MATH1131 Mathematics 1A Calculus Chapter 01 Sets, Inequalities and Functions
Dr. Thanh Tran
1
Sets of Numbers Solving Inequalities Functions Implicitly Dened Functions Continuous Functions
2
3 School of Mathematics and Statistics The University of New Sou
UCLA
MATH 131A, Analysis
Winter 2016
Martin Gallauer
Homework 6
Due: February 12, 2016
1
We saw in class that the square function x2 : R R is uniformly continuous on each
bounded subset S R. Prove that x2 is not uniformly continuous on R.
2
Prove each of
UCLA
MATH 131A, Analysis
Winter 2016
Martin Gallauer
Homework 4
Due: January 29, 2015
1
In class we proved the following implication:
Completeness Axiom
All bounded monotone sequences converge
Prove now the converse, establishing equivalence. (Admittedly,
UCLA
MATH 131A, Analysis
Winter 2016
Martin Gallauer
Homework 5
Due: February 5, 2016
1
Consider the series
n=1
(1)n+1
1 1 1 1 1 1
= 1 + + + = ln(2).
n
2 3 4 5 7 8
(1)
By the Alternating Series Test, it converges to a real number which we call ln(2). By t
UCLA
MATH 131A, Analysis
Winter 2016
Martin Gallauer
Homework 3
Due: January 22, 2015
1
Do problem 9.6 in Ross.
2
Prove the following claim made in class: If (sn ) is a sequence with sn > 0 for all n then
lim sn = lim(
1
) = 0.
sn
3
Do problem 9.12 in Ros
UCLA
MATH 131A, Analysis
Winter 2016
Martin Gallauer
Homework 2
Due: January 15, 2015
1
It was claimed in class that Q does not satisfy the completeness axiom. In this problem
you should prove this, i.e. show the existence of a subset S Q satisfying
(a) S
MATH1131 Mathematics 1A Calculus Chapter 10 The Hyperbolic Functions
Dr. Thanh Tran
School of Mathematics and Statistics The University of New South Wales Sydney, Australia
1 2 3 4 5 6 7 8
Hyperbolic sine and cosine functions Other hyperbolic functions Tr
MATH1131 Mathematics 1A Calculus Chapter 09 Logarithmic and Exponential Functions
1
The natural log function
Dr. Thanh Tran
School of Mathematics and Statistics The University of New South Wales Sydney, Australia 2
The exponential function
1
2
The natural
MATH1131 Mathematics 1A Calculus Chapter 08 Integration
Dr. Thanh Tran
School of Mathematics and Statistics The University of New South Wales Sydney, Australia
1 2 3 4 5 6 7 8
Riemann sums and Riemann integrals Riemann integrals and areas Basic properties
MATH1131 Mathematics 1A Calculus Chapter 07 Curve Sketching
Dr. Thanh Tran
School of Mathematics and Statistics The University of New South Wales Sydney, Australia
1
Curves dened by a Cartesian equation
2
Parametrically dened curves
3
Curves dened by pola
MATH1131 Mathematics 1A Calculus Chapter 06 Inverse Functions
Dr. Thanh Tran
1
Some examples One-to-one (or injective) functions Inverse functions The Inverse Function Theorem Inverse Trigonometric Functions
2
3 School of Mathematics and Statistics The Un
MATH1131 Mathematics 1A Calculus Chapter 05 The Mean Value Theorem and Its Applications
Dr. Thanh Tran
1
The Mean Value Theorem Applications of the MVT Sign of the Derivative Critical, Global Max. and Min. Points Counting Zeros of Functions Antiderivative
MATH1131 Mathematics 1A Calculus Chapter 04 Differentiable Functions
Dr. Thanh Tran
1
Gradients of tangents and derivatives Rules for differentiation Differentiability and continuity Implicit differentiation Derivatives and rates of change Local maxima an
MATH1131 Mathematics 1A Calculus Chapter 03 Properties of Continuous Functions
Dr. Thanh Tran
School of Mathematics and Statistics The University of New South Wales Sydney, Australia
1
Combining Continuous Functions
2
The Intermediate Value Theorem
3
The
UCLA
MATH 131A, Analysis
Winter 2016
Martin Gallauer
Homework 7
Due: February 19, 2016
1
Do problem 19.2 in Ross.
2
For each of the following functions do the following:
nd its domain,
for each a R not in the domain, determine the left-hand and right-ha