## anonymous 5 years ago Consider the function f(x) = x - x^3. determine one or more horizontal shifts that will change its form so that no linear term is present.

1. anonymous

hmmmm

2. anonymous

Send x to x-c, collect terms and solve for c such that the coefficient of the linear term is exterminated.

3. anonymous

$f(x-c)=(x-c)-(x-c)^3$$=(c^3-c)+(1-3c^2)x+3cx^2-x^3$If you let $c=\pm \frac{1}{\sqrt{3}}$and sub. in, your linear (x) term will disappear.

4. anonymous

how did you know that?

5. anonymous

Translating a 1D function to the left or right amounts to shifting the argument by a constant. Just look at y=x^2 versus y=(x-1)^2.

6. anonymous

right, so you moved it so that the roots changed

7. anonymous

The question asks for a horizontal shift that will eliminate the linear term.

8. anonymous

ok , the original function has 3 roots. so shouldnt you move it up vertically ? oh wait... by translating it the roots will average out or something

9. anonymous

ok i see what you did, i got confused because it says shifts* plural , but that was clever

10. anonymous

It's just an application of the definition of 'horizontal shift' along with the condition stipulated in the question that the shift is wanted so that the linear term is extinguished.

11. anonymous

It's a bizarre question.

12. anonymous

you know anything about gabriel horn

13. anonymous

the paradox is that there is no bottom to the horn, so in theory you can keep adding say water to it (ideal water) . but the volume is finite

14. anonymous

ideal water (quantum molecular issues aside)

15. anonymous

It is indeed a weird question, but a similar technique can be used to reduce, for example, a quartic equation to one with no cubic term which makes it easier to solve in some ways. Sounds little random but that's just about application of this thing I've seen before.

16. anonymous

Yes

17. anonymous

Same with suppressing a cubic.

18. anonymous

here is another crazy integral. if you find the surface area of y= ln x from 0 to 1 (so its below x axis), and revolve it about y axis , you get a finite surface area (even though y = ln x goes to negative infinity)

19. anonymous

its called depressed equation , but what you did is straightforward :)

20. anonymous

actually there might be another way, buts kind of long. you can use the symmetric polynomial reduction form . so x^3 - (a+b+c) x^2 + (ab +ac +bc)x - abc , where a,b,c are your roots

21. anonymous

yes, depressed.

22. anonymous

so you want ab + ac + bc = 0

23. anonymous

nevermind, you have it :)

24. anonymous

On the bright side, I like discussions on here with people who can do Maths, and don't spam exactly the same question over and over and do nothing themselves - makes a nice change.

25. anonymous

yup

26. anonymous

newton can you look over my solution for work problem, im stuck on the density g thing