Engineering mathematics 1

‫املنيا‬ ‫جامعة‬‫الهندسة‬ ‫كلية‬
‫والطاقة‬ ‫امليكانيكية‬ ‫ى‬‫القو‬ ‫هندسة‬‫قسم‬
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Engineering Mathematics I
1st Year, Mechanical Power & Energy Engineering Department
2013/2014
Lecturer: Dr. Mohamed R. O. Ali
Email: mohamedroali@mu.edu.eg
Office: Room No.: 210 Mechanical Power & Energy Department
References: •Numerical Analysis, Richard L. Burden & J. Douglas
Faires, 9th Edition, 2010.
•Thomas' Calculus: Early Transcendentals: Media Upgrade.
Weir, M.D., et al., 2008: Pearson Addison-Wesley.
References to chapters will be given from time to time

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The unit aims to:
The programme unit aims to provide a basic course in
calculus, algebra, and numerical analysis to be used in
Mechanical Power and Energy Engineering to students
with A-level mathematics.
Aim:

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Here we are going to:
1. Review the Single-variable calculus where a solid
knowledge of calculus is essential for an understanding
of the analysis of numerical techniques, and more
thorough review might be needed if you have been
away from this subject for a while.
2. Present an introduction to convergence, error analysis,
the machine representation of numbers, and some
techniques for categorizing and minimizing
computational error.

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• Mathematical Preliminaries
• Direct Methods for Solving Linear Systems
• Iterative Techniques in Matrix Algebra
• Boundary-Value Problems for Ordinary
Differential Equations
List of Contents

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The behaviour of ideal gas is assumed to follow a low
known as the gas low which combines the temperature (T),
pressure (P), number of moles (N), and volume (V)
occupied by the gas.
𝑃𝑉 = 𝑁𝑅𝑇
Two measurements were done the results of the first were
P= 1atm, V=0.1𝑚3 , N=0.0042 mol, and R=0.08206.
Using the low to calculate T gives T=290.15K or t=17℃,
while the measurements showed that t=15 ℃.
In the second case the pressure was doubled and the
volume was reduced to one half of the first case, applying
the law again gives t=17 ℃ , while the measured
temperature was 19 ℃. The question here where does this
differences come?
Mathematical Preliminaries
Introduction

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Review of Calculus
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Limits and Continuity
A function f defined on a set X of real numbers has the
limit L at , written
lim
𝑥→𝑥0
𝑓 𝑥 = 𝐿
if, given any real number 𝜀 > 0, there exists a real number
𝛿 > 0 such that 𝑓 𝑥 − 𝐿 < 𝜀 , whenever 𝑥 ∈ 𝑋and 0 <
𝑥 − 𝑥0 < 𝛿.
0x
Def. 1

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Fig. 1

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Fig. 2

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For a function f defined on a set of real numbers X and 𝑥0 ∈ 𝑋, f is
continuous at 𝑥0 if lim
𝑥→𝑥0
𝑓 𝑥 = 𝑓(𝑥0)
The function f is continuous on the set X if it is continuous at each
number in X.
The set of all functions that are continuous on the set X is denoted C(X).
When X is an interval of the real line, X can be replace by the definition
interval of it [a, b] like this C[a, b].
R is the set of all real numbers, which also has the interval notation (−∞,
∞). So the set of all functions that are continuous at every real number is
denoted by C(R) or by C (−∞, ∞).
The limit of a sequence of real or complex numbers is defined in a similar
manner.
Def. 2

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Let 𝑥 𝑛 𝑛=1
∞
be an infinite sequence of real numbers. This sequence has
the limit x (converges to x) if, for any 𝜀 > 0 there exists a positive integer
𝑁(𝜀) such that 𝑛 > 𝑁(𝜀) , whenever 𝑥 𝑛 − 𝑥 < 𝜀. .
The notation lim
𝑛→∞
𝑥 𝑛 = 𝑥, or 𝑥0 → 𝑥 as 𝑛 → ∞ means that the sequence
𝑥 𝑛 𝑛=1
∞
converges to x.
If f is a function defined on a set X of real numbers and 𝑥0 ∈ 𝑋, then the
following statements are equivalent:
a. f is continuous at 𝑥0;
b. 𝑥 𝑛 𝑛=1
∞
is any sequence in X converging to 𝑥0, then lim
𝑛→∞
𝑓(𝑥 𝑛) = 𝑓(𝑥0)
Def. 3
Theorem1

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For the function g(x) graphed here, find the following limits or explain
why they do not exist.
a. lim
𝑥→1
𝑔(𝑥) b. lim
𝑥→2
𝑔(𝑥) c. lim
𝑥→3
𝑔(𝑥)
Example
1
Fig. 3

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a. Does not exist. As x approaches 1 from the right, g(x) approaches 0.
As x approaches 1 from the left, g(x) approaches 1. There is no single
number L that all the values g(x) get arbitrarily close to as 𝑥 → 1.
b. 1
c. 0
Solution

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Which of the following statements about the function 𝑦 = 𝑓(𝑥) graphed
here, are true and which are false?
a. lim
𝑥→2
𝑓(𝑥) does not exist
b. lim
𝑥→2
𝑓 𝑥 = 2
c. lim
𝑥→1
𝑓 𝑥 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
d. lim
𝑥→𝑥0
𝑓(𝑥) exists
at every point 𝑥0 in (-1,1)
e. lim
𝑥→𝑥0
𝑓(𝑥) exists
at every point 𝑥0 in (1,3)
Example
2
a. False b. False c. True d. True e. True
Solution
Fig. 4

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Explain why these limits do not exist?
1. lim
𝑥→0
𝑥
𝑥
2. lim
𝑥→1
1
𝑥−1
Example
2
Solution
1. lim
𝑥→0
𝑥
𝑥
does not exist because
𝑥
𝑥
=
𝑥
𝑥
= 1 if 𝑥 > 0 and
𝑥
𝑥
=
𝑥
−𝑥
= −1 if 𝑥
< 0. As 𝑥 approaches 0 from the left,
𝑥
𝑥
approaches -1. As 𝑥 approaches
0 from the right,
𝑥
𝑥
approaches 1. There is no one number L that all the
function values get arbitrarily close to when 𝑥 → 0.
2. As 𝑥 approaches 1 from the left,
1
𝑥−1
become increasingly large and
negative, as 𝑥 approaches 1 from the right,
1
𝑥−1
become increasingly
large and positive. There is no one number L that all the function values
get arbitrarily close to when 𝑥 → 1, so lim
𝑥→1
1
𝑥−1
does not exist.

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Differentiability
Let f be a function defined in an open interval containing 𝑥0. The function
f is differentiable at 𝑥0 if
𝑓(𝑥0) = lim
𝑥→𝑥0
𝑓 𝑥 −𝑓(𝑥0)
𝑥−𝑥0
exists. The number 𝑓(𝑥0) is called the derivative of f at 𝑥0. A function
that has a derivative at each number in a set X is differentiable on X.
The derivative of f at 𝑥0 is the slope of the tangent line to the graph of f at
(𝑥0, 𝑓(𝑥0)) as shown in Figure 2.
Def. 4

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Fig. 5

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If the function f is differentiable at 𝑥0, then f is continuous at 𝑥0.
Return to Definition 2 where the function is continuous at a certain point
its limit at this point must exist and this is of some how is part of the
conditions of being differentiable.
Theorem2
The next theorems are of fundamental importance in deriving methods
for error estimation.
The proofs can be found in any standard calculus text.
The set of all functions that have n continuous derivatives on X is denoted
𝐶 𝑛(𝑥), and the set of functions that have derivatives of all orders on X is
denoted 𝐶∞(𝑥). Polynomial, rational, trigonometric, exponential, and
logarithmic functions are in 𝐶∞(𝑥), where X consists of all numbers for
which the functions are defined.

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Rolle’s Theorem
Suppose f ∈ C[a, b] and f is differentiable on (a, b). If f (a) = f (b), then a
number c in (a, b) exists with f (c) = 0. (See Figure 3.)
Theorem3
Fig. 6
Rolle’s Theorem says that a
differentiable curve has at least
one horizontal tangent between
any two points where it crosses a
horizontal line.

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Horizontal tangents of a cubic Polynomial
The polynomial function graphed in the following figure is continuous
at every point of [-3,3] and is differentiable at every point of (-3,3).
𝑓 𝑥 =
𝑥3
3
− 3𝑥
Example
3
Fig. 7
Rolle’s Theorem says that 𝑓’ must be
zero at least once in the open interval
between 𝑎 = −3 and 𝑏 = 3. In fact,
𝑓’(𝑥) = 𝑥2
− 3 is zero twice in this
interval, once at 𝑥 = − 3 and again at
𝑥 = 3
Solution

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Mean Value Theorem
If f ∈ C[a, b] and f is differentiable on (a, b), then a number c in (a, b) exists
with (See Figure 4.)
𝑓′
𝑐 =
𝑓 𝑏 − 𝑓(𝑎)
𝑏 − 𝑎
Theorem4
Fig. 8
If we think of the number
(𝑓(𝑏) − 𝑓(𝑎)) (𝑏 − 𝑎) as the
average change in ƒ over [a, b]
and 𝑓’ 𝑐 as an instantaneous
change, then the Mean Value
Theorem says that at some
interior point the instantaneous
change must equal the average
change over the entire interval.

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Find the value or values of c that satisfy the equation
𝑓 𝑏 −𝑓(𝑎)
𝑏−𝑎
= 𝑓′ 𝑐 for the function 𝑓(𝑥) = 𝑥2
if it is continuous for 0 ≤ 𝑥
≤ 2 and differentiable for 0 < 𝑥 < 2.
Since 𝑓(0) = 0 and 𝑓(2) = 4 , the mean
value theorem says that at some point 𝑐 in
the interval, the derivative 𝑓’(𝑥) = 2𝑥 must
have the value of (4 − 0)/(2 − 0) = 2. in
this (exceptional) case we can identify 𝑐 by
solving the equation 2𝑐 = 2 to get 𝑐 = 1.
Fig. 9
Example
4
Solution

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Temperature change: It took 20 sec for a mercury thermometer to rise
from -10℃ to 100 ℃ when it was taken from a freezer and placed in
boiling water. Show that somewhere along the way the mercury was
rising at the rate of 5.5 ℃ /sec.
If T 𝑡 is the temperature of the thermometer at any time 𝑡, then T 0
= −10℃ andT 20 = 100℃.
From The Mean Value Theorem there exists 0 < 𝑡0 < 20 such that
𝑇 20 −𝑇(0)
20−0
=
100−(−10)
20
= 5.5℃/sec=𝑇′
𝑡0 .
The rate at which the temperature was changing at 𝑡 = 𝑡0 as measured by
the rising mercury on the thermometer.
Example
5
Solution

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Free fall on the moon: On our moon, the acceleration of gravity is 1.6𝑚
/𝑠𝑒𝑐2
. If a rock is dropped into a crevasse, how fast will it be going just
before it hits bottom 30 sec later?
Ifa 𝑡 = 𝑣′
𝑡 = 1.6
⇒ 𝑣 𝑡 = 1.6𝑡 + 𝐶;
at (0,0) we have 𝐶 = 0
⟹ 𝑣 𝑡 = 1.6𝑡.
When t=30sec
then 𝑣 30 = 1.6 ∗ 30 = 48
m/sec.
Then the speed of the rock will
be 48m/sec just before it hits the
crevasses bottom.
Example
6
Solution
Fig. 10

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Extreme Value Theorem
If f ∈ C[a, b], then 𝐶1, 𝐶2 ∈ [a, b]
exist with 𝑓(𝐶1) ≤ 𝑓(𝑥)
≤ 𝑓(𝐶2), for all x ∈ [a, b]. In
addition, if f is differentiable on
(a, b), then the numbers 𝐶1 and
𝐶2 occur either at the endpoints
of [a, b] or where f is zero. (See
Figure below)
Theorem5
•There is a way to set the price of an item so as to maximize
profits.
•Among all ellipses enclosing a fixed area there is one with a
smallest perimeter. (The circle, in fact.)
•What goes up must come down.
Theorem5
Applications
Fig. 11

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Piping Oil from a Drilling Rig to a Refinery: A drilling rig 12 km
offshore is to be connected by pipe to a refinery onshore, 20 km straight
down the coast from the rig. If underwater pipe costs $500,000 per km
and land based pipe costs $300,000 per km, what combination of the
two will give the least expensive connection?
Example
7
Solution

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Integration
If f ∈ C[a, b], then 𝐶1, 𝐶2 ∈ [a, b] exist with 𝑓(𝐶1) ≤ 𝑓(𝑥) ≤ 𝑓(𝐶2), for all x
∈ [a, b]. In addition, if f is differentiable on (a, b), then the numbers 𝐶1
and 𝐶2 occur either at the endpoints of [a, b] or where f is zero. (See
Figure below)

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Direct Methods for Solving Linear Systems
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Start
Direct Methods for Solving Linear Systems
16/05/2014 28

Start
Iterative Techniques in Matrix Algebra
16/05/2014 29

Start
Iterative Techniques in Matrix Algebra
16/05/2014 30

Start
Boundary-Value Problems for Ordinary Differential Equations
16/05/2014 31

Start
Boundary-Value Problems for Ordinary Differential Equations
16/05/2014 32

Engineering mathematics 1

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