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Date: October 24, 2013 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.

No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.

Candidates suspected of any of the following, or similar, dis honest practices shall be immediately dismissed from the exam ination and shall be liable to disciplinary action: • Having at the place of writing any books, papers or memo randa, calculators, computers, sound or image players/record ers/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 candi dates or imaging devices. The plea of accident or forgetfulness shall not be received.

Candidates must not destroy or mutilate any examination ma terial; 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 di rections communicated by the instructor or invigilator.

For examiners’ use only Question Mark Possible marks 1 6 2 3 4 5 Total 4 6 4 5 25

Please note that your answers must be in “calculator-ready” form, but they do not have to be simplified. You do not have to use the δ − ε definition of limit unless you are explicitly asked to.

This page may be used for rough work. It will not be marked.

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1. (a) [3 marks] Prove, using the δ − ε 1 definition of limit, that lim x → 2 x = 1 2 .

(b) [3 marks] Prove, using the δ − ε 1 definition of limit, that lim x → 0 x = 0.

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2.

[4 marks] Imagine a fish on a cutting board. Prove that you can cut the fish into two portions of equal volume with a single straight cut. (You may assume that the fish is finite in extent. If you wish, you may also assume that the fish is made of mashed potatoes.) 4

3. (a) [3 marks] Evaluate lim t → 0 t + sin(2 t ) , or explain why the limit does not exist.

tan(7 t ) (b) [3 marks] Find f 0 (1).

Let f ( x ) = g cos πx 2 , g (0) = 1, g (1) = − 2, g 0 (0) = 3 and g 0 (1) = − 4.

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4.

[4 marks] or not Let f ( x ) = sin( | x | ). Use the limit definition of derivative to determine whether f 0 (0) exists.

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5.

[5 marks] A turkey (again, you may assume if you wish that the turkey is made out of mashed potatoes) is put into an oven that has a constant temperature of 200 ◦ C. A thermometer in the turkey registers its temperature. When the turkey is first put in the oven, the thermometer reads 20 ◦ C. Thirty minutes later, it reads 30 ◦ C. If the turkey is cooked when the thermometer reads 80 ◦ C, how long will the turkey have to be in the oven before it is cooked? Use Newton’s law of cooling (or warming) to obtain a differential equation for the temperature read by the thermometer, and solve the differential equation.

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