WORKSHEET ON PARTIAL DERIVATIVES

Problem 1 :

Find the partial derivatives of the following functions at the indicated points.

(1) f (x, y)  =  3x² − 2xy + y² + 5x + 2, (2, −5)

(2) g(x, y) = 3x² + y² + 5x + 2, (1, −2)

(3) h(x, y, z) = x sin(xy) + z²x, (2, π/4, 1)

(4) G(x, y)  =  e^(x+3y) log(x²+y²), (-1, 1)

Solution

Problem 2 :

For each of the following functions find the f_x and f_y and show that f_xy  =  f_yx

(1) f (x, y)  =  3x/(y+sinx)

(2) f(x, y)  =  tan^-1(x/y)

(3) f(x, y) = cos (x²-3xy)

Solution

Problem 3 :

If

U(x, y, z) = (x²+y²)/xy + 3z²y

find (∂u/∂x), (∂u/∂y) and (∂u/∂z)

Solution

Problem 4 :

If U(x, y, z) = log (x³+y³+z³), find 

(∂u/∂x) + (∂u/∂y) + (∂u/∂z)

Solution

Problem 5 :

For each of the following functions find g_xy, g_xx, g_yy,and g_yx.

(i)  g(x, y) = xe^y+3x²y

(ii)  g(x, y) = log(5x+3y)

(iii) g(x, y) = x²+3xy-7y+cos(5x)

Solution

Problem 6 :

Let

w(x, y, z) = 1/√(x²+y²+z²), (x, y, z) ≠ (0, 0, 0).

Show that (∂²w/∂x²) + (∂²w/∂y²) + (∂²w/∂z²)

Solution

Problem 7 :

If V(x, y) = e^x(x cos y - y sin y), then prove that 

∂²w/∂x²  = ∂²w/∂y²

Solution

Problem 8 :

If w(x, y) = xy + sin(xy) , then prove that

∂²w/∂y∂x  = ∂²w/∂x∂y

Solution

Problem 9 :

If V(x, y, z)  =  x³+y³+z³+3xyz, show that 

∂²v/∂y∂z  = ∂²v/∂z∂y

Solution

Problem 10 :

A firm produces two types of calculators each week, x number of type A and y number of type B . The weekly revenue and cost functions (in rupees) are

R(x, y) = 80x + 90y + 0.04xy − 0.05x² − 0.05y² and

C(x, y) = 8x + 6y + 2000 respectively.

(i) Find the profit function P(x, y) ,

(ii) Find ∂P/∂x (1200, 1800) and ∂P/∂y (1200, 1800) and interpret these results.

Solution

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