Calc 4 review! I need help with any of all of them!
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Justify your work, communicate clearly and mathematically. calculation not allowed. Unless decimal approximations are requested, numerical answers must Label your problems clearly and mark the work you want graded/not graded; also, box in your answers. Using techniques from this class, approximate the value of exy - cosx at (.1,2.3) Consider the vector field F(x, y) = (cos(sin x + y) cosx ex, cos(sin x + y) + y) the work done as you traverse the Archimedes spiral (r = Theta) from (x,y) = (0,0) to (x,y)= (2 pi. 0). (Hint: check to see if the vector field is conservative.) Find the absolute maximum and minimum of y ex-z on the ellipsoid9x2+4y2+36z2=36 Compute Consider the region bounded by 2 = 4 - x2 - y2 and 2 = 0. Let S be the boundary of this region. Compute the flux of F = (xy2z - xz, yz + ev, - zev) across S. If F represents a fluid flow, explain what the above computation means. First, sketch and describe the surface 5: r(u. v) - (ucosv, usinv.v,v), where u Second, let f(x. y. z) be the distance function from the point (x. y, z) to the z - axi% Integrate the function f over S. Pick 2 of the following and solve. Compute fc(:2 + yeI)dx+ (ez + yz)dy + (2xz + y2/2 )dz along the curve C: r(t)= 5 cost, sint, cost, sint). > Consider the curve coordinates this is just the wobbly circle r = 3 + cos St). Compute where F - (-y,x)/(x2 + y2). State and prove Clairaut's theorem. State and prove the chain rule. State and prow Green s theorem for simply-connected regions. Bonus if you can prove Green's theorem for non-simply-connected regions