* Partial Derivatives * Find the first partial derivatives of the function. F (x, y) = cos (et) dt 25 Answer F. (x, y) = cos (e"), Fy (x, y) = - cos (ev) 38 / (2, y) = ysin (xy); fy 1, 4 4 yse implicit differentiation to find ax/ Oz and as/ dy. . yz + x ly = z2 If f (x, y, z) = xy?23 + arcsin (xvz), find fory. [Hint: Which order of differentiation is easiest?] Answer 6yz Van der Waals Equation The Van der Waals equation for n moles of a gas is 85 + V 2 ) ( V - nb ) = n RI where P is the pressure, V is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are positive constants that are characteristic of a particular gas. Calculate OT/OP and ap/av. Answer + Or V - nb OP 2n a n.RT Op nR ' av V/3 (V - nb)2 93 The ellipsoid 4x2 + 2y2 + 22 = 16 intersects the plane y = 2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 2). Answer + x = 1+t, y = 2, z = 2 - 2t