1051 Assignment 2
Due Thursday 1st April 2010, 2 pm
Write your name, student number, tutors name, tutorial group (T1, T2, . . . ) and tutorial time clearly on the top of the front page of your assignment. Staple the pages of your assignment. Locate the bo
MATH1051 Past Exams: Continuity & Differentiability
1. Consider the function
2 3 + 2
() = cfw_ 1
,
,
1
=1
For which value(s) of is continuous at = 1? For these values of , is one-to-one?
[2010/1/#3]
2.
[2010/2/#3]
3.
[2011/1/#3]
4.
[2007/1/#3]
MATH1051/S
MATH1051 Past Exams: Limits
IMPORTANT NOTE:
Leave the following questions out for the moment. You will be able to do these when we cover
LHopitals rule.
2c, 3b, 4c, 7, 8c, 9c, 10, 13c, 15c
1.
[2012/1/ #2]
2.
[2011/2/ #1]
3. Determine the following limits,
14.1 Denition: Continuity
14
134
Continuity
14.1
Denition: Continuity
We say that a function f is continuous at a if
(i) f (a) is dened (that is, a is in the domain of f );
(ii) lim f (x) exists; and
xa
(iii) lim f (x) = f (a).
xa
If f is not continuous a
MATH1051 Sem.1 2015
Past Exam Question by Topic
Worked Solutions
Limits of Sequences
1.
2.
3.
Page 1
MATH1051 Sem.1 2015
Past Exam Question by Topic
Worked Solutions
Limits of Sequences
4.
5.
6.
Q6b
Page 2
MATH1051 Sem.1 2015
Past Exam Question by Topic
W
MATH1051 Sem.1 2015
Past Exam Question by Topic
Worked Solutions
Continuity/Differentiability
1.
2.
Page 1
MATH1051 Sem.1 2015
Past Exam Question by Topic
Worked Solutions
Continuity/Differentiability
3.
lim0+
(1+)(1)
= 1 and lim0
(1+)(1)
= 1
4. To show t
MATH1051 Past Exams (Topics)
Limits of Functions
Worked Solutions
1.
2.
Page 1
MATH1051 Past Exams (Topics)
Limits of Functions
Worked Solutions
3.
4.
Page 2
MATH1051 Past Exams (Topics)
Limits of Functions
Worked Solutions
5.
6.
Alternative solution for
MATH1051 Sem. 1 2015
Past Exam Questions by Topic
Worked Solutions
Derivatives
Question 1
Question 2:
(a) To find all local maxima and minima we solve () = 0. First write
() = ( + 2) +2 = ( 2 + 2) +2
Differentiating w.r.t. , we get () = ( 2 + 2) +2 (1) +
15.1 Tangents
15
140
Derivatives
Finding the instantaneous velocity of a moving object and other problems involving rates of change are
situations where derivatives can be used as a powerful tool. All rates of change can be interpreted as
slopes of approp
Taking the limit:
7.4 Theorem: p-test
For
, the p-series
Is convergent if
(since each term will get smaller and smaller), and divergent if
.
Actually, if
, this reduces to the harmonic series. If you wanted to change the base, you
could do so quite easily
A word of warning if Series B converges, but every term in Series A is larger then it could
very well diverge (or simply converge to a higher value). This test would tell you nothing!
Similarly if Series B diverges, but every term in Series A is smaller t
Chapter 2 - Functions
2.1 Notation
The notation
some input from
means maps from into . In other words, the function transforms
into an output which is in the set .
2.2 Definition: Function, Domain, Range
A function
is a rule which takes every element whic
2.9 Composition of Functions
If
and
are two functions, the composition of
and , denoted
(
, is given by:
)
This basically means you apply the function to some input first, then apply function . This
means that the domain (inputs) of
are given by the possi
MATH1051 Past Exams: Sequences
IMPORTANT NOTE:
Leave the following questions out for the moment. You will be able to do these after we
have completed LHopitals Rule in Section 15.9 of your workbook.
4c, 11c, 14b, 16c
1. Determine whether the following seq
1051 Assignment 2
?
?
?
?
?
?
Due Monday 29th August 2016, 3 pm
Make sure you include your personalized cover page with this assignment.
Staple the pages of your assignment.
Submit your assignment on level 3 of building 67 by the indicated deadline.
Late
MATH1051
Assignment 4
All questions must be submitted by 3 pm on Thursday 20 October. Assignments can be submitted at your
tutorial or to the assignment submission machine (3rd floor, Priestley Building #67). You do need a cover
sheet which will be emaile
1051 Tutorial Problems Sheet #1
Question 1. Find two numbers whose product is 5 and whose sum is 4.
Question 2.
(a) Find a complex number z = a + bi with z 2 = i.
(b) () Find all 6 complex numbers z satisfying z 6 = 1. Plot these 6 numbers in the complex
MATH1051 Sem.1 2013 Final Examination: Solutions
Question 1:
a)
( )
b)
the Squeeze theorem,
c)
( )
; Since
( )
.
, by
(LHopitals, ). Applying LHopitals again, we have
Question 2:
a)
(
)
(
)
. Therefore
b)
Generate a few terms to find a pattern:
Therefore
Semester One Final Examinations, 2013
MATH1051 Calculus and Linear Algebra I
This exam paper must not be removed from the venue
Venue
_
Seat Number
_
Student Number
|_|_|_|_|_|_|_|_|
Family Name
_
First Name
_
School of Mathematics & Physics
EXAMINATION
S
Chapter 1 - Numbers
1.1 Number Systems
Set of natural numbers
. Subset of , and .
: Set of integers
. Subset of and .
: Set of rational numbers, which are of the form where
and
. Subset of .
: Set of real numbers represented by a finite or infinite decim
Chapter 5 - Continuity
5.1 Definition: Continuity
We say that a function
is continuous at a point
if:
exists
The limit at
exists
Picture a graph of the function. The first criterion ensures theres a point somewhere on the
graph when
. The second criterion
6.8 Derivative of Inverse Function
Useful examples here, but mostly more practice. We do, however, have some more handy
derivatives (and by handy, we mean that itll probably be on your exams):
6.9 LHpitals Rule
I believe this is pronounced Le Pee-Tals the
6.12 Definition: Increasing / Decreasing
A function
is strictly increasing if:
A function
is strictly decreasing if:
6.13 Increasing / Decreasing Test
If
, then is strictly increasing on
to values getting larger (more positive).
. The positive rate of cha
Chapter 6 Derivatives
Derivative has a stigma to it, since its part of calculus and calculus is regarded as difficult.
Derivatives simply involve rates of change (which are slopes on a graph).
6.1 Tangents
Lets say we want to determine the tangent line at
Chapter 7 - Series
7.1 Infinite Sums (Notation)
Recall that an infinite sum is represented as:
The lower bound (
) may vary.
Notice that a sequence is simply a list of terms
but a series is a sum
.
7.2 Motivation
Math is generally fairly dry. So you know
This is true no matter what value of you have. You simply cant get a negative output from ,
since a negative exponent simply makes the output smaller, not positive. Accordingly, you cant
input a negative number into
its not a possible output to the norma
1051 Assignment 2
?
?
?
?
?
?
Due Monday 29th August 2016, 3 pm
Make sure you include your personalized cover page with this assignment.
Staple the pages of your assignment.
Submit your assignment on level 3 of building 67 by the indicated deadline.
Late
1051 Assignment 3
?
?
?
?
?
?
Due 19th September 2016, 3 pm
Make sure you include your personalized cover page with this assignment.
Staple the pages of your assignment.
Submit your assignment on level 3 of building 67 by the indicated deadline.
Late assi
1051 Assignment 3
?
?
?
?
?
?
Due 19th September 2016, 3 pm
Make sure you include your personalized cover page with this assignment.
Staple the pages of your assignment.
Submit your assignment on level 3 of building 67 by the indicated deadline.
Late assi