Question

The expression dy/dx = x(cbrt(y)) gives the slope at any point on the graph of the function f(x) where f(2) = 8. B. Write an expression for f(x) in terms of x.
I can't figure out what to do with that cubed root; it keeps throwing me off.

Answer 1

dy/dx = x y^(1/3)

y^(-1/3) dy = x dx

(3/2)y^(2/3) = (1/2)x^2 + c

when x = 2 , y = 8

(3/2)(4) = (1/2)(4) + c
c = 6 - 2 = 4

(3/2) y^(2/3) = (1/2) x^2 + 4

3 y^(2/3) = x^2 + 8

y^(2/3) = (1/3) x^2 + 8/3

Answer 2

And then you could simplify it out to get y = (x^2/3 + 8/3)^(3/2). Thank you so much, I went back and ran (2, 8) through and it finally worked out, thanks!!!

Answer 3

your wrong

Answer 4

Strange, I did not write that.

Answer 5

All I did was solve for y so I could get the y by itself. I need to right the expression in terms of x, so I need to isolate y completely. In order to do this, I just applied the 2/3 root to both sides of the equation. Put in different terms, I could also say that it is to the 3/2 power. I worked it through with the initial condition f(2)=8, and it worked out. So, would you put your stamp of approval on this simplification.

Answer 6

Yes, that is correct. One answer you could receive is y= (1/3)^(3/2) * (x^2 + 8)^(3/2). The 1/3 in the beginning is just taking the dividing of 3 from the other components of the equation just to make it simpler.