Sheet 11. Functions of two variables

11 Functions of two variables

Sheet 11. Functions of two variables

Exercise 11.1. Find the rst-order partial derivatives of the given functions:

a) f(x, y) = x³y + 2xy b) f(x, y) = x²y⁵− 3x²+ 3y² c) f(x, y) = x²y²+ 3xy d) f(x, y) = x⁵y¹⁰− x³sin y + y²e^x e) f(x, y) = y

x² + y² f) f(x, y) = x − y x + y g) f(x, y) = e^x(cos x + x sin y) h) u = e^x(x²+ y²) i) f(x, y) = x²sin y + x cos (xy) j) f(x, y) = sin (x²+ y²) k) f(x, y) = ln (x + y) l) z = x^y

m) z = e^xy2 n) u = e^{x sin y} o) z = arctany

x p) z = ln³

x +p

x²− y²

´

Exercise 11.2. Find the second-order derivatives of the given functions:

a) f(x, y) = x³+ xy²− 5xy³+ y⁵ b) f(x, y) = xy + x² y³ c) f(x, y) = x^y d) f(x, y) = e^xy e) z = lnx

y f) z = arctan xy

Exercise 11.3. Find (local) extrema, if exist:

a) f (x, y) = x²+ 2x + 3y²− 6y b) f (x, y) = −3x²+ 6x + 4y²+ 24y c) f (x, y) = −x²+ 4x − ¹₂y²− 5y d) f (x, y) = 3x³+ 3x²y − y³− 15x e) f (x, y) = 2x²+ 3xy + y²− 2x − y + 1 f) f (x, y) = x²− xy + 2y²− x + 4y − 5 g) f(x, y) = 4x²y + 8x²− ¹₃y³ h) f(x, y) = 3x³+ 3x²y − y³− 15x i) f(x, y) = x³+ y³− 3xy j) f(x, y) = x⁴+ y⁴− 2x²+ 4xy − 2y² k) f(x, y) = 2 −p

3x²+ y² l) f(x, y) = (x² + y)√ e^y m) f(x, y) = (2x + y²)e^x n) f(x, y) = e^x−y(x²− 2y²)

Last update: January 21, 2009 1