Assignment:
Q1. A corporation manufactures a product at two locations. The cost of producing x units at a location one and y unites at location two are C1(x)=.01x^2 + 2x + 1000 and C2(y)=.03y^2 + 2y + 300, respectively. If the product sells for $14 per unit, find the quantity that must be produced at each location to maximize the profit P(x,y)=14(x+y)-C1-C2.
Q2. Find the slope in the x-direction of the surface f(x,y)=xy^2 at the point (3, -2, 12).
Q3. Find the distance between the points (-1, 2, 5) and (1, -1, 2).
Q4. Find the least squares regression line for the points (0,1), (1,3), (2,2), (3,4), (4,5).
Q5. Use Lagrange Multipliers to maximize f(x, y, z)= 4x^2 + y^2 + z^2 with the constraint that 2x – y + z= 4
Q6. Find the distance between the point (1, -2, 3) and the plane 3x – 4y + 2z – 1=0
Provide complete and step by step solution for the question and show calculations and use formulas.