Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 8
Continue: Some Applications on Differentiation.
VI- Extrema & Inflection points, Optimization & curve sketching
[1] (i) f ( x ) 3 x
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 2
Functions.
1. Does the relation x2 + 9y 2 = 4 dene y as a function of x?
2. Does the expression 2y 6x = 2 denes y as a function of x?
3. Find the
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 6
Continue: Dierentiation.
1. Use the denition of the derivative to nd f (x):
i) f (x) = cos x
ii) f (x) = sinh x
2. Find the equation of the tange
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 5
Dierentiation.
1. Use the denition of the derivative to nd f (x):
i) f (x) = x3
ii) f (x) =
x+1
1
iv)f (x) =
x
2. (i) Show that the function f (
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 4
Some techniques for limits of functions.
1. Evaluate each of the following limits:
2
i) lim (3x4 + 2x2 x + 1)
2x +1
ii) lim x2 +6x4
x2
iii) lim
x
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 3
Types of functions.
1. Find the domain and range of the functions:
f (x) = 2 sin(x3 ).
1
g(x) =
.
1 + sin x
2. Graph each of the following func
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 1
Numbers and Inequalities.
1. Dene the following: N Z Q R.
2. Prove that 3 Q. (Hint: prove it by contradiction).
/
3. Represent 2, 3, 6, 7, 10 a
Department of Mathematics
Assoc. Prof. Tarek Emam
Winter 2014
Mathematics
for Engineering Students (MATH-301)
Classwork Nr. 9
Name :
ID:
Group:
Exercise 1
Use the Greens theorem to evaluate:
xy dx + (x2 y 3 ) dy
c
where C : The triangle with vertices (0,
Department of Mathematics
Assoc. Prof. Tarek Emam
Winter 2014
Mathematics
for Engineering Students (MATH-301)
Classwork Nr. 7
Name :
ID:
Group:
Exercise 1
Evaluate the following iterated integral:
2 x
2
(x + 2x2 y) dy dx
0 2x
Solution:
Exercise 2
Evaluat
Department of Mathematics
Assoc. Prof. Tarek Emam
Winter 2014
Mathematics
for Engineering Students (MATH-301)
Classwork Nr. 6
Name :
ID:
Group:
Exercise 1
Evaluate the following integral:
1 2
(t + 3t + 2
i
j
k)dt
1+t2
Solution:
Exercise 2
Find the vect
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 7
Continue: Some Applications on Dierentiation.
III- Linear approximation & IV- Dierential
1. Find the linear approximation of the function f (x) =
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 9
Not so short introduction to Integration.
1. Find the following indenite integrals
(
(
8
1)
4)
(a)
x + 2 dx.
(b)
+
dx.
x
x3 x
x
1
2 3x2
2x
e
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
for Engineering - MATH 103
Work Sheet Nr. 8
Continue: Some Applications on Dierentiation.
VI- Extrema & Inection points, Optimization & curve sketching
Denition: Critical numbers of a funct
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 9
Integration
3
x2
x 1
2
1
x dx
c x2 c
3
3
1
x
2
4
1
4
x 2
8
3
(b) 3 dx 8 x dx 4 dx 8
4 ln | x | c 2 4 ln | x | c
x
x
2
x
x
1
[1
Mathematics Department
Dr. Hany El Sharkawy
Winter 2012
Math103 - Engineering 1st semester
Solution of Inequalities
(I) Some Rules:
Rule 1: If x
y Then
(a) k x
ky
If k
0
(b) k x
ky
If k
0
Rule 2: If x
1
x
y
1
y
If x and y have the same sign,
i.e. +ve toge
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 3
Types of functions
[1] (i) f ( x ) 2 sin( x )
3
Df R
1 sin( x 3 ) 1
2 2 sin( x 3 ) 2
2 y 2
R f [2,2]
1
1 sin x
D g cfw_x : 1 sin
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 7
Continue: Some Applications on Differentiation.
III Linear approximation & IV- Differential
[1] The linear approximation of the func
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 5
Differentiation.
[1] (i) f ( x ) x , f ( x h) ( x h)
3
3
f ( x h) f ( x )
( x h) 3 x 3
[( x h) x][( x h) 2 x( x h) x 2 ]
lim
lim
h
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 6
Continue: Differentiation.
[1] (a) f ( x) cos x, f ( x h) cos( x h)
f ( x h) f ( x )
cos( x h) cos( x)
cos( x). cos(h) sin( x). sin(
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 4
Some Techniques for limits of functions
[1] (i) lim 3 x 2 x x 1 3( 2) 2( 2) ( 2) 1 59
x 2
(ii) lim
x2
4
2
4
2
2x 2 1
2(2) 2 1
9
2
2
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 2
Functions
4 x2
9
2
For each one input value we have two output values, x 9 y 2 4 is NOT a function.
[1] x 9 y 4
2
[2]
y2
2
2 y 6x
Department of Mathematics
Dr. El-Sharkawy
Winter 2012
Mathematics
For Engineering - Math 103
Solution for Work Sheet Nr. 1
Numbers and Inequalities
[1]
N cfw_1, 2, 3, 4, 5, .
Z cfw_. , 3, 2, 1, 0, 1, 2, 3, .
Natural numbers
Integers
Rational numbers Q x
Department of Mathematics
Assoc. Prof. Tarek Emam
Winter 2014
Mathematics
for Engineering Students (MATH-301)
Classwork Nr. 5
Name :
ID:
Group:
No Submission is Requested !
Exercise 1
Consider a thin rod of length L with an initial temperature f (x) throu
Department of Mathematics
Assoc. Prof. Tarek Emam
Winter 2014
Mathematics
for Engineering Students (MATH-301)
Classwork Nr. 4
Name :
ID:
Exercise 1
Find the Fourier series of the function:
cfw_
1
: x < 0
, f (x) = f (x + 2)
f (x) =
1
:0x<
Solution:
Page 1
Department of Mathematics
Assoc. Prof. Tarek Emam
Dr. Ahmed Mostafa
Winter 2014
Mathematics
for Engineering Students (MATH-301)
Classwork Nr. 1
Name :
ID:
Exercise 1
1. Solve the dierential equation: y + y + y = 0
Solution:
Page 1
Group:
2. Find the gener
Math 103 Class Work Sheet (8) Solution
[1] 4 16 dx 4 x (i) 2 x 16 dx 2 x 2 16 ln x 16 x c x x 1 1 (ii) sin(5 x 1) dx sin(5 x 1) 5dx cos(5 x 1) c 5 5 1 1 (iii) cosh(6 x ) dx cosh(6 x ) 6dx sinh(6 x) c 6 6 1 1 2 2 (iv) sec (6 x 5) dx sec (6 x 5) 6dx tan(6 x
Mathematics for Engineers I (Math103).student-intranet/faculties/math/2008 Fall/math103
Class Work Sheet 71.
1. Graph each of the following functions: (a) f (x) = x3 3x + 3 (b) f (x) = x4 2x2 (c) y = x( x 5)2 2 (d) y = x + sin x , (e) y = x 8 x2 (f) y = |
Mathematics for Engineers I (Math103).student-intranet/faculties/math/2009 Fall/math103
Class Work Sheet 61.
1. Using LHospital rule, evaluate each of the following limits: a) lim x2 1 x 1
x1
9
cos x b) lim+ 1sin x x 2 ex x x
x
c) lim e
t0
3t
1 t
d) lim
e