Question

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.

Answer 1

hmmmm

Answer 2

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

Answer 3

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

Answer 4

how did you know that?

Answer 5

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.

Answer 6

right, so you moved it so that the roots changed

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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.

Answer 11

It's a bizarre question.

Answer 12

you know anything about gabriel horn

Answer 13

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

Answer 14

ideal water (quantum molecular issues aside)

Answer 15

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.

Answer 16

Yes

Answer 17

Same with suppressing a cubic.

Answer 18

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)

Answer 19

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

Answer 20

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

Answer 21

yes, depressed.

Answer 22

so you want ab + ac + bc = 0

Answer 23

nevermind, you have it :)

Answer 24

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.

Answer 25

yup

Answer 26

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