. Find an equation for the plane containing the points P1, P2, and P3. (a) P1 =(1,1,1),13'2 = (1,2,2), P3 = (1,1, 1) (b) P1 = (2, 1,3), 132 = (1,1,1), P3 = (1,2, 3) . Let 7 : R > R3 be a parametric equation for a curve. Find a parametric equation for the the tangent line to the curve at the point 7(a)). (a') '7\") = (2t2t21 _3t3); t0 = 1 (b) 705) = (cos(t),sin(2t), ct); to = g . Sketch the cross sections for the equation f (2:, y, z) = c. (a) f(:c, y, z) = x2 + 312 Z2; 0 = 0; Cross sections: as = 1, 0,1,2 (b) at, 3;, z) = 23:2 + y2 + 322; c = 4; Cross sections: at = 1,0, 1, 2 . Convert the point (5c,y,z) = (1,1, \\/3) into cylindrical coordinates (136,2) and spherical coordinates (p, ()5, 6). . Describe the following region in three dimensional space using spherical coordinates: 220; x2+yzsz2; 222+y2+z2 S4. . Let f (2:, y) = iii132$. Show that lim f($,y) does not exist. (aw-HOB) . Find an equation for the tangent plane to the graph of the function f (2:,y) at the point P. 10. 11. 12. 13. 14. (a) f(-'I=a y) = 393 + 2y; P = (1: 2) (b) f(:l:,y,z) = 311:2 + 21:3; P = (1, 2) (6) at, :9) = Goa-'31!) + 832; P = (1: % Compute the (normalized) directional derivative of the function f in the direction of the vector v at the point P. Also nd the direction and rate of maximal increase of f at the point P. (a) ay) = 8\"\" + Saint? - 2y); I! = (12); P = (0: t) (b) aw) = 932 +29!2 2; v = (1,2,3); P = (1,1,2) Consider the level set of the function f passing through the point P. Find an equation for the tangent plane of the level set at the point P. (a) f($:yaz) =3m2+2y3z; P:(112:19) (b) 1:03:93 2) : 332 +93 _ Z2; P = (1: _1:\\/) (C) f(:L',y,z)=m2+y2z;P=(1,1,2) A ball is thrown from the origin (0,0, 0) with an initial velocity of on = (1, 1,2). Assuming that the acceleration of the ball due to gravity as a functions of time is a(t) = (0,0,g) (you can take 9 = 101rn/s2 for convenience), where does the ball land? (We are taking the ground to be the Qty-plane.) Where does the ball land if we double the initial speed? Are there vectors u of length 1 and vector v of length 2 so that the vector u x v has length 3? Let r1 = (t, tg) and r2 2 (2t, t2+1) be parametric curves describing the trajectories of two objects. (a) Compute the distance between the two objects at t = 3. (b) Show that the trajectories of the objects intersect, but show that the two objects do not collide. (c) Adjust the speed of 1-1 so that the two objects collide. (d) (Continuation of last part) What is the angle of impact? what is the speed of the two objects upon impact? (Choosing the order for computing higher partial derivatives.) Consider the function f(:c,y) = sin(m%) + my. Since 3' is innitely differentiable, fzy = fyx. However, computing the partial derivatives in one order is a lot of work and the other is not. Try to compute the mixed partial derivative in both orders. (Always keep on the look out for an easier order.)