Question

Find an equation of the tangent plane to the following parametric surface, r(u, v)  =  (u2 + 9) i + (v3 + 6u) j + (u + 2v) k , at the point (10, 5, −1). Write the equation in the form ax + by + cz + d  =  0, where a, b, c, and d have no common factors. Then enter the values of a, b, c, and d (in that order) into the answer box below, separated with commas

Answer 1

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Here are the given ruu) = (479)i + (W²+64)J +14+20) ī Let's first compute ru xro. two individual vectors rucu,o) qui tout 2014,-) 30²J + 2F will the cross product (which nomal vector ng is give us the Now 1 J 31 k 24 6 362 2 - 4 =) (19-36%) -J(44) + F(646). now, this is all time, but in order to use it will need to determine the value of ufo that will give us the point in question, we can easily do this by setting the individual components of the parametric representation equal to the coordinates of the point in question. doing this gives =) u=1 83 64 -) -) +1+20=-1 = 20 = -1 a u=o o=-) too posible values. To determine the correct value of ute into the third equation. u=-1, u=1 satisfies the eq 63 +6420 required utl values are 1,-). untq = 10 = 5 1993 - ut 20 here uf we have then

qi -uJ + GF : required tangent plane equation is 968-10) - 4 (7-5)+617 +1)=0 =) qx-90-49 +20 +62 +6=0 =) 94-444 62-64=0 ia=9, b==4 c=6, d =-64