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Name (print): ID number: University of British Columbia MIDTERM TEST: SCIE 001 MATHEMATICS Date: February 14, 2012 Time: 8:30 a.m. to 9:20 a.m. Number of pages: 7 (including cover page) Exam type: Closed book Aids: No calculators or other electronic aids Rules governing formal examinations: Each candidate must be prepared to produce, upon request, a UBC card for identification. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: • Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners; • Speaking or communicating with other candidates; • Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received. Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator. Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator. For examiners’ use only Question Mark Maximum mark 1 9 2 7 3 6 4 5 5 7 Total 34 1. Evaluate the following integrals. Z 1. (a) [3 marks] x cos 5x dx Z 1. (b) [3 marks] 0 Z 1. (c) [3 marks] 1 y2 y+1 dy + 2y − 5 sin3 ωt dt, where ω 6= 0 is a constant 2 2. [7 marks] Using the general slicing method, evaluate a definite integral to find the volume of a tetrahedron (pyramid with four triangular faces), all of whose edges have a length L. Remember to give justification. 3 3. [6 marks] Prove that 100 X k=1 1 ek/100 1 1 < 100 1 − . (Hint: consider the function f (x) = x .) e e 4 4. [5 marks] For a mass m kg at a height h ≥ 0 m above the surface of the earth, the gravitational force, in N, acting on the mass is given by F~ = −GM m(R + h)−2 r̂, where G is the gravitational constant in N m2 /kg2 , M is the mass of the earth in kg, R is the radius of the earth in m, and r̂ is the unit vector pointing away from the centre of the earth. Determine the work required to launch a rocket with a mass of 500 kg to a height of 2500 km above the earth’s surface. 5 5. Let f (x) = sin 0 1 x if x 6= 0 . if x = 0 5. (a) [2 marks] Let δ be any number in the interval (0, 1]. Consider a partition P of the interval [0, 1] whose first subinterval is [0, δ]. On that subinterval, what is the maximum value of δf (x)? What is the minimum value of δf (x)? 5. (b) [5 marks] Prove that f is integrable on the interval [0, 1]. You may use, without proof, the fact that if a function is continuous on [δ, b], then it is integrable on [δ, b]. 6 This page may be used for rough work. It will not be marked. 7