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Math 173 Final Review Exam 1: gray, Exam 2: buff, Exam 3: yellow Exam 4: one version (solutions on website)Find the distance from the point (4, -4, 3) to the plane 2x – 2y + 5z + 8 = 0. Ans: 3933A block of 100 N is suspended from a rope tied to two other ropes. One rope is horizontally attached to a wall and the other is fastened to the ceiling. The angle between ceiling and the rope is 60°. What are the tensions in the ceiling rope and the wall rope? For #3 A rectangular box has dimensions of 1 inch, 2 inches, and 3 inches as shown.a) Find the length of the diagonal as shown (from lower back left to upper front right)b) Find the angle the diagonal makes with the left-hand edge marked “x”.For 2 lines with parametric equations:Line 1:x = – 3tLine 2: x = –1 + 6t a) Show that the lines are parallely = 1 + ty = –2tz = 3 – 2tz = 2+ 4tb) Find an equation of the plane containing both linesFor the quadric surface x2 – y2 + 4z2 = 4a) Draw traces of the surface for each of the coordinate planes.b) Sketch and name the surface in ?3. a) Convert the point (5, 30o, 45o) in spherical coordinates to cylindrical coordinates. b) Convert (5, 30o, 45o) in spherical coordinates to rectangular coordinates.c) Find inequalities in cylindrical coordinates that describe the solid region above the plane z = 3 and below the surface z = 25-x2-y2For the curve rt=lnt, t3, cos?(πt) a) Give the domain of the curveb) Find the equation of the line tangent to the curve at t = 1, writing your final answer in parametric form.a) Find the point where the plane 4x + y – z = 4 intersects the line x=y+12=z-13 b) Find the point where plane 4x + y – z = 4 intersects line x=y=z5 and give a geometric interpretation.The acceleration of a particle is given by a(t)= sint, et, 3t2 . Find the velocity function, v(t) if the initial velocity is v(0)= -2, 5, 3 Find ?z?x and ?z?y for the equation z=yz+x lnyDetermine whether the following limits exist, and prove your conclusion.a) x,y→(1,1)lim x-yx2-y2b) x,y→(0,0)lim 5xyx2+y2a) For fx, y=3x2+2x-y2 , find the equation of the tangent plane at point (1, -2, 1)b) Use the linearization of f to approximate f at the point (1.1, -1.9)Find and identify (local max, local min, or saddle point) all critical points of fx,y=x3-3x+3xy2 a) Draw the region bounded by the curves y=4 , y=12x, and x=0.b) Find the absolute maximum and absolute minimum of the function fx,y=2x-y over the region above.A rectangular plate in the x-y plane is has temperature readings for 0 < x < 8, and 0 < y < 6, x & y in meters, in gray strips. The table is aligned with the first quadrant.65162687580445607280862385674879003052789896y x02468a) Estimate the values of the partial derivatives Tx 6,4 and Ty 6,4 b) Write the gradient vector, ? T c) For the unit vector u= 12, 12 find DuT6,4, the directional derivative of u at the point (6,4).d) Interpret the meaning of Tx 6,4 and Ty 6,4 and DuT6,4Evaluate the integral 02x24 xey2 dy dxLet r(t)= -2cost, sin?t a) Find rπ6 and Tπ6 the position and tangential vectors at the point where t= π6 b) Find the curvature at the point where t= π6c) Draw a rough sketch of r(t)= -2cost, sin?t and show the vectorsrπ6and Tπ6 in your sketch. Use LaGrange multipliers to find the maxima and minima of fx,y=x2+2y2-4y subject to the constraint x2+y2=9Evaluate the integral D y?(x2+y2) dA, where D is the half circle enclosed by y=4- x2 and the line y=0 a) Find the mass of a triangular lamina with vertices at (0,0), (0,2) and (2,0) if the density function is proportional to the distance from the y-axis. Include the proportionality constant, k. b) Using the mass from part a), set up the integral for x , the x-coordinate of the center of mass. Include limits of integration, and simplify constants, but DO NOT INTEGRATE (no bonus for this problem). For the integral 02x24 02-y/2fx,y,z dz dy dx find the correct limits for the new orders of integration below, where V is the volume defined by the original integral. DO NOT INTEGRATE.a) Vfx,y,z dz dx dy b) Vfx,y,z dx dy dz c) Vfx,y,z dy dz dx Set up the integral needed to find the surface area of the part of the half-cylinder x2+z2=1 that lies above the rectangle 0≤x≤12 and 0≤y≤1 in the x-y plane. DO NOT INTEGRATESet up the integral with appropriate limits. but DO NOT INTEGRATEE(x-y) dV , where E is the solid that lies between the cylinders x2+y2=1 and x2+y2=16 above the x-y plane, and below the plane z=y+4 DO NOT INTEGRATE. Convert to spherical coordinates, showing correct limits of integration:Esin((x2+y2+z2)3/2) dV , where E is the solid bounded above by the sphere x2+y2+z2=1 and below by the cone z= 3x2+3y2 A thin wire is bent into the shape of the semi-circle x2+y2=4 for y≥0 . The linear density of the wire at any point is proportional to the distance from the point to the x-axis. a) Write a function for the linear density, ?(x,y)b) Find the mass of the wire.DO NOT INTEGRATE: Set up the integral: Rx-2y3x-ydA where R is the parallelogram enclosed by the lines x – 2y = 0, x – 2y = 4, 3x – y = 1, and 3x – y = 8. Use the transformation u = x – 2y, v = 3x – y For Fx,y,z=y2, 2xy+1, z2 a) Is F a conservative vector field? If so find a potential function, f(x,y,z)b) Evaluate CF?dr for F above, using any method, where the curve C is C: x=cost, y=sint , z=t 0 ≤t≤π2 (all 3 coordinates)Evaluate using any method: C1+x2 dx+2xy dy where C is the counterclockwise path traced between the vertices (0, 0), (1, 0), and (1, 3), returning to the origin.For F(x,y,z)= x2y, y2z , xy2 a) Find curl F (0, 1, 5)=?×F (0, 1, 5). b) Find div F (0, 1, 5)=??F (0, 1, 5). c) What is the direction of rotation of F at (0, 1, 5), if any? Let ru,v= ucosv, usinv, va) Write an equation of the plane tangent to r at the point (-1, 0, π)b) DO NOT INTEGRATE: Set up an integral that would give the surface area of r for 0≤u≤2, 0≤v≤π , using an appropriate coordinate system.Evaluate CF?dr; F =-12y2, x, z2. C is the intersection of the plane y + z = 4 and the cylinder x2+y2=1 Find the surface integral of F over the boundary of E if Fx,y,z=x2, z2-x, y3, and E is the solid bounded by z=x2+y2, and z=2-x2-y2. Spherical coordinates recommended. #31 Answer: ? # 32 Answer: 0 ................
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