The School of Computing, Engineering and Maths
300463 Fundamentals of Mechanics
2016 Tutorial Questions (Dr Jonathan Vincent)
Week 8 Due date:
at your tute class on 19 September to 23 September.
1. Read the following sections from the text book.
Article
Tute Question Set 5
Questions 5-7 will be explained in
Lecture-tutorial
1. Determine the reaction at the roller F and the support reactions at A and E for the
frame loaded as shown.
2. Calculate the magnitude of the force acting on the pin D. Pin C is
fix
300463 Fundamentals of Mechanics
Tutorial Exercises. 2016
Week 7
Your tutorial exercises have been marked for completeness rather than correctness marks awarded dont necessarily indicate a correct solution.
You should therefore check your answers agains
Problems in composite materials
Niklas Jansson
Henrik Oxfall
Rodney Rychwalski
Anders Sjgren
Staffan Toll
Bjrn Voigt
Compiled by R. Rychwalski
Edited by H. Oxfall & B. Voigt
Part A Introduction
A.1
Proceed from the rule of mixtures (ROM) and calculate the
300672 Mathematics 1A
Lecture 12 Integration
(This section is non-examinable but is included to give a taster for 300673 Maths 1B)
Antiderivatives
A function F is called an antiderivative of f on an interval
I if
F' x f x
for all x in I.
Examples
(1)
F x
300672 Mathematics 1A
Lecture 11 Still more on
Applications of
Differentiation
Applied Maximum and Minimum Problems
The maximum or minimum values of a function can
occur at the end points (if the domain is restricted) or at
the turning points. Use of this
300672 Mathematics 1A
Lecture 9 Applications
of Differentiation
Maximum and Minimum Values
A function f has an absolute maximum at c if f(c)
all x in the domain, D, of the function f.
The number f(c) is the maximum value of f in D.
f(x) for
A function f h
300672 Mathematics 1A
Lecture 10 More of
Applications of
Differentiation
Consider two functions f and g such that
f a 0, g a 0
f ' , g' are continuous
g' a 0
f x f a
f x f a
f ' x f ' a x a
xa
xa
lim
lim
x a g' x
g' a lim g x g a x a g x g a
xa
xa
xa
300672 Mathematics 1A
Lecture 8 Last one on
Differentiation
Related Rates
If we are concerned with rates of change with respect to
time we differentiate implicitly with respect to t.
Sometimes different quantities are related to each other
through time. C
300672 Mathematics 1A
Lecture 5 - Differentiation
CALCULUS
Calculus is divided into two main areas, namely:
Differential calculus
Differentiation is used to compute the rate of change at
which one variable changes in relation to another, at any
particular
300672 Mathematics 1A
Lecture 7 Still more on
Differentiation
Using techniques from lecture 7,
d yx
a
dx
d ln a y x
e
dx
d ln a y( x )
e
dx
dy
ln a. eln a y( x )
dx
So:
d yx
dy
a
a y( x ) ln a
dx
dx
Derivatives of Logarithms
Let
So
300672 Week
300672 Mathematics 1A
Lecture 3 - Limits
Limits
We write
lim f ( x ) L
x c
if the function, f, has a limit, L, if x approaches the value
c from the left hand side.
We write
lim f ( x ) L
x c
if the function, f, has a limit, L, if x approaches the value
c
300672 Mathematics 1A
Lecture 6 More on
Differentiation
Product Rule
Consider two differentiable functions u = f(x) and
v = g(x). The derivative of their product is given by:
d
dv
du
( uv ) u v
dx
dx
dx
or
d
d
d
( f ( x )g( x ) f ( x ) [ g( x )] g( x ) [
300672 Mathematics 1A
Lecture 4 More on
Limits
Review of Limits
involving x approaching infinity
Last week we noted the following:
lim
Note:
x
Also:
lim
x
1
0
x
4
0
x
lim
1
0
x2
lim
5
0
x2
x
x
Examples
(1)
5x
5x
5
5
5
x lim
lim
lim
x 3 7x
x 3
x 3
7x
300672 Mathematics 1A
Lecture 1 Functions and
Inverse Functions
This lecture revises/introduces the functions and their
inverses which will be used throughout this unit.
FUNCTIONS AND INVERSE FUNCTIONS
Consider:
y=x2
x
y
2 1
4 1
y = 2x 1
0
0
1
1
2
4
x
y
3
300672 Mathematics 1A
Lecture 2 Complex
Numbers
COMPLEX NUMBERS
Complex numbers came into existence when it was found that
the solution to the simple quadratic equation
x2
1
0
had no solution in the real number system.
The general quadratic equation ax bx