Maths 1A20
Calculus
2013-14
Sheet 3
Functions: Domains, ranges, graphs and inverses
Easy Questions
1. (Homework parts (a), (b), (c) For each of these functions, say whether or not it is: polynomial, odd,
even, periodic. If it is periodic, what is the pe
Maths 1A20
Calculus
2013-14
Sheet 4
More on Functions; Exponential and Related Functions
Easy Questions
1. (Homework, parts (c), (d) For each of the following, evaluate it if it makes sense:
(a) sin1 [sin(2/3)];
(b) cos(cos1 1 );
2
(c) cos(cos1 2);
(d)
Maths 1A20
Calculus
2013-14
Sheet 5
Dierentiation. Taylor Polynomials
Easy Questions
1. (Homework, parts (c), (d) Dierentiate the following, stating for what values of the variable the result
holds:
(a) (1 x2 )2 ; (b) sin2 t; (c) sin2 t2 ; (d) 2 tan ; (
Maths 1A20
Calculus
2013-14
Sheet 2
Sequences, Series, Limits
Easy Questions
1. (Homework, parts (b), (d), (e)
Do the following sequences converge or diverge as n ? If convergent, what is the limit?
(a) (1)n ;
(b)
5n + 2
;
n1
sin( 1 n)
2
;
n
(c)
(e) 2n
Maths 1A20
Calculus
2013-14
Sheet 6
Taylor Series and Applications
Easy Questions
1. (Homework, part (a) Find the rst few terms of the Maclaurin series for the following functions. For
example, the rst 3 non-zero terms of the Maclaurin expansion of sin
Maths 1A20
Calculus
2013-14
Sheet 1
Basic Algebra and Trigonometry
Doing mathematics is a skill, and requires a lot of practice. The problem sheets for this course are designed to
give you this practice. They are often divided into three sections: some ea
3.8. LIMITS OF FUNCTIONS
3.6
CHAPTER 3. THE THEORY OF FUNCTIONS
Inverse functions
3.7.1
Cosine inverse cos1 or arccos
Arccos(x)
cos is monotonic on [0, ]
with a range [1, 1].
Hence, there is an inverse
We understand that a function f takes a value x from
UNIVERSITY OF BRISTOL
Examination for the Degree of B.Sc. and M.Sci. (Level 1)
MATHEMATICS 1AS/1AM/1A20 PAPER ONE
MATH 10100/10300/11004
(Paper Code 10100/10300/11004)
May/June 2013, 3 hours
This paper contains two sections, Section A and Section B. Answe
UNIVERSITY OF BRISTOL
Examination for the Degree of B.Sc. and M.Sci. (Level 1)
MATHEMATICS 1AS/1AM/1A20 PAPER ONE
MATH 10100/10300/11004
(Paper Code 10100/10300/11004)
May/June 2012, 3 hours
This paper contains two sections, Section A and Section B. Answe
UNIVERSITY OF BRISTOL
Examination for the Degree of B.Sc. and M.Sci. (Level 1)
MATHEMATICS 1AS/1AM/1A20 PAPER ONE
MATH 10100/10300/11004
(Paper Code 10100/10300/11004)
May/June 2011, 3 hours
This paper contains two sections, Section A and Section B. Answe
6.4. POWER SERIES AND CONVERGENCE
6.3.1
Maclaurin Series
CHAPTER 6. TAYLOR SERIES
is well behaved for x > 0, the Taylor series is only valid
for 0 < x < 2.
Defn: A Maclaurin series is a Taylor series about the
point x = 0 so the Maclaurin series for f (x)
Chapter 5
Dierential Calculus
5.1
Basics
5.1.2
The chain rule
The same as function-of-a-function rule. It says
Defn: The derivative of a function f is dened by
d
f (g(x) = f (g(x) g (x)
dx
which can be extended.
dy
df
d
and
and f (x) all
Notation: If y =
Chapter 3
The Theory of Functions
3.1
Introduction
3.2
Intervals and sets
Sometimes need care and precision handling functions in Defn: The set of all numbers between a and b and includorder to avoid errors. For example:
ing the endpoints a and b is denot
Chapter 2
Series and Limits
2.1
Sequences
which reads as L is the limit of the sequence cfw_an as n
tends to .
Defn: A sequence is an ordered list (of numbers), for Defn: If a sequence does not converge, we say the seexample 1, 2, 3.
quence diverges. (In
Lecture notes for Mathematics 1A20
2013-14
Dr Sean Collins1
1
with acknowledgments to David Griel and Richard Porter
Chapter 1
Review of Algebra and Trigonometry
1.1
Basic Ideas
1.1.3
Powers
These ideas should all be familiar to you; the language Rules, f
Maths 1A20 Calculus
2013-2014
Solution Sheet 6
Taylor Series and Applications
4. (a) f (x) = 4x3 4x so the stationary points satisfy
x(x2 1) = 0 giving x = 0 or x = 1. Next, f (x) =
12x2 4, so f (0) = 4, f (1) = 8. Hence there
is a local max. at x = 0 and
Maths 1A20 Calculus
2013-2014
Solution Sheet 5
Dierentiation. Taylor Polynomials
1. (a) 2(1 x2 )(2x). Holds for all x;
(b) 2 sin t cos t. Holds for all t;
2
2
(c) 2 sin t cos t (2t). Holds for all t;
(d) 2 tan + 2 sec2 provided = (n + 1 ) for some
2
integ
Maths 1A20 Calculus
2013-2014
Solution Sheet 4
(b) coth x = 1/ tanh x < 0 for x < 0, so we cant have
x < 0. Also coth x doesnt exist for x = 0. So domain
is cfw_x : x > 0;
More functions; Exponential
and related function
(c) Same domain as in (b). Here ta
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Maths 1A20 Calculus
2013-2014
Solution Sheet 3
Functions:
Domains, Ranges, Graphs, Inverses
(c)
(d)
1. (a) Polynomial. Odd. Not periodic;
(b) Not polynomial. Odd. Function repeats itself
when 3 increases by 2, and this is the smallest repetition length. S
Maths 1A20 Calculus
2013-2014
Solution Sheet 2
Sequences, Series and Limits
1. (a) Divergent: it oscillates between 1 and 1;
(b) Dividing top and bottom by n gives
(d) tan (n)
close to /2;
since arctan of a large number is
2n
2 2 2 2
2
(e) 2n /n! 0. Well