course : engineering mathematics 1 code : bas 001

Faculty of Engineering course :Math1 by Prof H N Agiza 1

Course : Engineering Mathematics 1Code : BAS 001Coordinator : Prof Hamdy Agiza

Room 226Credit hours 3Teaching hours 2 H/W lectures+2H/w tutorial


1- work with functions represented in a variety of ways: graphical, numerical,

analytical, or verbal.

2- understand the meaning of the derivative in terms of a rate of change and local linear

approximation and should be able to use derivatives to solve a variety of problems.

3- communicate mathematics both orally and in well- written sentences and should be

able to explain solutions to problems.

4- use technology to help solve problems,, interpret results, and verify conclusions.

5- develop an appreciation of calculus as a coherent body of knowledge and as a human

accomplishment.

Course Objectives:By completeting the course students should be able to:


Course ILOS مخرجات التعلم من المقررOn succesful completion of the course, the students should be able to:

recognise properties of functions and their inverses;

recall and use properties of polynomials, rational functions, exponential, logarithmic,

trigonometric and inverse-trigonometric functions;

understand the terms domain and range;

sketch graphs, using function, its first derivative, and the second derivative;

use the algebra of limits, and l’Hôpital’s rule to determine limits of simple expressions;

apply the procedures of differentiation accurately, including implicit and logarithmic

differentiation;

apply the differentiation procedures to solve related rates and extreme value problems;


Differentiation:

Function Concept – Function Classification – Inverse Function –Elementary Functions (Trigonometric, logarithmic, Exponential, Hyperbolic and its inverse functions)- Limits – limits theorem – mean value theorem- Derivation, derivation rules – First function derivations – Series theorem – Parametric derivation – higher order derivation – partial derivation – applications on the differentiation –l'Hopital rule -Taylor expansion – Maclauren Series – Curve drawing – Maximum and Minimum values.

Linear algebra and matrix theory.

Course Contents:

5

Assessments:

Assessment method weight

quizzes Examination 20%

Semester work 10%

Oral Examination 10%

Mid-Term Examination 20%

Final-Term Examination 40%

Total 100%

Assessment schedule

Week 4, 12

All the term

Week 14

Week 8

Week 15

100%

Faculty of Engineering course :Math1 by Prof H N Agiza

A function f is a rule that assigns to each element x in a set D

exactly one element, called f(x) , in a set E.

The set D is called the domain of the function.

The range R of f is the set of

all possible values of f(x) as x

varies throughout the domain.

x is called independent variable

and f is dependent variable



REPRESENTATIONS OF FUNCTIONS

1-verbally (by a description in words)

2- numerically (by a table of values)

3- visually (by a graph)

4- algebraically (by an explicit

formula)


Examples :


Domain = R

Range = [0, ∞]

Overview


Faculty of Engineering course :Math1 by Prof H N Agiza12

Describe the objective(s) of the exercise.

New product or service ideas?

New feature ideas?

Feature/product naming?

Promotion ideas?

New process for doing something?

Define top requirements or restrictions.

Brainstorming Objectives


Let . Determine the domain of f. 2 3 2f x x x

Domain is: ,1 2,

1 2

+ + + + + - - - - - - + + + + +

Example

Solution


The graph of a function f is shown in figure.

(i) Find the value of 1f and the zeros of the function.

(ii) What are the domain and range of f.

1 4f zero is at 6x

3 6x 2 4y

Example

Solution


Symmetry

Even function graph

is symmetric about

the y-axis.

If a function satisfies

for every number x

in its domain,

then f is called an

even function

f x f x

If

satisfies for

every number x

in its domain,

then f is called

an odd function

f x f x

its graph is

symmetric

about the

origin.


Transformations of Functions

h(x) = af(x) vertical stretch or shrink

h(x) = f(ax) horizontal stretch or shrink

h(x) = f(x) + k vertical shift

h(x) = f(x + h) horizontal shift

h(x) = -f(x) reflection in the x-axis

h(x) = f(-x) reflection in the y-axis


Increasing and decreasing function

Composition of Functions

Let and .

Calculate g ◦ f and f ◦ g . 2 1f x x 3 4g x x

( )g f x g f x

23 1x

23 1 4x

( )f g x f g x

29 24 15x x

2

3 4 1x

g f x f g x

Example

Solution

2 1g x 3 4f x



Trigonometric Functions

22Faculty of Engineering course :Math1 by Prof H N Agiza


EXPONENTIAL FUNCTIONS

In general, an exponential function is a function of the form

𝑓 𝑥 = 𝑎𝑥 where a is a positive constant.

Let’s recall what this means.

If n, a positive integer, then

and

And if x=p/q then


Rules For ExponentsIf a > 0 and b > 0, the following hold true for all real numbers x and y.

yxyx aa. a 1

yx

y

x

aa

a. 2

xyyx aa. 3

xxx (ab)b. a 4

x

xx

b

a

b

a.

5

16 0 . a

x

-x

a. a

17

q pq

p

a. a 8

INVERSE FUNCTIONS AND LOGARITHMS


Find the inverse of the function

We solve the equation

1f f x x

13 1f x

1 1

3

xf

3 1f x x

Example

Solution

13 1f x


Find the inverse of the function

We solve the equation

1f f x x

5

12 f x

1 5

2

xf

52f x x

Example

Solution

5

1

2

xf


1f

y x

The graph of

is the reflection of the graph of f about the line

(a, b)

(b, a)

Another useful fact is this: Since an invertible function must be one-to-one,

two different x values can not correspond to the same y value. Looking at

the figure, we see that this means In order for f to be invertible,

no horizontal line can intersect the graph of f more than once.

x

y

Note


Determine whether each of the following functions is even,

odd neither even nor odd.

4 3f x x 3f x x x 2 3f x x x

4

3f x x

4 3x

f x

3

f x x x

3x x

3x x

2

3f x x x

2 3x x

f x

even odd neither even nor odd

Example



Assignment (1)ExercisesChapter

Ex 1.1 (3,4,7-10,31-37,38,47,55-56,62,73-78)

1Ex 1.3 (3,4,9-24,31-36,41-46,51)

Ex 6.1(5-8,18,20,23-28,31-32, 39-41)

Ex 6.3(3-8,27-38)

6

course : engineering mathematics 1 code : bas 001

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