calculus 2, integral, double integral, derivatives, Exercises of Calculus

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(b) (c) (d) Solutions to Exam 2 ‘ 3 L (25 points) Let T(% y, z) = Pry ee” (x,y,z) ahs (0, 0,0). (a) Find the equations of the tangent plane and normal line at the point (—2,1,0) to the surface F(e, y, 2) — 5 Solution: We compute oy, z)= ___ ot oF z)= by oF 1 z)= 6 On Gaye zee? By Gay pee 92°" ~ "Gay ys ep At (—2,1,0) we have oT 12 OT 6 OT —2,1, — 1, —(—2,1,0) =0. Bq (210) = 35> Ge(-21,0) =—pe, GE (-2,1,0) =0 Therefore the normal line which has the direction of the gradient is given by 12 (2, y, 2) = (—2, 1,0) + ¢(5:, 55 ), teER. The equation of the tangent plane is 32(x + 2) — #(y— 1) + 0(z —0) =0 or 2x -y+5=0. Find the rate of change of T at (1,2,2) in the direction toward the point (2,1,3). Solution: To compute this directional derivative, we must first compute the unit vector u with the given direction. This is given by (2,1, 3) — (1,2, 2) 1 "=12.1,3)-(.2.2 var Then 2 4 4 1 10 D,T(1, 2,2) = VT(1, 2,2) > u = (-=, -=, -=) - &0.,-1,1) = -— =. (1,2,2) (152.2) 0 = (95) o9 oy)” =o Show that at any point (except the origin) the direction of greatest increase is given by a vector that points towards the origin. Solution: The direction of greatest increase at any point is given by the gradient, hence it is - 6a 7 by _ 6z —_ 6 (sy,2) (a? + y? + 22)2’ (a? + y? =3 zy?’ (x? + y* + z*)? na (x? + y? = a} TY; Note that the vector (x,y,z) is the posaan vector that points away from the origin, but since the scalar in front of the vector — -@ayeyt is always negative, the vector will point towards the origin. Let A := {x y) € R?: T(z, y,0) = sh, Use the implicit function theorem to find the points in d A where y can be expressed as a function of x, i.e. y = y(x); at those points compute = Solution: The equation T(x, y,0) = ; is equivalent to x? + y? = 9. Take f(x,y) = 2? + y* and we see that its partial derivatives are continuous everywhere and f,(x,y) = 2y 4 0 everywhere where y 4 0. This means that y can be written as a function of © everywhere, except for y = 0 (i.e. for the points (+3,0)).