1. Sketch the following set of points in the z-y plane: {(x, y) € R2 :(y...
Consider the following subsets of R2:
C1 ={(x,y)∈R2 :x+y≤3,x≥0,y≥0}
C2 ={(x,y)∈R2 :4x+y≤4,x≥0,y≥0}
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
Draw the set of all points (z, y) E R2 satisfying the following equations: 5. xy 0
question #6
1. Sketch the following surfaces: (a) z-+y2/9 (b) a2 =y2 +22 (c) 2/4+(y-1)2+(z+1)/9 1 (d) r2+y-22+1 0 (e) -y2+-1 0. 2. Find an equation for the surface consisting of all points that are- point (1,-3, 5) and the plane r = 3. 3. Sketch the curve F(t)<t cos(t), t sin (t), t > 4. Find a vector equation that represents the curve of the intersec r2y =9 and the plane y + z = 2. 5. Find a...
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y > 0} is a 1-dimensional manifold with boundary and the hemisphere D= {(x, y, z) | 22 + y2 + z2 = 1, 2 > 0} is a 2-dimensional manifold with boundary. (7) Suppose X is an n-dimensional manifold with boundary. Let ax denote the set of points in the boundary of X. Show that ax is an (n-1)-dimensional manifold.
Sketch the following region in the complex plane: the set of z such that z (32i) 2
(a) (2 points) In the plane below, sketch the region corresp below, sketch the region corresponding to: {(x, y) - 1<y <2}. Use the convention that, if the boundary of a region is included in the dicated using a solid line. Otherwise, use a dashed/dotted line. Clearly show any 2 and y intercepts on your graph.
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
Show your work! No work, no credit! 1. Given T[c x, y, z >)-< x-z, y >. Complete the following a. Check if T is a linear transformation. Show your work! b. Find the domain and range of T. c. If T is a linear transformarion, find the matrix A that induced T. (6 points) 1)
(1 point) Calculate ſls f(x, y, z)ds For y = 4 – z2, Is $(x, y, z) ds = 0 < x, z <7; f(x, y, z) = z