MAT 137Y, 20082009 Winter Session, Solutions to Supplementary Problem Set #3 1. (SHE 12.5)
1 1 4. Given ak = (1)k k ln k , we know that |ak | = k ln k diverges by the integral test (see 11.2 #21), but ak converges by the alternating series test, so the se
University of Toronto, MAT 137Y A Note About Integration Techniques for Term Test 3 One of the most common questions asked is whether one is responsible for memorizing the table of integrals that is provided in the front and back covers of the text. The a
Department of Mathematics, University of Toronto MAT 137Y, 2008-09 Winter Session
Problem Set Supplement #1 Not to be handed in.
Assignment Posted/Revised: November 4, 2008, 11:03
While the following problems do not have to be handed in, you will be respo
Department of Mathematics, University of Toronto MAT 137Y, 2008-09 Winter Session As a reminder, the following rules will be strictly enforced.
Problem Set #11 Deadline: Thursday March 26, 6:10 p.m.
Assignment Posted/Revised: March 17, 2009, 22:16
1. The
Department of Mathematics, University of Toronto MAT 137Y, 2008-2009 Winter Session
Problem Set #1 Deadline: Thursday September 25, 6:10 p.m.
Assignment Posted/Revised: September 22, 2008, 16:23
Read the following instructions carefully! It contains guide
MAT 137Y, Limit Proofs and Some Examples We will do our first example at great length with much commentary and explanation. You should not regard it as a model for your proofs but as a guide to thinking through your solution. Example 1. Find a number > 0
MAT 137Y An Introduction to Proofs One of the challenges that most students encounter in this course is the ability (or lack thereof) to write proofs. This is because most high schools tend to focus on the computational aspect of mathematics. However, uni
Department of Mathematics, University of Toronto Term Test 3 March 10, 2004 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes
Examiners: V. Blomer, K. Consani, M. Harada, G. Leuschke, D. Miller, M. Pinsonnault, P. Rosenthal, S. Uppal, R. Wendt
1. Comput
Department of Mathematics, University of Toronto Term Test 3 March 12, 2003 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes
Examiners: B. Begun, K. Consani, J. Colliander, A. del Junco, J. Korman, R. Rotman, D. Slepcev, R. Sreekantan, S. Uppal
1. Eval
Department of Mathematics, University of Toronto Term Test 3 March 13, 2002 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes 1. Evaluate the following integrals. (6%) (i) (6%) (ii) (7%) (iii) (7%) (iv) (7%) (v) (7%) (vi)
31
1
12
5
dx . xx4
1
0
x arctan
MAT 137Y, 2003-2004 Test 3 Solutions 1. Compute the following integrals.
Z
(10%) (i)
Z Z
52+x dx. 5
2+x
Z
dx = 25
5x dx = 25
5x + C. ln 5
(10%) (ii)
x3 (ln x)2 dx.
2 ln x x
Let I be the integral and integrate by parts. We let u = (ln x)2 , dv = x3 dx. The
MAT 137Y, 2008-2009 Winter Session, Solutions to Term Test 1 1. Evaluate the following limits. (Do not prove them using the formal definition of limit.) (10%) (i) lim x sin x . 1 - cos x Multiplying top and bottom by 1 + cos x, we have
x0
x sin x(1 + cos
Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y Calculus! Friday, April 23, 1999 Time Alloted: 3 hours Examiners: G. Baumgartner, A. Hwang, B. Khesin, B. Madore, R. Pyke 1. For each of the series below, mark the appropria
Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y Calculus! Monday April 27, 1998 Time Alloted: 180 minutes Examiners: M.D. Choi, F. Recio, P. Rosenthal, F. Sottile (15%) 1. Sketch the graph of the function
clearly stating
Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y Calculus! Thursday May 1, 1997 Time Alloted: 180 minutes Examiners: E. Hironaka, A. Hwang, B. Khesin, P. Rosenthal (12) 1. An aluminum can in the form of a cylinder is to co
Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y Calculus! Monday May 6, 1996 Time Alloted: 180 minutes (12) 1. What is the largest area of a rectangle having two vertices on a semicircle of radius 10 and one side along th
Faculty of Arts and Science University of Toronto MAT 137Y1Y Calculus! April/May Examinations; May 4, 2006 Time Alloted: 3 hours
Examiners: I. Alexandrova, M. Harada, V. Ivrii, G. Lynch, M. Sparykina, A. Savage
1. Evaluate the following expressions; simpl
Faculty of Arts and Science University of Toronto MAT 137Y1Y Calculus! April/May Examinations; May 4, 2005 Time Alloted: 3 hours
Examiners: P. Blue, M. Branker, D. Cheliotis, N. Derzko, G. Karali, A. Igelfeld, F. Latremoliere, S. Uppal
1. Evaluate the fol
Faculty of Arts and Science University of Toronto MAT 137Y1Y Calculus! April/May Examinations; April 29, 2004 Time Alloted: 3 hours
Instructors: V. Blomer, K. Consani, M. Harada, G. Leuschke, D. Miller, M. Pinsonnault, P. Rosenthal, S. Uppal, R. Wendt
(10
University of Toronto MAT 137Y1Y Calculus! April/May Examinations; April 29, 2003 Time Alloted: 3 hours
Instructors: B. Begun, J. Colliander, K. Consani, A. del Junco, J. Korman, R. Rotman, D. Slepcev, R. Sreekantan, S. Uppal
1. (a) Find the derivatives o
Faculty of Arts and Science University of Toronto April/May 2002 Examinations MAT 137Y, Calculus! Time Alloted: 3 hours 1. Evaluate the following expressions.
(6%) 2. Find the volume of the solid generated by revolving the region bounded by the curves
abo
MATH 137 (U of T) Study Sheet
Properties of Real Numbers
Properties of real numbers: associativity, commutativity, existence of identity, existence of inverses Subsets of real numbers: natural numbers (e.g. 1,2,3,), integers (e.g. -2,-1,0,1,2,), rational
MATH 137 Exam Equation Sheet
PROPERTIES OF REAL NUMBERS Properties of real numbers: associativity, commutativity, existence of identity, existence of inverses Subsets of real numbers: natural numbers (e.g. 1,2,3,), integers (e.g. -2,1,0,1,2,), rational nu