University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2010
Solutions 2
Coordinate geometry
1.
3
5
a) y = x + ,
2
2
2.
a) The equation is (x 4)2 + (y 2)2 = 25, or in expanded form, x2 8x + y
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Solutions 1
Polynomials
1. a) The quotient is x2 + 4x + 4 and the remainder is 10.
b) The quotient is 3x 1 and the remainder is 3x
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Exercise 1, hand in on Monday 8th October
Please hand in at the lecture; or deliver it to my oce in Mathematics
(9.18.q) by 4 p.m.
University of Leeds
School of Civil Engineering
CIVE1619, January 2009
Solutions
1. a) By long division we get the quotient 3x3 + 13x 1 and the remainder 45x 15.
Hence
3x5 + 4x3 x2 + 6x 12
45x 15
3
=
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Solutions 4
More about dierentiation
1.
a) We have y = sin1 u where u =
x. Thus
dy
dy du
1
1
1
=
=
=
.
dx
du dx
2 x 1x
1 u2 2 x
University of Leeds
School of Civil Engineering
CIVE1619, January 2010
Solutions
1. a) By long division we get the quotient 2x + 5 and the remainder 3. Hence
2x3 + 3x2 9x 7
3
= 2x + 5 + 2
.
2x2
x
x x2
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Solutions 2
Coordinate geometry
1.
4
1
a) y = x + ,
3
3
2.
a) The equation is (x 1)2 + (y 2)2 = 16, or in expanded form, x2 2x + y
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Exercise 2, hand in by Monday 22nd October
Please hand in at the lecture; or deliver it to my oce in Mathematics (9.18.q) by 4
p.m
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Exercise 3, hand in by Monday 5th November
Please hand in at the lecture; or deliver it to my oce in Mathematics (9.18.q) by 4
p.m
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Exercise 5
This exercise sheet is not to be handed in, and will not be marked. However, the material
on it is just as important as
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Solutions 3
Trigonometry
1.
a)
By expanding the squares we get
(sin + cos )2 = sin2 + cos2 + 2 sin cos = 1 + 2 sin cos
and
(sin c
7.
More about dierentiation
7.1.
More notation
Given a function y = f (x), its derivative is also a function of x; we sometimes need to
evaluate it at a particular value, say x = a.
When using the not
e = 2.71828 . . .
e = lim (1 + 1/n)n ;
n
n (1 + 1/n)
n
e
n
(1 + 1/n)n
n = 10
n = 100
n = 10000
n = 1000000
a y = ax
y = ex
y = exp(x)
10
8
6
y
4
2
10
8
6
4
2
0
2
4 x
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2010
Solutions 4
More about dierentiation
1.
a) We have y = sin1 u where u =
x. Thus
dy
1
dy du
1
1
=
.
=
=
dx
du dx
2 x 1x
1 u2 2 x
1. Polynomials
1.1. Denitions
A polynomial in x is an expression obtained by taking powers of x, multiplying them by
constants, and adding them. It can be written in the form
c0 xn + c1 xn1 + c2 xn2 +
2.
Rational functions and partial fractions
2.1.
Rational functions
A rational
function
is a function of the form
f (x) =
p(x)
q (x)
where p(x) and q (x) are polynomials in x with q 0. For example
x+3
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2010
Solutions 1
Polynomials
1. a) The quotient is x2 + 3x + 2 and the remainder is 10.
b) The quotient is 3x + 5 and the remainder is
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2010
Solutions 3
Trigonometry
1.
a) By using the denitions of tan and cot, and writing the fractions over a common denominator gives
si
3.
Coordinate geometry in the
3.1.
(x, y)-plane
Lines
Recall that the gradient or slope of a line is
m=
change in y
.
change in x
If the line passes through the points (x1 , y1 ) and (x2 , y2 ) then
m
sin
cos
tan =
sin cos
1
sin
1
sec =
cos
cos
1
cot =
=
tan
sin
cosec =
h
y
x
sin =
x
y
h
h
x
y
, cos = , tan = , cosec = , sec = , cot = .
h
h
x
y
x
y
(cos
6.
Dierentiation
6.1.
Denition
Given a function f (x), its graph is the curve y = f (x).
The tangent at a point on the curve, is the line through the point which touches the curve.
The gradient of the
10.
Ordinary Dierential Equations
A rst order dierential equation for y is one only involving x, y and dy/dx, and no
higher derivatives: d2 y/dx2 , etc.
A rst order dierential equation with separable
9.
Integration
9.1.
Standard integrals
If u is a function of x, then the indenite integral
u dx
is a function whose derivative is u. The standard integrals are:
(1) If a is a constant, then
a dx = ax
University of Leeds
School of Civil Engineering
CIVE1619, Autumn 2012
Exercise 4, hand in by Monday 19th November
Please hand in at the lecture; or deliver it to my oce in Mathematics (9.18.q) by 4
p.