4. [0.38/0.76 Points] DETAILS PREVIOUS ANSWERS SCALCET9 16.XP.3.008.MI. Consider F and C below. F(x, y) = xi+ y j, C is the arc of the parabola y = 4x-from (-1, 4) to (0, 0) (a) Find a function f such that F = Vf. 3 f( x, y) = (b) Use part (a) to evaluate - vf . dr along the given curve C. c 21 X5. [0.38/0.76 Points] DETAILS PREVIOUS ANSWERS SCALCET9 16.XP.3.009. Consider F and C below. F (x, y) = 7xy2i + 7x2yj C: r(t) = (t + sin\\_ nt ), t + cos - nut Outs1 2 (a) Find a function f such that F = Vf. 2. f ( x, y) = 7x .2 + c 2 (b) Use part (a) to evaluate Vf . or along the given curve C. 21 2 X9. [0.19/0.76 Points] DETAILS PREVIOUS ANSWERS SCALCET9 16.3.035. Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous first-order partial derivatives, then the following is true. ap aQ Op OR aQ OR ay dx Oz dx az ay Since F is conservative, there exists a function f such that F = Vf, that is, P, Q, and R are defined as follows. (Enter your answers in terms of f, f, and f.) P= X Q = X R = X OR says that ap aQ op OR =fry = fyx =fx= = fzx and - fuz = fzy = v dz ox az ay Since P, Q, and R have continuous first-order partial derivatives, | Clairaut's theorem ay Ox