Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 8 IN
MATH102 INTEGRAL CALCULUS AND
DIFFERENTIAL EQUATIONS
[total 24 marks]
Question 1. [8 marks]
First we need to determine the distribution function for the radii as
r1.8
2.21.8 if r [1.8, 2.2]
FR (r) =
0
if
r < 1.8
1
if
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 9 IN
MATH102 INTEGRAL CALCULUS AND
DIFFERENTIAL EQUATIONS
[total 20 marks]
Question 1. [8 marks]
(a) From a basic property of the density, we see that
1=
f (x)dx = c
xe
x2
2
dx
0
= c
0
d x2
(e 2 )dx
dx
= c lim [e
x2
2
X
]X
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 11 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS
[total 20 marks]
Solve the following dierential equations. Where applicable, nd the solution
which satises the given initial values.
Question 1.
y + 10y + 25y = 0
[
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 10 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS
[total 20 marks]
Question 1. Solve (that is, nd all solutions of) the dierential equation
dy
xy
.
=
dx
1 + x2
[3 marks]
Solution. Separating the variables gives
dy
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SOLUTIONS FOR ASSIGNMENT 1 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL
EQUATIONS
[total 20 marks]
Question 1. (a) Let f (x) = 1 + x2 , 0 x 3. Find the smallest and
the biggest Riemann sum for f on [0, 1] with partition x0 = 0, x1 =
1, x2 = 2, x3 = 3.
[2
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 6 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS
[total 20 marks ]
Question 1. Write down the binomial expansion for (1 + x)k . Hence derive series
expansions for
(a)
1
.
1x2
1
(b) sin
x=
[5 marks]
x
dt .
0
1t2
[
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 7 IN
MATH102 INTEGRAL CALCULUS AND
DIFFERENTIAL EQUATIONS
[total 16 marks]
Question 1. [8 marks] A guided missile has ve distinct sections
through which a signal must pass if the missile is to operate properly.
Each of the
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 2 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL
EQUATIONS
[total 24 marks]
Question 1. Evaluate
(a)
x
by substituting
u = 2x2 + 1
[2marks],
by substituting
2x2 + 1 dx
u = a2 + x2 .
[2marks]
and
x dx
a2 + x2
(b)
Solution. (a
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 3 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS
[total 22 marks]
Question 1. [6+2 marks] Evaluate
x2 dx
by substituting u = x3
1 + x6
x dx
by a suitable substitution
1 x4
/2
sin
d.
1 + sin2
0
(a)
(b)
(c)
Solutio
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
Chapter 1
Continuous Random Variables
So far we have discussed probabilities in terms of sets. The concept of a random variable
allows us to apply calculus to probability theory.
1.1
Random Variables
We begin the formal denition: A random variable is a re
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
Chapter 1
Probability
If the weather report predicts rain with probability of 0.6 (say), then we may interpret this
as:
(a) 0.6 is a measure of the forecasters belief that it will rain, or
(b) on days when similar atmospheric conditions have been observed
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
Contents
i
Chapter 1
SecondOrder Dierential Equations
1.1
Background
In this section we will study second order linear dierential equations with constant coefcients, i.e. equations of the form
Ay + By + Cy = f (x).
An alternative exposition based on line
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 5 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL
EQUATIONS
[total 25 marks]
Question 1. Find the length of the curve x = t sin(t), y = 1 cos(t) for
0 t 2.
[5 marks]
Solution. We have
dx
dt
= 1 cos(t) and
dy
dt
= sin(t). Henc
Integral Calculus, Differential Equations and Introductory Statistics
MATH 102

Winter 2013
SAMPLE SOLUTIONS FOR ASSIGNMENT 4 IN MATH102
INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS
[total 20 marks]
Question 1. Express in partial fractions and integrate with respect to x:
4x + 3
[4marks].
(x + 2)2
Solution. Since we have a root of multiplicity t