(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

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1 MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere. i) :,, ); r = 7 ii) x ) +y ) +x +) =7 b) Given the vectors a =i j +k, b =i +j k, c = i +k. i) alculate a b c). ii) Determine the vector projection of c onto b. iii) Find the cosine of the angle between a and b. iv) Find a unit vector that is perpendicular to the plane determined by a and c. i) ii) 4 i +j k) iii) 4 iv) i j + k Problem. Given the planes P : x ) y +) z ) =, P : 4x y +z =, and the point Q :, 7, 4). a) Determine whether P and P are parallel, coincident, perpendicular, or none of the preceding. N N = implies P P b) Find an equation for the plane through Q which is parallel to P. x +) y 7) z 4) = c) Determine scalar parametric equations for the line through Q which is parallel to the line of intersection of P and P. x = 9t, y =7 8t, z =4 Problem. The position of an object at time t is given by: rt) =e t i + e t j t k, t<. a) Determine the velocity v and the speed of the object at time t. velocity: v = e t i + e t j k; speed: v = e t + e t b) Determine the acceleration of the object at time t. acceleration: a = e t i + e t j

2 c) Find the distance that the object travels during the time interval t<ln. 8 Problem 4. a) A curve in the plane is defined by the parametric equations: x = t +,y= 4 t. i) Find the length of from t =to t =. ii) Find the curvature of at t =. i) 6 [7] / ) units ii) κ = b) The vector function rt) = sin t i cos t j + t k determines a curve in space. i) Find the unit tangent vector T and the principal unit normal N. ii) Determine the curvature of at time t. iii) Determine the tangential and normal component of the acceleration vector. i) T = cos t i + sin t j + k; N = sin t i + cos t j ii) 4 9 iii) a T =, a N =4 Problem. Let fx, y) =x ln x/y)+xy. a) alculate f xx and f yz. f xx = x, f yx =y y b) Determine the directional derivative of f at the point, ) in the direction of the vector a = i j. 9 c) Suppose that x = st e t and y =se t. alculate f t. f t = [ +lnx/y)+y ] [se t + ste t ]+ [ xy +xy ] se t d) Determine an equation for the tangent plane to the surface z = fx, y) at the point,, 8) on the surface. x ) + 7y ) z 8) = Problem 6. Let F x, y, z) =xy +yz +x z. a) Determine the maximum directional derivative of F at the point,, ).

3 maximum directional derivative: F = 44 b) Find the directional derivative of F at the point,, ) in the direction parallel to the line x =+4t, y = t, z =t. 8 6 c) Determine symmetric equations for the normal line to the level surface F x, y, z) = at the point,, ). x = +4t, y = 6t, z =+t d) Suppose the x = t +,y=t, z = t. alculate df dt. df dt = y +4xz ) t + 4xy +z ) + 4yz +x ) t Problem 7. a) Let fx, y) =x y y x 8y +. i) Find the stationary points of f. ii) For each stationary point P found in i), determine whether f has a local maximum, a local minimum, or a saddle point at P. i), ),, ),, ) ii), ) loc. max, ±, ) saddle points b) Determine the minimum value of fx, y) =x + xy y + subject to the constraint x +y = 6. f 7, ) = 7 minimum value Problem 8. a) Find the absolute maximum and absolute minimum values of fx, y) =x +y x + on the closed disk D : x + y 4. absolute min: f, ) = ; absolute max: f, ± ) = b) Find the absolute maximum and absolute minimum values of fx, y) =+x +y x y closed triangular region bounded by the lines x =,y=,x+ y =9. on the absolute min: f, 9) = f9, ) = 6; absolute max: f, ) = 4 Problem 9. a) An open-topped rectangular container is to have a volume of cubic meters. Find the dimensions

4 of the container having the smalllest surface area. length 4 meters, width 4 meters, height meters b) The temperature T at a point x, y, z) on the sphere x +y +z = is given by T x, y, z) =4xyz. What are the maximum and minimum temperatures? max temp: T min temp: T,, ± ) =,, ± ) = T,, ± ) = Problem. a) Given the repeated integral 4 x the order of integration reversed. Evaluate one of the two integrals. 4 x x cosy ) dy dx = x cosy ) dy dx. Determine an equivalent repeated integral with 4 y x cosy ) dx dy = sin 6 b) Express the area of the region bounded by the curves y =x and y = x by a repeated integral integrating: i) first with respect to y, then with respect to x; ii) first with respect to x, then with respect to y. A = x y/ x dy dx = 4 y dx dy = 4 c) Use a double integral to find the volume of the solid S in the first octant that is bounded above by the surface z =4 x y, below by the x, y-plane, and on the sides by the planes y = and y = x. V = 4 y 4 x y ) dy dx = π/4 y 4 r )r dr dθ = π Problem. a) Evaluate xy dx dy dz where T is the solid in the first octant bounded above by the cylinder T z =4 x below by the x, y-plane, and on the sides by the planes x =,y=x and y =4. 4 y/ 4 x 4 4 x xy dz dx dy = xy dz dy dx = 8 x b) Set up a triple integral in cylindrical coordinates that gives the volume of the solid in the first octant that is bounded above by the hemisphere z = x y, below by the paraboloid z = x + y and on the sides by the xz- and yz-planes. x x y V = dz dy dx = x +y 4 π/ r r r dz dr dθ

5 c) Set up a triple integral in spherical coordinates that gives the volume of the solid that lies outside the cone z = x + y and inside the hemisphere z = x y. π π/ π/4 ρ sin φ dρ dφ dθ Problem. a) Let hx, y, z) =xy i + y j yz k, and let be the curve given by ru) =u i + u j +uk, u. alculate hr) dr. 4 b) Show that hx, y) =6xy y ) i +4y +x xy) j is the gradient of a function f. Use this information to calculate hr) dr where is the curve consisting of the line segment from, ) to, 4) and the parabola y = x from, 4) to, 9). 6 Problem. a) Let hx, y) =xy i +4x y j. alculate hr) dr where is the boundary of the triangular region in the first quadrant bounded by the x-axis, the line x = and the curve y = x. b) Let gx, y) =xy + e x ) i +x y + sin y) j. alculate 4x +9y = 6. Hint: check to see if g is a gradient. gr) dr where is the ellipse c) Use Green s Theorem to find the area enclosed by the astroid ru) =cos u i + sin u j, u π. π 8