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Name (print): ID number: University of British Columbia MIDTERM TEST: SCIE 001 MATHEMATICS Date: October 20, 2011 Time: 8:30 a.m. to 9:20 a.m. Number of pages: 8 (including cover page) Exam type: Closed book Aids: No calculators or other electronic aids Rules governing formal examinations: For examiners’ use only 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. Question Mark Maximum mark 1 6 2 6 3 6 4 6 5 5 6 6 Total 35 1. Calculate the following limits, or explain why they do not exist (δ − ε proofs are not necessary, but show steps in the calculation). x 1. (a) [3 marks] lim q x→∞ 2 (2x + 1) x2 − 14x + 49 x→7 sin (x − 7) 1. (b) [3 marks] lim 2 2. Find the derivatives of the following functions (after differentiating, simplification is not necessary). 2. (a) [3 marks] f (x) = p x+ √ x 2. (b) [3 marks] e3x ln(5x + 1) 3 3. [6 marks]Let f (x) = x 1 2x +1 if x ≤ 2 . if x > 2 3. (a) Sketch the function, making sure to label your axes. 3. (b) Let ε = 34 . Find some δ > 0 such that |f (x) − 2| < ε whenever 0 < |x − 2| < δ. Explain why your choice of δ is suitable. You may refer to your picture in part (a). 4 4. [6 marks] Let f be a function such that |f (x)| ≤ x2 for all x. 4. (a) Prove that f is continuous at 0 (a δ − ε argument is not necessary). 4. (b) Prove that f is differentiable at 0 (a δ − ε argument is not necessary). 5 5. [5 marks] A function f is said to have a fixed point at c if f (c) = c. For example, the function f (x) = x2 has fixed points at 0 and 1. Let f be any continuous function with domain [0, 1] and range [0, 1]. Prove that f has a fixed point. (Hint: consider the function g(x) = f (x) − x.) 6 6. [6 marks] Find all nonnegative values of the constant E ≥ 0 such that there is a function Ψ(x) which is: not identically zero, differentiable on the interval [0, 1], twice differentiable on the interval (0, 1), and which satisfies the second-order ordinary differential equation Ψ00 + EΨ = 0, 0 < x < 1, and the two boundary conditions Ψ0 (0) = 0, Ψ0 (1) = 0. 7 This page may be used for rough work. It will not be marked. 8