Engineering mathematics 1

كلية الهندسةجامعة المنياقسم هندسة القوى الميكانيكية والطاقة

Start04/13/2023 1

Engineering Mathematics I1st Year, Mechanical Power & Energy Engineering Department

2013/2014

Lecturer: Dr. Mohamed R. O. Ali

Email: [email protected]

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


Start04/13/2023 2

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:


Start04/13/2023 3

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.


Start13/04/2023 4

Ch. 1

• Mathematical Preliminaries

Ch. 2

• Direct Methods for Solving Linear Systems

Ch. 3

• Iterative Techniques in Matrix Algebra

Ch.4

• Boundary-Value Problems for Ordinary Differential Equations

List of Contents


Start04/13/2023 5

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.

Two measurements were done the results of the first were P= 1atm, V=0.1 , 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 PreliminariesIntroduction


Start13/04/2023 6

Review of CalculusMathematical Preliminaries

Limits and Continuity

A function f defined on a set X of real numbers has the limit L at , written

if, given any real number , there exists a real number such that , whenever and .

0xDef.

1


Start13/04/2023 7


Fig. 1


Start13/04/2023 8


Fig. 2


Start13/04/2023 9


For a function f defined on a set of real numbers X and , f is continuous at if

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


Start13/04/2023 10


Let be an infinite sequence of real numbers. This sequence has the limit x (converges to x) if, for any there exists a positive integer such that , whenever .

The notation , or as means that the sequence converges to x.

If f is a function defined on a set X of real numbers and , then the following statements are equivalent:

a. f is continuous at ;

b. is any sequence in X converging to , then

Def. 3

Th

eore

m 1


Start13/04/2023 11


For the function g(x) graphed here, find the following limits or explain why they do not exist.a. b. c.

Example 1

Fig. 3


Start13/04/2023 12


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 .

b. 1

c. 0

Solution


Start13/04/2023 13


Which of the following statements about the function graphed here, are true and which are false?a. does not exist

b. existsat every point in (-1,1)e. existsat every point in (1,3)

Example 2

a. False b. False c. True d. True e.

True

Solution

Fig. 4


Start13/04/2023 14


Explain why these limits do not exist?1. 2. Exampl

e 2

Solution

1. does not exist because if and if . As approaches 0 from the left, approaches -1. As approaches 0 from the right, approaches 1. There is no one number L that all the function values get arbitrarily close to when.

2. As approaches 1 from the left, become increasingly large and negative, as approaches 1 from the right, become increasingly large and positive. There is no one number L that all the function values get arbitrarily close to when, so does not exist.


Start13/04/2023 15


Differentiability

Let f be a function defined in an open interval containing The

function f is differentiable at if

exists. The number is called the derivative of f at . A function

that has a derivative at each number in a set X is

differentiable on X.

The derivative of f at is the slope of the tangent line to the

graph of f at as shown in Figure 2.

Def. 4


Start13/04/2023 16


Fig. 5


Start13/04/2023 17


If the function f is differentiable at , then f is continuous at .

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.T

heore

m 2

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 , and the set of functions that have derivatives of all orders on X is denoted . Polynomial, rational, trigonometric, exponential, and logarithmic functions are in , where X consists of all numbers for which the functions are defined.


Start13/04/2023 18


Rolle’s TheoremSuppose 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.)T

heore

m 3

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.


Start13/04/2023 19


Horizontal tangents of a cubic PolynomialThe polynomial function graphed in the following figure is continuous at every point of [-3,3] and is differentiable at every point of (-3,3).

Example 3

Fig. 7

Rolle’s Theorem says that must be zero at least once in the open interval between and . In fact, is zero twice in this interval, once at and again at

Solution


Start13/04/2023 20


Mean Value TheoremIf f ∈ C[a, b] and f is differentiable on (a, b), then a number c in (a, b) exists with (See Figure 4.)

Th

eore

m 4

Fig. 8

If we think of the number as the average change in ƒ over [a, b] and 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.


Start13/04/2023 21


Find the value or values of c that satisfy the equation for the function if it is continuous for and differentiable for .

Since and , the mean value theorem says that at some point in the interval, the derivative must have the value of . in this (exceptional) case we can identify by solving the equation to get .

Fig. 9

Example 4

Solution


Start13/04/2023 22


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 is the temperature of the thermometer at any time , then andFrom The Mean Value Theorem there exists such that/sec=The rate at which the temperature was changing at as measured by the rising mercury on the thermometer.

Example 5

Solution


Start13/04/2023 23


Free fall on the moon: On our moon, the acceleration of gravity is . If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later?

If

at we have

When t=30secthen m/sec.

Then the speed of the rock will be 48m/sec just before it hits the crevasses bottom.

Example 6

Solution

Fig. 10


Start13/04/2023 24


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

Th

eore

m 5

•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.T

heore

m 5

Applic

ati

ons

Fig. 11


Start13/04/2023 25


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


Start13/04/2023 26


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


Start13/04/2023 27

Direct Methods for Solving Linear Systems


Start13/04/2023 28

Direct Methods for Solving Linear Systems


Start13/04/2023 29

Iterative Techniques in Matrix Algebra


Start13/04/2023 30

Iterative Techniques in Matrix Algebra


Start13/04/2023 31

Boundary-Value Problems for Ordinary Differential Equations


Start13/04/2023 32

Boundary-Value Problems for Ordinary Differential Equations

Engineering mathematics 1

Engineering

Transcript of Engineering mathematics 1