(5) Let D = {(x, y) E R? ||r| + |y| < 1}. The region D is a square diamond () in the ry-plane with corners at the points (1,0), (0,1), (-1,0), (0, -1). (a) Let f(r, y) = x²y?. Compute // f(x, y) dA. %3D Exploit symmetry: f(-2, y) = f(x, -y) = f(-x,-y) = f(r, y).

Q: W(s, t) = F(u(s, t), v(s, t)), where F. u. and v are differentiable. If u (6,-6)= -2, u,(6, −6) = 3,…

A: The objective of the question is to find the partial derivatives W_s(6,-6) and W_t(6, -6) for the…

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: State the domain and range of each function a) b) c) 3₂ f(x) 44 3+ 2 1+ 1+ 2 3 1

A: Domain : Domain of a function are those x-values for which the function is defined.Range : Range of…

Q: Let F(t) = (- t + 1, t – 2t", - 2t). Find the line (L) tangent to r (t) at the point (- 15, – 24, –…

A:

Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: Compute fe F·dr

A: Green's theorem states that  ∮C(Ldx+Mdy)=∬D∂M∂x-∂L∂ydxdy…

Q: 14. Find the absolute maximum and minimum values of f(x,y) = x²y² + xy³ - 8xy² on the closed…

A: The given problem is to find the absolute maximum and absolute minimum values of f(x,y) over the…

Q: Find a linearization L(x, y, z) of f(x, y, z) = x²y + 4z at (1,-1,2).

A:

Q: 9. Let F be the same as in the previous problem, but this time let P be the line segment from (5,0,…

A:

Q: Let f (x, y, z) = x + yz and let C be the line segment from P = (0, 0, 0) to (8, 6, 6). (a)…

A: Given function is  and let  be the line segment from .And  for .Here, Now,Hence, .Now, Now,Hence, .

Q: Find the maximum value of f(x, y) = 11zy(9 – 99x – 9y) over the triangle with vertices (0, 0). (†,0)…

A: Given: The triangle vertices are (0,0),(111,0) and (0,1) f(x,y)=11xy(9-99x-9y)…

Q: et f(z, y) = for (z, y) # (0,0). Find the level curves of 2 + y²

A:

Q: There is a set of parallel lines in the scene. How would i Compute mathematically their vanishing…

A: Given that:A 3D line is represented by a set of points where the 3D point  is one point on the…

Q: Find the global extrema of f(x, y) = x3 - xy - y2 + y on the square [0, l] x [0, 1]

A:

Q: 5. Find the linear approximation of f (x, y, z) = x³y² + 22 at the point (2, 3,4). Use this to…

A:

Q: Show that the function f defined by (x, y) = (1, –1) 22 + y r + y f(x, y) = (x, y) # (1, –1) is not…

A:

Q: The range of f(x,y) = 2 - x2 + 4y² is : O [0,2] O xy-plane Joo+7] O

A:

Q: c) Compute fF.dr directly for the following. i) F(x, y) = (-y, x + 4), r(t) = (cost, sint), 0≤ t…

A:

Q: Let 7(t) = (- 5t + 2, t² – 4t³, 3tª). Find the line (L) tangent to 7(t) at the point (7, 5, 3) L:(x,…

A:

Q: The shaded region is a domain for the function eq1 eq2 -1 -1 A) ƒ(x, y) = V4 – x² – y² + Vx – y² +…

A:

Q: (b) Let C₁, C₂ and C3 be the curves in R³ with parametrisations C₁: (SER) (s − 2)i + (2s² − 1)j + k…

A: We check that the curves are not intersect at a single point. So i can't say about the second and…

Q: I grad(f)•dr where C is graph of y = x² + 3x from (0,0) to (2, 14) and f (x,y) = ev/lx*•1}

A: The function is fx, y=eyx+1. Find: ∫Cgradf·dr…

Q: 1 (vz,xz,xy) dr where C is the union of the curve y=x* from (0,0,0) to 1+xyz Evaluate (11,0) and the…

A:

Q: The absolute maximum value of ƒ(x, y) = 4xy² — x²y² – xy³ on the triangular region in the xy-plane…

A: Given that, f(x,y) = 4xy2 - x2y2 - xy3 and a triangular  region D with vertices (0,0) ,(6,0) and…

Q: W(s, t) = F(u(s, t), v(s, t)), where F. u. and v are differentiable. If u (6,-6)= -2, u,(6, −6) = 3,…

A: The objective of the question is to find the partial derivatives of W with respect to s and t,…

Q: Let f(x, y, z) = x + yz and let C be the line segment from P = (0, 0, 0) to (4, 5, 5). (a) Calculate…

A: Given function is  and let  be the line segment from . We have to calculate  and for , .Since,…

Q: Find the linearization L(x,y) of the function f(x,y) = 3x - 3y+6 at the points (0,0) and (1,1). The…

A: We have to find the linearization L(x,y) of the following function , at the points (0,0) and (1,1)…