MATH 2150 Real and Complex Analysis Semester 1, 2021School of Science MATH2150 Assignment 2 Due: 17:30pm, 2nd September, 2021 questions week 1 ～ 4 4 5 ～ 8 5 9 ～ 10 6 (5 + 5 + 5 + 5 + 4 + 5 + 6 + 3 (bonus)+ 5 + 5 (bonus) = 48 marks) Your final mark for Assign 1 = min{40, total mark (inc. bonus)} 1. (5 points) Let f : D R→ R be given as the following rational function f(x) = 2×4 3×2 + x 1 x2 + x 6 . Identify the largest possible domain of f . 2. (5 points) Let f : [ 1, 1] → R be defined as f(x) = x (i.e. the ceiling function, which gives the smallest integer greater than or equal to x). Sketch the graph of this function, identify its range and the points at which it is discontinuous on [ 1, 1]. Note that there is no need for a formal proof here, just a brief explanation is enough. 3. (5 points) Prove that the function f(x, y) given below is discontinuous at (0, 0). f(x, y) = { x y , y = 0, 0, y = 0, 4. (5 points) Prove that if f : D Rn → R is continuous at xˉ ∈ D, then the function h(x) = c · f(x), where c is an arbitrary constant in R, is also continuous at xˉ. 5. (4 points) Calculate the derivative using the definition directly fo
r f(x) = √ x for x > 0. 6. (5 points) Let f(x) = 3 lnx, g(x) = ln 2 · (x2 1). Prove that there exists a point c ∈ (1, 2) such that f ′(c) = g′(c). Hint: use Rolle’s theorems of differential calculus from Week 5 lecture slides 7. (6 points) Using L’Hospital’s rule, find the limit lim x→0 ln cosx ln cos 3x . Hint: you may also use the fact that limx→0 u(x) · v(x) = limx→0 u(x) · limx→0 v(x), whenever the individual limits on the right hand side exist. 1/2 8. (optional, 2 bonus marks) Let F (x, y, z) = z3 xz + y. Verify that around the point (xˉ, yˉ, zˉ) = (2, 1, 1) the equation F (x, y, z) = 0 defines an implicit function z = f(x, y) and conclude that by the implicit function theorem this function f is differentiable at (2, 1). Find both partial derivatives of f at (2, 1). 9. (5 points) Using properties of Riemann integral show that if f(x) is an integrable odd function on [ a, a], then ∫ a a f(x)dx = 0. 10. (optional, 3 bonus marks) Compute the integral K = ∫∫ P (√ x+ √ y )3 dxdy, where P = { (x, y) ∣∣x ≥ 0, y ≥ 0, √x+√y ≤ 1} . Hint: let x = r cos4 t, y = r sin4 t (0 ≤ r ≤ 1, 0 ≤ t ≤ pi/2). Page 2 2/2