MA2001 Calculus of Several Variables

Section A

1. Find and sketch the domain of the (real-valued) function f(x, y) = p y − x 2 . Find its range explaining your answer. [8]

2. Find the limit, if it exists, or show that the limit does not exist:[12]

3. Using the definition of differentiability show that the function f(x, y) = x 2 + xy + y 3 is differentiable at the point (1, 1). [9]

4. Find the greatest and the smallest values that the function f(x, y) = x 2 y takes on the ellipse x 2 + 2y 2 = 6. [10]

5. Sketch the region of integration and evaluate the integral

where the domain D is the triangle bounded by the lines x = 0, y = x, y = 1. Choose the order of integration wisely. [11]

Section B

6. (a) Let h = (h1, h2). Define what is meant by g(h) = o(|h|) as h → 0 and show that the function g(h) = h1 √ h2 + h 2 1h2 is o(|h|) using the definition. [5]

(b) Find the directional derivative of the function f(x, y) = e xy at the point (2, 3) in the direction of an angle of 2 3 π from the positive x-direction. [10]

(c) Find the tangent and the normal lines to the curve y = x 3 − 3x 2 + x − 1 at the point (3, 2). [10]

7. Consider the integral I = Z Z Z Ω p x 2 + y 2 dx dy dz, where Ω = {(x, y, z) : p x 2 + y 2 ≤ z ≤ p 2 − x 2 − y 2}.

(a) Sketch the domain of integration. Identify two surfaces forming the boundary of Ω and write their equations in spherical coordinates. [5]

(b) Write down the integral I as a triple integral using spherical coordinates. [10]

(c) Evaluate the integral I from part (b) of the question. [10]