Question

f(x, y) = 2x2 + 4y2 + 3xy 44x 95y +580 barrels of fertilizer are produced when x kilograms and y liters of ingredient Y are combined. Check: f(2,3) = 269 Use this context and the graph provided below to answer the next FOUR questions. y liters (3.5, 4.5) Sbarrels x kilograms

Answer 1

Solution: f(x, y) = 2x2 + 4y2 + 3xy – 44x – 95y + 580 (1) at (3.5, 4.5) df = 4x + 3y – 44 dx At (3.5, 4.5) df dx (3.5,4.5) 4(3.5) + 3(4.5) – 44 = -16.5 df (2) Units of Barrels kilogram dx So, Barrels per kilogram. (3) dy Liters dx (3.5,4.5) kilogram dy dx Slope of the tangent line at given point we can find it by linear approximation of the f(x, y). f(x,y) ~ L(x,y) = f(xo, yo) + fx(xo, yo)(x – xo) + fy(xo, yo)(y – yo) d = fx -(f(x, y)) = 4x + 3y – 44 dx d (f(x, y)) = 8y + 3x – 95 dy = fy - day (f(x,y)) = = At (x, y) = (3.5,4.5) df fx = 4(3.5) + 3(4.5) – 44 = -16.5 dx (3.5,4.5) df fy = 8(4.5) + 3(3.5) – 95 = -48.5 = dy (3.5,4.5)

f(xo, yo) = 151.25 L(x, y) = 151.25 +(-16.5)(x – 3.5) + (-48.5)(y – 4.5) L(x,y) = 151.25 + 57.75 – 16.5x + 218.25 – 48.5y F(x,y) = L(x, y) = 427.25 – 16.5x – 48.5y F(x,y) will be zero because we have x - y plane. dy So, slope = Slope of above plane dx (3.5,4.5) 427.25 – 16.5x – 48.5y = 0 Differentiate both sides wrt x. 0 – 16.5 – 48.5y' = 0 -16.5 y' = -0.34020 ~ -0.340 48.5 dy Units of dx Litres kilogram somLiters per kilogram.