1. Compute all first and second partial derivatives of the following functions: Osin(z +v) () A++...

 

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1. Compute all first and second partial derivatives of the following functions: Osin(z +v) () A++ VI++ (a) +-2r y? (b) cos(zy") (e) V+y 2. If u show that ip 3. If az + 2hry +by 1,thed find %3D dr dz OIf z= tan-1 where z= Int and y = e', find %3! dt dz 5. If z = r - y?, where z= e'cost and y= e'sint, find %3D %3D dt 6. In each of the following functions, find the critical point and classify it. f(z,v) +y - 6zy (c) f(z, y) = z – 3ry + 6y? @f(z, y) = rlny + - 4x - 5y + 8. (b) f(z, y) = 2r – 2ry + 3y? - 4z - 8y + 20 %3! %3D %3D %3D The packaging department of an Engineeringing company has been asked to design a rectangular box with no top and a partition down the middle. The box must have a volume of 48 cubic inches. Find the dimensions that will minimise the amount of material used to construct the box. The amount of material M used to construct the box is given as M ry + 2xz +3yz and the volume of the box is %3D V = ryz = 48. [Hint: Considerz and y as independent variables, with z depending on z and y] %3D 1. Compute all first and second partial derivatives of the following functions: Osin(z +v) () A++ VI++ (a) +-2r y? (b) cos(zy") (e) V+y 2. If u show that ip 3. If az + 2hry +by 1,thed find %3D dr dz OIf z= tan-1 where z= Int and y = e', find %3! dt dz 5. If z = r - y?, where z= e'cost and y= e'sint, find %3D %3D dt 6. In each of the following functions, find the critical point and classify it. f(z,v) +y - 6zy (c) f(z, y) = z – 3ry + 6y? @f(z, y) = rlny + - 4x - 5y + 8. (b) f(z, y) = 2r – 2ry + 3y? - 4z - 8y + 20 %3! %3D %3D %3D The packaging department of an Engineeringing company has been asked to design a rectangular box with no top and a partition down the middle. The box must have a volume of 48 cubic inches. Find the dimensions that will minimise the amount of material used to construct the box. The amount of material M used to construct the box is given as M ry + 2xz +3yz and the volume of the box is %3D V = ryz = 48. [Hint: Considerz and y as independent variables, with z depending on z and y] %3D