Q1 Q2 Q3 Q4 Q5 Total The Australian National University Final Examination Solutions – June 2015 MATH 1014 – Mathematics and Applications II. Book A – Calculus Questions 1 – 5 Student No: U Important notes: One A4 sheet hand written on both sides permitted. This sheet is to cover both calculus and linear algebra. No calculators or books permitted. Answer all questions. Calculus in Book A, Linear Algebra in Book B. Questions have different weights, the value of each is shown. The maximum grade on this part of the paper is 50. Place your answers in the space provided on the test sheet. If you need more space, use the back of the pages. You must justify your answers and show your work. Please be neat. You have 15 minutes reading time. You have 3 hours to complete the entire exam. MATH 1014 – Book A, Page 1 of 14 Question: 1(Calculus: Taylor series) 10P (a) Determine the centre, radius, and interval of convergence of the power series 1X n=1 ( 1)n n32n (x + 1) n : (5P) Begin Your Solution Here MATH 1014 – Book A, Page 2 of 14 Continue Your Solution Here MATH 1014 – Book A, Page 3 of 14 (b) Consider the function f (x) = e x e x 2 : Find T4, the 4th degree Taylor polynomial, for f (x) centred at 0. (5P) Begin Your Solution Here MATH 1014 – Book A, Page 4 of 14 Question: 2(Calculus: parametric curve) 14P Consider the curve given by the parametric equations for 0< < 2 : ( x = sin ; y = 1 cos: (a) Find the slope of the tangent line at = 4 . (4P) (b) For what values of is th
e tangent line horizontal (2P) (c) Find the second order derivatived2ydx2 . Is the curve concave up or concave down (4P) (d) Find the area A under the curve. (4P) Begin Your Solution Here MATH 1014 – Book A, Page 5 of 14 Continue Your Solution Here MATH 1014 – Book A, Page 6 of 14 Continue Your Solution Here MATH 1014 – Book A, Page 7 of 14 Question: 3 (Calculus: polar curve) 6P Consider the polar curve r ( ) = 2(sin + cos ); 0 = 2: Find the arc length L of the curve. Begin Your Solution Here MATH 1014 – Book A, Page 8 of 14 Question: 4 (Calculus: partial derivatives) 6P Show that f (x; y; z) = 1p x2 + y2 + z2 satisfies the Laplace equation @2f @x2 + @2f @y2 + @2f @z2 = 0 for all (x; y; z) 6= (0; 0; 0). Begin Your Solution Here MATH 1014 – Book A, Page 9 of 14 Continue Your Solution Here MATH 1014 – Book A, Page 10 of 14 Question: 5 (Calculus: maxima and minima) 14P Consider the function f (x; y) = x3 + y2 3x 2y: (a) Find the critical points of f . (5P) Begin Your Solution Here MATH 1014 – Book A, Page 11 of 14 (b) Classify the critical points in Part (a) as local maxima, local minima, or saddle points. (5P) Begin Your Solution Here MATH 1014 – Book A, Page 12 of 14 (c) Consider the parametric curve ( x = t2; y = t3 5: Using the chain rule, find dfdt , where f is the same function as before: f (x; y) = x3 + y2 3x 2y: Express your answer as a function of t. (4P) Begin Your Solution Here MATH 1014 – Book A, Page 13 of 14 Extra page intentionally left blank MATH 1014 – Book A, Page 14 of 14