Problem Set 2 - Due by Thursday, October 4, 2012
Posted/Revised: September 26, 2012
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character t
Problem Set 5 - Due by Thursday, Nov. 22, 2012
Posted/Revised: November 16, 2012
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tuto
Problem Set 7 - Due by Thursday, Jan. 31, 2013
Posted/Revised: January 24, 2013
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tutor
Problem Set 8 - Due by Thursday, Feb. 14, 2013
Posted/Revised: February 7, 2013
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tutor
Problem Set 8 - Solutions
* 2. * (a)
Find all values for c such that the area between the graphs of f (x) = cx 1,
x = 0, x = 2, and the x-axis is equal to 1.
Solution: First we note that c 0, since that would create a rectangle which has
area equal to 2.
Problem Set 9 - Due by Thursday, March 7, 2013
Posted/Revised: February 25, 2013
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tuto
Problem Set 10 - Due by Thursday, March 28, 2013
Posted/Revised: March 21, 2013
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tutor
Problem Set 9 Solutions
* 2. (a) SHE 8.2 # * 34 , * 48 , * 76
x
cos( x)dx.
Solutions: 8.2 #34
cos( x)dx =
x
cos( x)
1
dx. Then du = 2x dx and v = 2 sin( x). This
Let u = x and dv =
x
cos( x)
dx.
last expression can be found by substituting u and du into
Problem Set 10 - Solutions
* 2. SHE 11.3 # * 52 , SHE 11.4 # * 48
Solution:
# 52:
Let m bn M , since cfw_bn is bounded. Then m an an bn an M . We can
apply Pinching Theorem for sequences.
# 48:
Since the sequence cfw_ak converges, we can nd for any
ak L
TERM TEST 4 ANNOUNCEMENT
Last posted/revised on Wednesday, March 13, 2013
Term Test 4 will be written on Wednesday, March 20, 2013 from 6-8pm, with the early
sitting at 4-6pm. Locations will be announced on the course website.
The test will cover sections
DEPARTMENT OF MATHEMATICS
University of Toronto
MAT137Y1
Calculus!
Practice for Test 1 Solutions
Duration: 1 hour 45 minutes
Total: 50 marks
Surname:
(Please Print)
First Name(s):
(Please Print)
Tutorial Code (e.g., MZA, etc):
Student ~
Problem Set 4 - Solutions
2. * (a)
Give an example of a function that is continuous on an interval (a, b), but
not dierentiable on the entire interval. It may, however, be dierentiable at some of
the points in the interval.
x
when x 0
. The derivative of
Problem Set 4 - Due by Thursday, Nov. 8, 2012
Posted/Revised: November 2, 2012
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tutori
Problem Set 2 - Solutions
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
* 2. (a) SHE 1.3 # * 52 , * 58
Solution to 52:
a b = a + (b) a + b by the triangle inequality, which
is equal to a + b . Therefore, a b a + b .
1
1
a = ( a)2 = a a a
Problem Set 5 - Due by Thursday, Nov. 22, 2012
Posted/Revised: November 16, 2012
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tuto
Problem Set 4 - Due by Thursday, Nov. 8, 2012
Posted/Revised: November 5, 2012
Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen
Problems marked with * are to be handed in during lecture or in SS1071.
Please write your three character tutori
TERM TEST 1 ANNOUNCEMENT
Last posted/revised on Sunday October 21, 2012
Term Test 1 will be written on Wednesday, October 31, 2012 from 6-8pm, with the early
sitting at 4-6pm. Locations will be announced on the course website by the Wednesday
before the t
TERM TEST 3 ANNOUNCEMENT
Last posted/revised on Tuesday, January 15, 2013
Term Test 2 will be written on Wednesday, January 23, 2013 from 6-8pm, with the early
sitting at 4-6pm. Locations will be announced on the course website.
The test will cover sectio
TERM TEST 2 ANNOUNCEMENT
Last posted/revised on Wednesday November 21, 2012
Term Test 2 will be written on Wednesday, November 28, 2012 from 6-8pm, with the early
sitting at 4-6pm. Locations will be announced on the course website by the Wednesday
before
DEPARTMENT OF MATHEMATICS
University of Toronto
MAT137Y1
Calculus!
Practice for Test 2
Duration: 1 hour 45 minutes
Total: 50 marks
Surname:
(Please Print)
First Name(s):
(Please Print)
Tutorial Code (e.g., M2A, etc.):
Student ID Number:
PLEASE CIRCLE THE
DEPARTMENT OF MATHEMATICS
University of Toronto
MAT137Y1
Calculus!
Practice for Part B of Test 3
Part B Long Answer
B1.
Let f (x) =
x
.
1 + x2
(a) Find f (x).
[2 marks]
(b) On which interval(s) is f (x) increasing? On which interval(s) is it decreasing? [
Supplementary Material 1: Logic Quantiers
In mathematics, we frequently wish to talk about elements in a set. For example, when
we say n is an integer, we are letting n be an element of the set of integers. If we want to
say something about all integers,
Supplementary Material 2: Bounds on Sets of Real Numbers
111
Consider the sequence 1, , , , . . . Each successive number is smaller than the previous.
234
In fact, the terms in this sequence get innitely small, while remaining positive. In other
words,
cf
Supplementary Material 3: Conditional Statements
In mathematics, many theorems are phrased as an if-then statement. For example, the
Fundamental Theorem of Algebra for polynomials with real coecients can be stated as:
If f (x) is a polynomial of degree n