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 2 4y = 5.
b) y = 5x 14,
c) y = x + 8,
1
d) y = x + 4.
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 + 3.
c) The quotient is 2y 3 6y 1 and the remainder is
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. on that day, in which case you need to ll out a Cover
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
= 3x + 13x 1 +
.
x2 3
x2 3
b) We know that 0 = f (1) = 2
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
b) We have y = cos1 u where u = x2 . Thus
dy
2x
dy du
1
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
b) We know that 0 = f (2) = 8 + 4a + 2b 4 and that 0 =
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 2 4y = 11.
c) y = 2x + 11,
b) y = 3x + 3,
1
d) y = x +
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. on that day, in which case you need to ll out a Cover
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. on that day, in which case you need to ll out a Cover
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 the other exercise sheets.
Integration (Week 9 work)
1
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 cos )2 = sin2 + cos2 2 sin cos = 1 2 sin cos .
Adding th
CIVE 1619 Preparatory Mathematics for Engineers
January 2012 Solutions
1. (a) By long division, the quotient is x2 + 2x + 9 and the remainder is 9x + 3.
Hence
x4 + x3 + 7x2 + 3
9x + 3
= (x2 + 2x + 9) + 2
.
2x
x
x x
(b) f (1) = 1 + a + b 6 = 0 and f (2) =
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 notation f (x) for the derivative, the value at x = a is d
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 6
y = ex
8
10
ex x ex
x ex ex
ax a > 1 ekx
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
b) We have y = cos1 u where u = 2x. Thus
dy
dy du
1
2
=
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 + + cn1 x + cn
where n is an integer 0, and c0 , c1 , .
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
,
x7
x2
,
3 + x2 x
2x
x2 + 3x + 2
.
1
The last is the
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 2x 6.
c) The quotient is 2y 3 2y 1 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
sin
cos
sin2 + cos2
tan + cot =
+
=
.
cos
sin
cos 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=
y2 y1
.
x2 x1
Negative gradient means that it slopes
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 , sin )
sin(150 ) = sin(30 ) = 1/2 sin(225 ) = sin(45
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 curve at a point is the gradient of the tangent.
The d
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 variables is one of the form
dy/dx = f (x)g(y) .
To nd
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 + c
where c is a constant, called the constant of integ
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.m. on that day.
These exercise sheets will be used for