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York University (9,811)
MATH 1505 (56)
Final

# 2008exam.pdf

18 Pages
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School
York University
Department
Mathematics and Statistics
Course
MATH 1505
Professor
Alfred Pietrowski
Semester
Fall

Description
York University Faculty of Arts, Faculty of Science and Engineering Final Examination April 17, 2008 Mathematics 1505.06 Mathematics for Life and Social Sciences NAME (print): (Family) (Given) SIGNATURE: STUDENT NUMBER: Section A, MWF @ 8:30 (CLH A) - Prof. Chawl Section B, MWF @ 9:30 (CLH A) - Prof. Pietrowsk Section C, MWF @ 9:30 (VH C) - Prof. Grigull Section Section D, T @ 7 (CLH G) - Prof. Mohammed Section E, MWF @ 10:30 (CLH E) - Prof. Chawla Section G, MWF @ 10:30 (CLH K) - Prof. Raguimov Instructions: 1. You have to answer all questions and show all your work. 2. Put answers and rough work on the question paper, using the back pages if necessary. 3. Nonprogrammable and nongraphing calcula- tors are allowed. 4. Exam is for 3 hours. 5. This exam has 18 questions. Make sure you have everything. 6. Total number of mark is 200. Page 1 Mathematics 1505.06 Final Exam. April 17, 2008 1. (8 marks) (a) Solve |3x + 5| < 1. Express your answer in interval notation. √ (b)Find the largest possible domain of the function f(x) =2 − 3x. Express your answer in interval notation. continues ... Page 2 Mathematics 1505.06 Final Exam. April 17, 2008 2. (8 marks) 2 (a) Solve ln(4x − 2x ) − ln(x) = ln(2x − 1). (b) Find all values of α in the interval [0,2π) that satisfy the equation 2sinαcosα = cosα. continues ... Page 3 Mathematics 1505.06 Final Exam. April 17, 2008 3. (24 marks) Determine the following limits: 2e + 3e −x (a) lim x −x x→∞ 5e − 7e 4x − 1 (b) lim x→ 12x + 5x − 3 2 continues ... Page 4 Mathematics 1505.06 Final Exam. April 17, 2008 √ √ (c) lim 2x − 1 − 2x − 3x x→∞ 3 2 − ! (d) lim x2 x x→∞ 2 7 2+ x x continues ... Page 5 Mathematics 1505.06 Final Exam. April 17, 2008 4. (8 marks) A toxin is introduced into a bacterial colony, and t hours later, the population is given by N(t) = 10000(8 + t)e −0.1t (a) What was the population when the toxin was introduced? (b) When is the population maximized? Justify your answer. (c) Find the maximum population. continues ... Page 6 Mathematics 1505.06 Final Exam. April 17, 2008 5. (18 marks) Given a function y = f(x) = x −21 . x (a) Determine intervals on which the function is strictly increasing, strictly decreasing. Find the coordinates of all local (relative) maximum and minimum points (if any). (b) Determine intervals on which the function is concave up, concave down. Find the coordinates of all inﬂection points (if any). continues ... Page 7 Mathematics 1505.06 Final Exam. April 17, 2008 (c) Sketch the graph of the function. 6. (5 marks)Find a function F(x) such that F (x) = x and F(2)=0. continues ... Page 8 Mathematics 1505.06 Final Exam. April 17, 2008 7. (30 marks) Calculate the derivatives for the following functions: (a) 8ex2 2 (b) (1 + 3x) (1 + 2x) 3 (c) ln(x) · cos(a + bx), where a and b are constants. continues ... Page 9 Mathematics 1505.06 Final Exam. April 17, 2008 x/2 (d) 2 (e) Let f(x) = 2x + e . Find d (f−1 (4 + e )). Note that f(2) = 4 + e .
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