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 assignm
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]
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/
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
MATH1051 Past Exams: Derivatives
The table lists the types of questions for this topic:
Question Type
Finding Derivatives
Characteristics of Function: Max/Min,
Increasing, Decreasing etc
Optimisation
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
P
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
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 Fun
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
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
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
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 Ser
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, Ra
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
m
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.
4
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
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)
(
)
(
)
. Therefor
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
_
Fir
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
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 p
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 LHp
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 get
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 Tan
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.
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 negat
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
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 leve
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 leve
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 Buildi
MATH1051
CALCULUS
AND
LINEAR ALGEBRA I
Semester 1, 2016
Lecture Workbook
How to use this workbook
This book should be taken to lectures, tutorials and lab sessions. In the lectures, you will be expect