Test2012.pdf

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Department
Mathematics
Course
MATA30H3
Professor
Sophie Chrysostomou
Semester
Winter

Description
University of Toronto Scarborough Department of Computer & Mathematical Sciences MATA30: Calculus I - Midterm Test Examiner: Sophie Chrysostomou Date: Friday, November 23, 2012 Duration: 110 minutes DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO. FAMILY NAME: GIVEN NAMES: STUDENT NUMBER: DAY AND TIME OF YOUR TUTORIAL: SIGNATURE: CIRCLE THE NAME OF YOUR TEACHING ASSISTANT: Fazle Chowdhury Kathleen Smith Jianhao Yang NOTES: • No calculators, cell phones or any electronic aids are permitted at the test room. • No books, notebooks or scrap paper are permitted near your examination table. • There are 12 numbered pages in the test. It is your responsibility to ensure that, at the start of the test, this booklet has all its pages. The last two pages (11 and 12) are empty. • Please leave all the pages of this booklet stapled. Do not remove any pages. • Answer all questions in the space provided. Show your work and justify your answers for full credit. FOR MARKERS ONLY 1 / 10 2 /10 3 a / 5 3 b / 5 3 c / 5 3 d / 5 3 e / 5 4 / 6 5 / 7 6 / 10 7 / 7 8 a / 5 8 b / 5 8 c / 5 8 d / 5 8 e / 5 TOTAL /100 MATA30Y page 1 1. (a) [ 4 marks] Give the definition of g(x) = arctanx and give the domain and its range. (b) [ 2 marks] Give the graph of g(x) = arctanx. (c) [ 4 marks] Find the exact value of the expression: ▯ ▯ ▯▯ ▯√ ▯ 1 cos arctan 3 + arcsin√ 2 MATA30Y page 2 −lo4 (x)+l9g (3) 2. (a) [ 4 marks] Find all x satisfying: 4 = 3 sinx + sin(2x) (b) [ 6 marks] Prove the identity: = tanx. 1 + +cos(x)cos(2x) MATA30Y page 3 3. [5 marks each; 25 marks total] Evaluate each of the following limits, else explain why the limit does not exist. Justify your answer. Do notˆspital’s rule. 2 (a) lim√x − 9 x→3 3 − √x ▯ ▯ ▯ ▯ 2 1 1 (b) lim x sin cos x→0 x x
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