Paper 4: Fundamentals of Business Mathematics & Statistic

3.42 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 94 :

Find the first partial derivatives of f(x, y, z, u, v) = 2x + yz – vx + vy^2.

Solution:

Fx = 2 – u fy = z + 2vy fz = y fu = -x fv = y^2

Example 95 :

If f(x, y) = 3x^2 y – 2x^3 + 5y^2 , find fx(1, 2) and fy(1, 2).

Solution:

Fx = 6xy – 6x^2.

Hence, fx(1, 2) = 12 – 6 = 6. Fy = 3x^2 + 10y.

Hence, fy(1, 2) = 3 + 20 = 23.

Example 96 :

For f(x, y) = 3x^2 y – 2xy + 5y^2 , verify that fxy = fyx.

Solution:

fx = 6xy – 2y, fxy = 6x – 2. Fy = 3x^2 – 2x + 10y, fyx = 6x – 2.

Example 97 :

If f(x, y) = 3x^2 – 2xy + 5y^3 , verify that fxy = fyx.

Solution:

fx = 6x – 2y fxy = -2 fy = -2x + 15y^2 fyx = -2

Example 98:

For f(x, y) = 3x^4 – 2x^3 y^2 + 7y, find fxx, fxy, fyx, and fyy.

Solution:

Fx = 12x^3 – 6x^2 y^2 , fy = -4x^3 y + 7, fxx = 36x^2 – 12xy^2 , fyy = -4x^3 , fxy = -12x^2 y, fyx = - 12x^2 y.

Example 99 :

If f(x, y, z) = x^2 y + y^2 z – 2xz, find fxy, fyx, fxz, fzx, fyz, fzy.

Solution:

fx = 2xy – 2x, fy = x^2 + 2yz, fz = y^2 – 2x. fxy = 2x, fyx = 2x, fxz = -2, fzx = -2, fyx = 2y, fzy = 2y.