Problem 1(i)
Basic Partial Derivatives
Find ∂f/∂x and ∂f/∂y for f(x,y) = 3x² - 2xy + y² + 5x + 2 at (2, -5)
∂f/∂x Solution:
∂f/∂x = 6x - 2y + 5
At (2, -5): 6(2) - 2(-5) + 5 = 12 + 10 + 5 = 27
∂f/∂x = 27
∂f/∂y Solution:
∂f/∂y = -2x + 2y
At (2, -5): -2(2) + 2(-5) = -4 - 10 = -14
∂f/∂y = -14
Problem 1(ii)
Basic Partial Derivatives
Find ∂g/∂x and ∂g/∂y for g(x,y) = 3x² + y² + 5x + 2 at (1, -2)
∂g/∂x Solution:
∂g/∂x = 6x + 5
At (1, -2): 6(1) + 5 = 11
∂g/∂x = 11
∂g/∂y Solution:
∂g/∂y = 2y
At (1, -2): 2(-2) = -4
∂g/∂y = -4
Problem 1(iii)
Three-Variable Function
Find ∂h/∂x, ∂h/∂y, ∂h/∂z for h(x,y,z) = x sin(xy) + z²x at (2, π/4, 1)
∂h/∂x Solution:
∂h/∂x = sin(xy) + xy cos(xy) + z²
At (2, π/4, 1): sin(π/2) + (π/2)cos(π/2) + 1 = 1 + 0 + 1 = 2
∂h/∂x = 2
∂h/∂y Solution:
∂h/∂y = x² cos(xy)
At (2, π/4, 1): 4 cos(π/2) = 0
∂h/∂y = 0
∂h/∂z Solution:
∂h/∂z = 2zx
At (2, π/4, 1): 2(1)(2) = 4
∂h/∂z = 4
Problem 1(iv)
Exponential and Logarithmic
Find ∂G/∂x and ∂G/∂y for G(x,y) = eˣ⁺ʸ log(x²+y²) at (-1,1)
∂G/∂x Solution:
∂G/∂x = eˣ⁺ʸ log(x²+y²) + eˣ⁺ʸ (2x)/(x²+y²)
At (-1,1): e⁰ log(2) + e⁰ (-2)/2 = log(2) - 1 ≈ -0.3069
∂G/∂x = ln(2) - 1
∂G/∂y Solution:
∂G/∂y = eˣ⁺ʸ log(x²+y²) + eˣ⁺ʸ (2y)/(x²+y²)
At (-1,1): e⁰ log(2) + e⁰ (2)/2 = log(2) + 1 ≈ 1.6931
∂G/∂y = ln(2) + 1
Problem 2(i)
Mixed Derivatives Verification
Find fx, fy and show fxy = fyx for f(x,y) = 3x/(y+sin x)
First Derivatives:
fx = [3(y+sin x) - 3x cos x]/(y+sin x)²
fy = -3x/(y+sin x)²
Mixed Derivatives:
fxy = [-3(y+sin x) + 6x cos x]/(y+sin x)³
fyx = [-3(y+sin x) + 6x cos x]/(y+sin x)³
✓ fxy = fyx
Problem 2(ii)
Inverse Tangent Function
Find fx, fy and show fxy = fyx for f(x,y) = tan⁻¹(x/y)
First Derivatives:
fx = (1/y)/(1 + (x/y)²) = y/(x² + y²)
fy = (-x/y²)/(1 + (x/y)²) = -x/(x² + y²)
Mixed Derivatives:
fxy = (x² - y²)/(x² + y²)²
fyx = (x² - y²)/(x² + y²)²
✓ fxy = fyx
Problem 3
Three-Variable Function
Find ∂U/∂x, ∂U/∂y, ∂U/∂z for U(x,y,z) = (x²+y²)/(xy) + 3z²y
Solution:
Simplify: (x²+y²)/(xy) = x/y + y/x
∂U/∂x = 1/y - y/x²
∂U/∂y = -x/y² + 1/x + 3z²
∂U/∂z = 6zy
Problem 4
Logarithmic Function
Find ∂U/∂x + ∂U/∂y + ∂U/∂z for U(x,y,z) = log(x³+y³+z³)
Solution:
∂U/∂x = 3x²/(x³+y³+z³)
∂U/∂y = 3y²/(x³+y³+z³)
∂U/∂z = 3z²/(x³+y³+z³)
Sum = 3(x²+y²+z²)/(x³+y³+z³)
Problem 5(i)
Second Order Derivatives
Find gxy, gxx, gyy, gyx for g(x,y) = xeʸ + 3x²y
Solution:
gx = eʸ + 6xy
gy = xeʸ + 3x²
gxx = 6y
gyy = xeʸ
gxy = eʸ + 6x
gyx = eʸ + 6x
✓ gxy = gyx
Problem 10
Business Application
A firm produces calculators A (x units) and B (y units) with:
R(x,y) = 80x + 90y + 0.04xy - 0.05x² - 0.05y²
C(x,y) = 8x + 6y + 2000
Profit Function:
P(x,y) = 72x + 84y + 0.04xy - 0.05x² - 0.05y² - 2000
Marginal Profits at (1200,1800):
∂P/∂x = 72 + 0.04y - 0.10x = 24
Each additional type A calculator increases profit by ₹24
∂P/∂y = 84 + 0.04x - 0.10y = -48
Each additional type B calculator decreases profit by ₹48