The vector field F(x, y)= is conservative. Find such that F= ∇f

f = tanx *i   + y³*j

Step-by-step explanation:

F(x,y) = ??

Fx,y) = gradf

Then

f(x,y)  = ∫ sec²x dx  + ∫ 3y²dy

∫ sec²x dx    =   ∫  1/ cos²x  dx

By change of variables

if  z  =  tan x      then   z  =  sinx/cosx

dz/dx = cosx*cosx - (-sinx*sinx) / cos²x

dz/dx = cos²x  +  sin²x /cos²x

dz/dx=  1 /cos²x

dz = dx/ cos²x

∫  1/ cos²x  dx    =   ∫dz = z = tanx

And  ∫3*y²   = y³

Then  f = tanx *i   + y³*j

the formula for the equation of a circle is (x – h)2+ (y – k)2 = r2, where (h, k) represents the coordinates of the center of the circle, and r represents the radius of the circle. if a circle is tangent to the x-axis at (3,0), this means it touches the x-axis at that point.