University of Waterloo
MATH 104 Winter 2013
Final Exam
Monday, April 15, 2013 (12:30-3:00 pm)
Instructor: Shahla Aliakbari
: 2< x 0
1
[8 marks] 1. Let f ( x) = x
: 0 < x 1 . Sketch this function, and indicate its domain.
2 x :
x >1
[12 marks] 2. Evaluate
MATH 104 Fall 2014
Assignment 4
Due: Tuesday, October 21 (4:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clear
MATH 104 Fall 2014
Assignment 3
Due: Monday, October 6 (04:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clearl
MATH 104 Fall 2014
Assignment 1
Due: Monday, September 22 (4:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clea
MATH 104 Winter 2014
Assignment 2
Due: Monday, September 29 (4:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
cl
MATH 104 Fall 2014
Tutorial 4
[Short summary of main topics and extra examples.]
Rate of changes
f ( a + h) f ( a )
h
f ( a + h) f ( a )
, provided the limit exits
The instantaneous rate of change = lim h0
h
The average rate of change =
Geometrical interp
MATH 104 Fall 2014
Tutorial 3
[Short summary of main topics and extra examples.]
Logarithmic functions: If a > 0 and a 1 , then the logarithmic function with base a is defined
as y = log a x x = a y .
Note that log a a = 1 , log a 1 = 0 , log a a x = x ,
MATH 104 Fall 2014
Assignment 5
Due: Monday, October 27
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clearly you expr
MATH 104 Winter 2013
Assignment 2, Solutions
1. Textbook, Page 154, Exercises 3.4: # 32, 38, 44
We could use a few points as well:
2. Textbook, Page 189, Exercises 4.1: # 28
3. Textbook, Page 197, Exercises 4.2: # 8
4. Sketch the graph of f ( x) = ( x + 1
MATH 104 Fall 2014
Assignment 5, Solutions
1. Textbook, Exercises 11.7: # 28, 48, 64
2. Textbook, Exercises 11.8: # 36, 46, 72
3. The average molecular velocity of a gas in a certain container is given by v = 29 T (in meters
per second), where T is the te
University of Waterloo
MATH 104 Winter 2013
Final Exam, SOLUTIONS
Monday, April 15, 2013 (12:30-3:00 pm)
: 2 x 0
1
[8 marks] 1. Let f ( x) x
: 0 x 1 . Sketch this function, and indicate its domain.
2 x :
x 1
Solution: The domain is D f (2, ) .
[12 marks]
MATH 104 Fall 2014
Assignment 7, Solutions
1. Textbook, Exercises 12.3: # 26, 40, 48
2. Textbook, Exercises 12.6: # 4, 16
3. Consider a company with cost and price functions:
C ( x) = 12000 + 4 x + 0.0002 x 2 and p( x) = 12 0.0001 x
(a) Determine the prod
MATH 104 Fall 2014
Assignment 9, Solutions
1. Textbook, Exercises 13.4: # 34, 40, 52
2. Textbook, Exercises 13.5: # 16, 26, 34
3. Assume that the average value of the function y = f (x) from x = a to x = b is given by:
f ave =
1
ba
b
f ( x)dx
a
The tempe
MATH 104 Fall 2014
Assignment 8, Solutions
1. Textbook, Page 753, Exercises 13.1: # 26, 42, 50
2. Textbook, Page 761, Exercises 13.2: # 6, 16, 48
3. A ball is thrown vertically upward with an initial velocity of 25 meters per second (m/s) from
an initial
MATH 104 Fall 2014
Assignment 9
Due: Monday, December 1 (4:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clearl
MATH 104 Fall 2014
Assignment 6, Solutions
1. Textbook, Page 688, Exercises 12.1: # 12, 38, 56
2. Textbook, Page 700, Exercises 12.2: # 24, 36, 42
3. Let f ( x) =
x+3
x 2 +1
.
1 3x
and find the intervals where f ( x) is increasing or decreasing.
( x 2 +1)
MATH 104 Fall 2014
Assignment 8
Due: Monday, November 24 (4:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clear
MATH 104 Fall 2014
Assignment 7
Due: Monday, November 17 (4:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clear
MATH 104 Fall 2014
Assignment 6
Due: Monday, November 3 (4:30 PM)
[Note: Although you are allowed to discuss the problems with other students, the TA, or the
instructor, the solutions you submit must be your own. Your mark will be influenced by how
clearl
MATH 104 Fall 2014
Tutorial 8
[Short summary of week 8 main topics and extra examples.]
The function F (x) is said to be anti-derivative of f (x) if F ( x) = f ( x) .
Indefinite integral:
f ( x) dx = F ( x) + C , where F ( x) = f ( x) (or F ( x) dx = F (
MATH 104 Fall 2014
Tutorial 5
[Short summary of main topics and extra examples.]
Composition of functions: The composition of f (x) and g (x) is defined by
( fog )( x) = f ( g ( x) . The domain of ( fog )( x) is defined such that x Dg and g ( x) D f .
Cha
MATH 104 Fall 2014
Tutorial 7
[Short summary of main topics and extra examples.]
Guideline for sketching the graph of y = f (x) :
Domain, Intercepts, Asymptotes
First and second derivatives
- Intervals of increasing or decreasing
- Local minimum and maxim
MA104 Week 6 Lecture Notes
Sequences [Text Reference: 11.1]
S a sequence [of real numbers] can be thought of as a list a 1 , a 2 , a 3 , . . . , a n , . . . of real
numbers, usually indexed by the natural numbers [ elements of 3 1, 2, 3, . . . ] and denot
MA104 Week 3 Lecture Notes
Arc Length [Text Reference: 8.1]
S Suppose a curve . is defined by an equation y fx, where a J x J b What is the length of
.?
S As usual, we can partition the interval a, b using points along the x-axis
a x 0 x 1 C x i1 x i C x
MA104 Week 5 Lecture Notes
Polar Coordinates [Text Reference: 10.4]
S the Cartesian coordinate system is our familiar x-y system
S polar coordinate system:
a point O in the plane is chosen, called the pole or origin
a ray (half-line) emanating from O is
MA104 Week 4 Lecture Notes
Tangents and Areas (Parametric Form) [Text Reference: 10.2]
Tangent Lines
S Assume that for a parametric curve x f t , y g t , the parameter may be eliminated, so that
the curve may be defined by an equation y F x . If the funct
MA104 Week 7 Lecture Notes
Series [Text Reference: 11.2]
S we have considered infinite sequences and their convergence
S we now want to give meaning to the sum of an infinite sequence
.
n1 a n
a1 a2 a3
.
C
we call a n a [infinite] series.
n1
S Defini
Lecture 1
Introduction
Recall that the velocity v of a moving object is given by the dierential equation:
dx
,
dt
where x and t respectively denote distance and time. This is a typical example of a dierential
equation. There are numerous situations in the