5) Find the region in the xy-plane for which (sin x – y)y' = vx- y satisfies the nonlinear existence and uniqueness theorem.
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- Check that the equation x³ + y³ + z³ = x + y + z implicitly defines at least one differentiable function z = z (x, y). Express az/ ax and az/ay in terms of x, y, zConsider F and C below. F(x, у, 2) %3D yzе^?i + extj + хуеX-k, C: r(t) = (t2 + 1)i + (t2 – 1)j + (t2 – 4t)k, 01. (+y + ) (1+? + +) 9. Integrate the function f(r,y, 2) = over the entire 3D coordinate system, i.e., all of R'.Find the absolute maximum and absolute minimum values of the function f(2, 3) = 2æ° + 3æy+ 2g over the coordinate axes and the closed region bounded by the line x + y = -1.a) Suppose you are given model with two explanatory variables such that: Y₁ = a + B₁x₁i + B₂x₂₁x + u₁, i = 1,2,...n Using partial differentiation derive expressions for the expressions for the intercept and slope coefficients for the model above.Suppose that f(r, y) = 2x + 2y“ – zy Then the minimum isLet f be a function admitting continuous second partial derivatives such that Vf(x, y) = (a.x² – x, y? – a²) - | with a < 0. It is possible to assemble with certainty that 1 a) The point (÷, a, f(÷, a)) is a saddle point of f and f reaches a relative maximum at the point (0, a). a (-0). b) The point a)) is a saddle point of f and f reaches a relative maximum at the point а). 1 c) The point (0, a, f(0, a)) is a saddle point of f and f reaches a relative minimum at the point (÷, a). d) The point (0,-a, f(0,-a)) is a saddle point of f and f reaches a relative minimum at the point (0, a).If the partial derivatives of A, B. U, and Vare assumed to exist, then I. V(U + V) = VU + VV or grad (U+ )3grad u+ grad V 2. V (A +B) = V-A+V B or div (A + B) +div A + div B 3. Vx (A +B) = VxA+VxB or curl (A + B) = curlA+ curl B 4. V.(UA) = (VU) - A+ U(V A) 5. Vx (UA) = (VU) xA + U(V x A) 6. V.(A x B) = B (Vx A)-A (Vx B) 7. Vx (A x B) = (B V)A- B(V A)-(A V)B+ A(V B) 8. V(A B) (B V)A+ (A V)B+ Bx (Vx A) + A x (V x B) 9. V.(VU) = VU= is called the Laplacian of U. +. and V =. ar dyaz is called the Lapacian operator. 10. Vx (VU) =0. The curl of the gradient of U is zero, 11. V.(Vx A) = 0. The divergence of the curl of A is zero. 12. Vx (Vx A)= V(V. A)-V AThe paraboloid z = x2 + y2 - 4x + 2y + 5 has a local minimum at (2, -1). Verify the conclusion of as shown for this function.