SA M PL E EX AM Student Number SAMPLE EXAM, Semester 1 2020 School of Mathematics and Statistics MAST10006 Calculus 2 This sample exam consists of 12 pages (including this page) Instructions to Students • This sample exam is intended to give an indication of the length and format of the written MAST10006 Calculus 2 exam in 2020 semester 1. • The questions on the real exam will be different to this sample exam. • This sample exam may not be indicative of the difficulty or topics covered in the real exam. • You are recommended to try to complete this exam within 2 hours, and under exam conditions for practice. • Write your answers in the spaces provided. • Answers to this sample exam are available on Canvas. Full solutions are not available. • The questions in this sample exam are drawn from the 2020 summer semester MAST10006 exam. • There are 8 questions with marks as shown. The total number of marks available is 60. Supplied by download for enrolled students only— c©University of Melbourne 2020 SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 1 (10 marks) In this question you must state if you use any standard limits, limit laws, continuity, l’Hoˆpital’s rule or the sandwich theorem. Let f : R→ R be given by f(x) = 1 x − sinxx2 , x < 0 kx, 0 ≤ x ≤ 1 cosec ( pix 2 ) , x > 1 where k ∈ R is a constant. (a) Find lim x→0 f(x), or explain why it does not exist. Page 2 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 1 (continued) (b) For which value(s) of k is f continuous at x = 1? Show all your working. Page 3 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 2 (10 marks) In this question you must state if you use any standard limits, limit laws, continuity, l’Hoˆpital’s rule, the sandwich theorem or series convergence tests. Let an = 4n n3 + r2n , where r ∈ R is a constant. (a) If r = 2 then does the sequence {an} converge or diverge? Justify your answer. (b) If r = 2 then does the series ∞∑ n=1 an converge or diverge? Justify your answer. Page 4 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 2 (continued) (c) If r = 3 then does the sequence {an} converge or diverge? Justify your answer. (d) If r = 3 then does the series ∞∑ n=1 an converge or diverge? Justify your answer. Page 5 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 3 (5 marks) Evaluate d47 dx47 ( e−x sinx ) . Page 6 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 4 (5 marks) Evaluate the integral ∫ 2×4 − x3 + 4×2 − x− 2 x3 + 2x dx Page 7 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 5 (5 marks) Find the general solution y(x) of dy dx = x ( ex 2 − 2y ) Page 8 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 6 (8 marks) Find the solution of the differential equation y′′ + 2y′ + y = 25 sin(2x) subject to the boundary conditions y(0) = −4, y(pi) = pi − 4. Page 9 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 7 (7 marks) Let S be a surface in R3 given by z = cosh √ x2 + y2 for (x, y) ∈ R2. (a) Find an expression for the level curve of this surface when z = c. For what value(s) of c does the level curve exist? (b) Sketch the cross section of the surface in the yz plane. Label each axis interept with its value. (c) Sketch the surface S in R3. Label each axis intercept with its value. Page 10 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 8 (10 marks) Let f : R2 → R, f(x, y) = 3×2 − 2×3 − 3y2 + 6xy. (a) Find the directional derivative of f at (0, 1) in the direction from (0, 1) towards (1, 0). (b) Find the equation of the tangent plane to the surface z = f(x, y) at the point where (x, y) = (0, 1). Page 11 of 12 pages SA M PL E EX AM MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020 Question 8 (continued) (c) Find all stationary points of f , and classify each point as a local maximum, local minimum or saddle point. End of Exam—Total Available Marks = 60 Page 12 of 12 pages
