizes as

x^3 - 225x = x(x^2 - 225) = x(x - 15)(x + 15),

you'll want the x-domain to go at least between something like - 20 to + 20, in order to show all three zeros of the function, which are of course at x = - 15, 0, and + 15.

Then there's the question of including the TURNING POINTS of the function:

df/dx = 3 x^2 - 225, so that the relative minima or maxima are at

x^2 = 225/3, that is x = +/- 15/√3 or +/- 5√3.

For x = - 5√3, f(x) = - 125*3*√3 + 225*5√3 = 750√3, while

for x = + 5√3, f(x) = + 125*3*√3 - 225*5√3 = - 750√3.

Thus you want the range of the function to at least encompass from - 750√3 to + 750√3. That's a VERY LARGE RANGE, from - 1299.04... to + 1299.04... !

Note that if x goes from - 20 to + 20, f(x) will lie between +/- 3500, perhaps a bit too big. Let's try x between +/- 18: then f(x) lies between +/- 1782. Even x between +/- 17 only gives f(x) between +/- 1088, not really enough to show that the function is heading off beyond the previous limits of its turning points. One last shot: x between +/- 17.5 gives f(x) between +/- 1421.875. So I'd go for something like the x-domain between +/- 18 (and +/- 20 at most), and the y-range between 2000 or so, and 3500 if necessary.

[One last shot: x between +/- 17.5 gives f(x) between +/- 1421.875. Now THAT looks like a reasonable value compared to the extrema of +/- 1299.04..., so how about x limits of +/- 17.5 and y limits of +/- 1400 or 1500?]

At this point, however, it's more a matter of aesthetic judgment as to what looks nice.

Good luck!

Live long and prosper.

P.S. Note that the now edited but inconsistent solution below completely fails to address how large f(x) will become in that proposed HUGE domain of x between +/- 30. You'd need a range between +/- 20250 to plot all that, whereas the suggestion is to use +/- 30.

Answer 2

x^3-225x =x(x^2-225) = (x -15)(x+15)

so roots are -20, 0, 20.
So I would set x-min to -30, x-max =+30 and xSCl to 2.
Then set the y values to the same as for x values.