Questions on Functions of Several Variables with Answers

Practice Questions on Functions of Several Variables with Answers

Questions on Functions of Several Variables with Answers

Q1. A function f(x,y) on R2 has the following partial derivatives

∂f(x,y)/∂x = x2  , ∂f(x,y)/∂y = y2

Then

(a) f has directional derivatives in all directions everywhere
(b) f has derivative at all points
(c) f has directional derivative only along the direction (1,1) everywhere
(d) f does not have directional derivatives in any direction everywhere

Q2. A function f:R2 → R is defined by f(x,y) = xy. Let v = (1,2) and a = (a1, a2) be two elements of R2. The directional derivative of  f  in the direction of v at a is

(a) a1 + 2a2
(b) a2 + 2a1
(c) a2/2 + a1
(d) a1/2 + a2

Q3. Let f:R2 → R2 be given by f(r,θ) = (rcosθ , rsinθ). Then for which of the subset U of R2 given below, f restricted to U admits an inverse?

(a) U = R2
(b) U = {(x,y)∈R2 : x > 0 , y > 0}
(c) U = {(x,y)∈R2 : x2 + y2 < 1}
(d) U = {(x,y)∈R2 : x < -1 , y < -1}

Q4. Consider the function f(x,y) = x2/y2 , (x,y) ∈ [1/2,3/2]×[1/2,3/2]. Then derivative of the function at (1,1) along the direction (1,1)

(a) 0
(b) 1
(c) 2
(d) -2

Q5. Let f(x,y) = xy/(x2 + y2) ,(x,y) ≠ (0,0) ; f(0,0) = 0 . Then

(a) f is continuous at (0,0) and the partial derivatives fx’fy exist at every point of R2
(b) f is discontinuous at (0,0) and fx’f exist at every point on R2
(c) f is discontinuous at (0,0) and fx’f exist only at (0,0)
(d) None of these

Q6. Let f:R2 → R and g:R2 → R be defined by f(x,y) = |x| + |y| and g(x,y) = |xy| . Then

(a) f is differentiable at (0,0), but g is not differentiable at (0,0)
(b) f is differentiable at (0,0), but f is not differentiable at (0,0)
(c) Both f and g are differentiable at (0,0)
(d) Both f and g are continuous at (0,0)

Q7. Consider the map f:R2 → R2 defined by f(x,y) = (3x – 2y + x2, 4x+5y+y2). Then

(a) f us continuous at (0,0)
(b) f is continuous at (0,0) and all directional derivatives exists at (0,0)
(c) f is differentiable at (0,0) but the derivative Df(0,0) is not invertible
(d) f is differentiable at (0,0) but the derivative Df(0,0) is invertible

Q8. The map L:R2 → R2 given by L(x,y) = (x-y) is

(a) Differentiable everywhere on R2
(b) Differentiable only at (0,0)
(c) DL(0,0) = L
(d) DL(x,y) = L for all (x,y) ∈ R2

Q9. Let f:[π, 2π] → R2 be the function f(t) = (cost, sint) . Which of the following are necessarily correct ?

(a) There exists to ∈ [π, 2π] such that f'(to) = 1/π [f(2π) – f(π)]
(b) There does not exist any to ∈ [π, 2π] such that f'(to) = 1/π [f(2π) – f(π)]
(c) There exists to ∈ [π, 2π] such that || f(2π) – f(π) || ≤ || f'(to) ||
(d) f'(t) =  (-sint, cost) for all t ∈ [2π, π]

Q10. Let f:Rn → Rn be the function defined by f(x) = x||x||2 for x ∈ Rn . Which of the following statements are correct ?

(a) (Df)(0) = 0
(b) (Df)(x) = 0 for all x ∈ Rn
(c) f is one-one
(d) f has an inverse

Q11. Let f:A∪E → R2 be differentiable, where

A = {(x,y) ∈ R2 : 1/2 < x2 < y2 < 1} and
E = {(x,y) ∈ R2 : (x-2)2 + (y-2)2 < 1/2} .

Let Df be the derivative of the function  f . Which of the following are necessarily correct ?

(a) If (Df)(x,y) = 0 for all (x,y) ∈ A∪E, then f is constant
(b) If (Df)(x,y) = 0 for all (x,y) ∈ A, then f is constant on A
(c) If (Df)(x,y) = 0 for all (x,y) ∈ E, then f is constant on E
(d) If (Df)(x,y) = 0 for all (x,y) ∈ A∪E, then for some (xo, yo), (x1, x2) ∈ R2,
f(x,y) = (xo, yo) for all (x,y) ∈ A and
f(x,y) = (x1, y1) for all (x,y) ∈ E

Q12. L:Rn → R be the function L(x) = <x, y>, where <•,•> is some inner product on Rn and y is a fixed vector in Rn. Further denote by DL, the derivative of L. Which of the following are necessarily correct ?

(a) DL(u) = DL(v) for all u,v ∈ Rn
(b) DL(0,0,…,0) = L
(c) DL(x) = ||x||2 for all x ∈ Rn
(d) DL(1,1,…,1) = 0

Q13. Let A = {(x,y) ∈ R2 : x+y ≠ -1} define f:A → R2 by f(x,y) = [x/(1+x+y) , y/(1+x+y)]. Then

(a) The Jacobian matrix of f does not vanish on A
(b) f is infinitely differentiable on A
(c) f is injective on A
(d) f(A) = R2

Q14. Let f:Rn → R be the map f(x1,…,xn) = a1x1 + …+anxn , where a = (a1,…an) is a fixed non-zero vector. Let Df(0) denote the derivative of f at 0. Which of the following are correct ?

(a) (Df)(0) is a linear map from Rn to R
(b) [(Df)(0)](a) = ||a||2
(c) [(Df)(0)](a) = 0
(d) [(Df)(0)](b) = a1b1+…+anbn for b = (b1,…,bn)

Q15. Let f(x,y) = √(|xy|) . Then

(a) fx and fy do not exist at (0,0)
(b) fx(0,0) = 1
(c) fy(0,0) = 0
(d) f is differentiable at (0,0)