MSA
Faculty of Engineering
Sheet ( 1 )
Functions, Limits & Continuity
Eng. Math. I ( MAT 151 )
1 - Solve the following inequalities:
a)
d)
4  6  x  11
g)
x( x  3)  0
b)
e)
25  x 2  0
h)
8  x 2  2 x  15
9 x 5
c)
f)
6  5x  x 2
i)
6  5 x  9
4 x3 6
x 2  2x  8  0
2 - Find the domain and the range of the following functions:
a)
f ( x) 
9 x
b)
f ( x)  1 /
d)
f ( x) 
4 x
e)
f ( x) 
g)
f ( x) 
x2  3
h)
f ( x) 
c)
f ( x) 
x 2
f)
f ( x)  3 x 
x6
i)
f ( x) 
4 x 2
x 2  5x  6
4 x
x 2
3 – Identify the following functions:
y  5x  8
a)
y5 x
d)
x 1
f ( x)  3
x 1
e)
f ( y) 
g ( x)  21/ x
h)
g ( y)  log 5 ( y  1/ y)
b)
y3
y 4
g)
i)
j)
g (u)  5 u  8  sin(2u)  u 5u  (u  1) /(u  2)
y  ( x 2  1) /( x  6)
k)
m)
y  log 2 ( y 1  4 y)
n)
y  3 x2
x4
y 7 3
x  x
c)
y  2 x 3  5x  8
f)
f ( z)  5 tan(3z   / 4)
l)
y  2 x 3  5x  8
o)
f ( z)  2 sin( z)  cos(3z)
4 - Sketch the original and shifted graphs together labeling each graph with its equation:
a) y  x 3 , right 2, down 1
b) y  2  x  1
c) y  3 x  2  1
d) y  x , left 1, up 2
e) y  1  ( x  2) 2
f) y  3 x  1  2
g) y  x 2 , right 2, down 1
h) y  2  3 x  1
i) y  x  2  1
5 - Graph the following functions ( labeling each step with its equation ):
a)
d)
y   x2 3
b)
e)
y  (1  x)  2
3
y  3  2  x
y  2 x  3
c)
f)
y   2x  1
y  3 x  2 1
6 – Which of the following functions are even, odd , or neither? , and then discuss the
symmetry of their graphs(if it exist):
a)
d)
g)
f ( x)  x 4  3 x 2  5
y  3x  x 5
y  x 2 ( x  2x3 )
b)
e)
h)
g ( x)  x 3  5x  1 / x
y  x2  x
y  (1  x) /(1  x)
- 1 / 11 -
c)
f)
i)
h( x)  2 x 3  5x  8
y  3x 2  x 6  5
f ( x)  x / ( x 2  1)
7 - Study the domains of the following functions and the symmetry of their graphs:
a)
f ( x) 
e)
g ( x)  x 3  3x  14 x
x 2
b)
g ( x)  x 3  x x 2  4
d)
f ( x)  x / x
f)
f ( x)  3x 2  x  1
g)
g ( x)  3  x
8 - Sketch each equation, and then discuss the following if it represents a function:
i) Domain and range.
ii) Increasing and decreasing intervals.
iii) Whether it is one-to-one or not?.
iv) Whether it is even, odd or neither?.
a)
y  2x  3
b)
y
x
c)
x  y2  3
d)
y  x 1  2
e)
y x 3 2
f)
y  x /( x  1) 2
g)
y  x2 1
 2 x  1

f ( x)   x 2
x  4

h)
y  2 / x3
x0
i)
k)
j)
0x2
2 x5
0.5 x  1

f ( x)   x 2
8

x  2

f ( x)   x 3
 x  3

x  1
x 1
x 1
0  x 1
1 x  2
2 x5
9 - Find the domain of the following functions:
x4 5
x  5x  6
a)
f ( x) 
c)
f ( x)  x  2 
e)
g ( x) 
x  6 1
b)
g ( x) 
4x
x 1
d)
g ( x) 
4  3x  x 2
 x4
x2  4
f)
f ( x)  x  5 
2
x2  4  5
x
x
x 2
2
x 3
10 - Evaluate the composite functions ( f  g )( x) & g  f ( x) and their domains when:
a) f ( x)  x  1
c) f ( x)  x  1
,
,
e) f ( x)  x  2
,
b) f ( x)   x 2
d) f ( x)  x  1
g ( x)  x
g ( x)  x  2
1
g ( x) 
x 1
f) f ( x)  x 2 ,
,
,
g ( x)  x  1
g ( x)  1 / x
g ( x)  x  1
11 - Study the domain and the symmetry for each of the following functions:
a)
f ( x) 
c)
f ( x) 
e)
f ( x) 
1  cos( x)
1  cos( x)
tan x  3 csc 2 x  4
1  sec( x)
sin( x)
tan( x)
x cos( x)  sin( x)

2  cos( x)
x2 1
b)
f ( x) 
d)
f ( x)  3 tan(2 x)  1  2  cos( x)
f)
f ( x) 
- 2 / 11 -
x sin( x)  cos( x) x  2
 2
1  cos( x)
x 4
12 - Find the inverse function f 1 (x) ( if exists) for the following functions, then find :
D( f 1 ) and R( f 1 )
f ( x)  3 x  2
x
f ( x) 
, x 1
x 1
a)
d)
b)
f ( x)  x 2  1 , x  0
c)
f ( x)  x 3  8
e)
f ( x)  x  4
f)
f ( x)  3 x  1
13 - Graph the following functions ( labeling each step with its equation ):
f ( x)  3  e x 1
a)
d)
g ( x)  1 / ln( x  2)
g ( x)  1  ln( x  2)
b)
e)
f ( x)  1  e x  2
f ( x)  2  1 / e x
c)
f)
g ( x)  1 / ln( x  3)
14 – Find the domain and the range of the following functions:
a)
f ( x)  x  e x1
b)
g ( x)  ( x 2  4) / ln(4  x 2 )
c)
f ( x)  e x /( x  2)
d)
g ( x)  x  1 ln x
e)
f ( x)  e x  2 x  4
f)
g ( x)  4  x 2 / ln x
c)
lim
15 - Evaluate the following limits:
a)
lim
x 2
16 - If
lim
( x  2)( x  1)
b)
x  7  3 x 1
f ( x)  3 ,
x 2
lim
x 3 
lim f ( x)  2
x 5
and
lim ( x
b)
x 2
c)
lim
x 5
d)
 1) f ( x) g ( x)  cos( x  2)

lim  sin

e)

f ( x)
 0.4  x 
x


g ( x )6
lim x  f ( x)( x  1)
x 2
f)
2
 f ( x)  3 g ( x) 
lim  f ( x) g ( x)  3 
x 2
17 - Given the following graph, compute each of the following:
a) f (4) , f (1) , f (6)
b)
lim f ( x) , lim f ( x)
x  4
c)
lim x f ( x)
x6
d)
x5 2
find the following limits:
x 5
 f ( x)  2  x 
2 x
x 5 
x2
x 2
x 2
2
3 x 3
lim g ( x)  4 ,
lim  f ( x)  2 g ( x)  x
a)
x 2  15
lim
x1
x  2 f ( x)
x 6
- 3 / 11 -
and
lim
x6
f ( x)
18 - For each of the following functions, find the limit of f (x) at the given points.
2 x  1

a) f ( x)   x 3  2
 x  3

x  1
,
x 1
at x = -1, 0, 1
1 x
x0
 2 x  3 sin( x)
 2
b) f ( x)   x
0 x2
 x  3
x2

,
at x = 0, 1, 2
19 - Find the values of a and b so that each of the following functions has a limit everywhere:
a  3 cos( x)

a) f ( x)   x 2  b

a x
x  b

b) f ( x)   x 3  a
ax  b

x0
0 x4 ,
4x
x 1
1 x  3
x3
20 - Use the sandwich theorem to:
a) Find
i)
ii)
lim f ( x)
lim
3 f ( x) 2  4 f ( x)  5 ,
x
3  cos x  f ( x)  x    2
If
b) Prove that
lim
x 
i)
iii)
x
x
c) Find
 x f ( x)
lim
lim
 
1
cos x 2  0
2
x
ii)
f ( x)
lim x f ( x)
iii)
x 0
x0
x0
lim
x4
,
( f ( x)) 2  4
5  x 2  f ( x)  5  x 2
If
21 - Find the following limits:
a)
lim
x 2
d)
lim
x 2  3x  2
x3  2 x 2
x2
x2  5  3
x  2
g)
lim
x 0
j)
lim
x 0
x2  8  3
lim
x 1
x  1
4
x  81
e) lim 2
x 3 x  9
b)
x
x
h)
lim
x  2
x 2  4 x  sin( x)
2x
tan (5 x)
m) lim 2
2
x 0 3 x  2 sin ( x)
lim
x 3
k)
sin 2 x
lim x  3 tan(x)

sin x  9
x3
2

2x  7x  1
3
 2x 2  4
lim x
x 
lim
x2  1
2x  4
x 9
f)
x 1
i)
x  2
l)
x 0
n)
lim
x 
q)
lim
x 0
1
x sin 
 x
(1  cos( x))
x  sin( x)
t)
lim
x 
xx
3x  4
- 4 / 11 -
x csc( 2 x)
lim 3 cos(5x)
x0
sin 2 (5 x)  sin(5 x 2 )
o) lim
x2
x 0
r)
lim
x 0
1
3
s)
lim
x 3
x9
x2  x  2
x2  x
lim
2
2
p)
x 1
2 x  42
c)
u)
sin(5 x) cos(5 x)  3 tan(7 x)
x  6 tan( x)
lim 
x 
x2  x  x2  x

22 - Find the points of discontinuity of the following functions:
a) f ( x) 
sec x  5 sin x
x2  4
x
e) f ( x) 
 sin( x)
x
sin x  x 2
cos x  1
b) f ( x) 
d) f ( x)  3 x sec x  5sin x
x 1
 x 1
x2  9
1
f) f ( x)  tan(x) 
x2
c) f ( x) 
23 - Discuss the continuity of the following function at the given points:
x 2  1

2 x
a) f ( x)  1
 2 x  4

0
x  1
x
c) f ( x)  
 5 sin x  5 x
1  x  0
 tan x
 2
0  x 1
x

b) f ( x)  0.5
x  1 at x  0,1,2
 x  2 1  x  2

2 x3
0


 sin x
x0
d) f ( x)   x
at x  0

x0
1
1  x  0
0  x 1
x 1
at x  0,1,2
1 x  2
2 x3
x0
x0
at x  0
24 - Find the values of a, b, and c at which the following functions are continuous
everywhere.
a)
c)
x2  1
x3
f ( x)  
x3
2ax
 sin ax

 x

f ( x)  5  x

2
c x  bx  12 

 
x3

b)
x0
0 x3
3 x
Assignments ( Sheets )
Assignment 1
1( g → i ), 2( h , i), 3( m → o),
4( g → i ), 5( e , f ), 6( g → h ),
7( e , f ), 8( g , h, k ), 9( e , f ), 10( e , f )
Assignments 2
11( e , f ), 12( e , f ), 13( e , f ), 14( e , f )
15(c), 16(c, f), 18(b), 20( c ),
22(e, f), 23(d), 24(b)
- 5 / 11 -
x
f ( x)   2
bx
x  2
x  2
MSA
Faculty of Engineering
Sheet ( 2 )
Differentiation
Eng. Math. I
( MAT 151 )
1 – Find the equations of the tangents to the following functions at x = 0 (if they exist):
a)
1  x
f ( x)  
2
1  x  x
c)
e)
x0
b)
f ( x)  x
f ( x)  1  x
d)
f ( x) 
f ( x) 
f)
f ( x)  cos x
0x
x
4 x
2 - Find the first derivative from the first principles (Definition):
a)
b)
c)
f ( x) 
4  3 x2
1
f ( x) 
3x  1
d)
f ( x)  sin( x)
e)
f ( x)  cos( x)
f ( x)  2 x 2  3
f)
f ( x) 
1
x 1
3 - Discuss the differentiability of the following functions at the given points:
a)
b)
c)
d)
 x2  x

f ( x)  sin x  cos x  1
 tan x

 x cos x

g ( x)   2 x  tan x
 2x

 3  cos x
f ( x)   2
x 2
 x 2  2x  1
g ( x)  
 1  2 tan x
x0
0  x  /4
at
x0
and
x  / 4
at
x0
and
x 
at
x0
at
x0
 /4  x
x0
0 x 
x
x0
0x
x0
0 x
4 – Find y  for the following functions:
a) y  x 3  x 
5
d) y  3x 2  4 x 
g)
x
3
y  2x 5  7 x 
1
5
b) y  x 6  4 x 2  5 3x  2  8
c) y 
x5  1
2x  3
6
e) y  x 4  8x 4   x  2  48x
f) y 
2x 7  6
x3  3
 16
h) y  7 x 2  x 2  4 x  2  3x 8
i) y 
x9  4
4x  3
7
x
1
x
5 – Find y  for the following functions:
a) y  x sin x 
sec x
 cot x
x
b) y 
- 6 / 11 -
sin(3x) cot(3x)
 3x  csc x
cos(3x)
c) y 
1  cos( x)
1  cos( x)
6 - Find y  for the following functions:
b) y  cot  x sin x 2 
c) y  tan 4 x sec(3x 2  5)
d) y  3x 3  6 6 x 2  5x  2
e) y  sec( x sin( x 2 ))
f) y  x tan 2 x  sin 5x
g) y  (7 x  4) 5 sin 5 (3x 4  2 x)
h) y  tan(3x 2 ) sec(3x 2 )
i) y  3 sin(cos( x))  csc( x)  1
a) y  5x 4  4 x 3  3x  2
6
4
3
7 - Find y  for the following functions:
a) x y 2  x 2 sin y  3x  5 y 3  6
c) x  
x2
2y 1
b) x 2  y  tan(x y)
 sin( x y)  2 x
d) cot( y  3)  sin 2  tan(x  y)
f) x 3  y 3  x  y 2  3( x  y) 2  sec(1.5)
e) x 3  y 2  cos(4 x  y)  2
8 - Find y  for the following functions:
a)
b)
c)
y  cos(4  5)   2
t 2 1
t 2 1
y  5t 2  2t  1
y
&
x  sec 2 ( 2  1)
&
x  t 3  5t  3
&
x  sin(t )
9 – Find y  for the following functions:
a)
c)
d)
f)
h)

y  2 x  3x 2  4 3

4
y   2  2  3
1  cos( x)
y
1  cos( x)
b) x  y  tan(x y)
&
y  cot 7 sin x 
y  sin( x  cos( x 2 ) )
x  cos 
sin(3x) cot(3x)
 5x 2
cos(3x)
e)
y
g)
i)
x y 3  x sin y  3 y  5x 3  2
x 3  y 3  x  y   3( x  y) 2  1
2
10 - Find y  in a simple form ( if possible ):
a)
ye
x  x tan x 3
 1
 x 
 x
b)
y  ln(2 ln x 2 )  ln 2 ( x  cot 2 x)
e)
y  ln 4 x 2  1  4 x tan 2 x 
2
d)
ye
g)
y  e x sec 3 x
h)

y  ln x

3

 sin 3x  1  4 x csc 5x
- 7 / 11 -

c)
y  ln e x
f)
y  ln
i)
y  e ln(x
6
sin 2 x
x2  5
3x  4
6
sin 2 x )

11 - Find y  in a simple form ( if possible ):
x
3
2 x
 x  sin x
2
a)
y4
c)
e)
g)
y  10 x  x10  10 x
e
2 x
5
 3x 
3
y  xx
i)
sin xcos y  sin y cos x
3

x  3 e 2 x 1 sec
2
y 
k)
y  tan x 
3x
2

1
2/5
x
x
x  33 / 4 2 x  13 / 5 sin
b)
y
d)
f)
h)
y   x  x  e
x1 y  y1 x  1
j)
y  5  6 x  7 x  8x 9
l)
sin xcos y  sin y cos x  3
4 x
2

1
2/5
x
ex
y  ln sin x 
sec x
ln(e )
sec2 x
12 - Prove that:
a)
b)
c)
y   y  2  1  0
y   e 2 y
x y  xy  y  0
2
y  ln(sin x  cos x)
If
If
If
y  ln(cos x)
y  cos(ln x)  sin(ln x)
13 - Evaluate the following expressions:
a)

 x 
sec tan 1   
 2 

b)


x  

cos csc 1 


2
x

9



b)
y  tan 2 sec 1 x
14 - Find y  in a simple form:
a)

y  sin 2 cot 1 x



c)

 x
sin tan 1 

2
 1 x

c)
y  cos 2 cos 1 1  x






15 - Find y  in a simple form( if possible ):
  
a)
y  sin 1 x  sin x 1  sin x 
b)
y  tan 1 cos x 2
c)
y  sin 1 sin x
d)
y  cos 1 (2 x)  cos(3x)
e)
y  cot( x sin 1 x)
f)
tan y  x 2  x
g)
y  x 1  x 2  sin 1 x
h)
y  tan 1
1  cos x
1  cos x
i)
 x 1 
y  cos 1 

 x 1
1


3
3
16 - Prove that:
a)
y  0
if
b)
y  0
if
c)
y 3 y  1  0
if
d)
y  4 x cos 1 x
if
e)
f ( x)  g ( x)
if
c)
2 yy  1
if
y
sin(cos 1 x)
1 x2
.
2
 x 
1  1  x 

y  tan 1 

tan

2
 x  .
1 x 


 t 
 t 
x  cos
y  sin
,

1  t 
1  t 
y  sin 1 ( x)  2 x 2 cos 1 ( x)  x 1  x 2 .
 x 1 
1
f ( x)  sin 1 
x.
, g ( x)  2 tan
 x 1

y  sin 1 ln e sin
x
.
- 8 / 11 -
17 - Prove that:
a)
cosh 2 x  sinh 2 x  1
b)
c)
sinh 1 x  ln( x  x 2  1)
d)
e)
cosh(a ln x) 
g)
sinh(2 x)  2 sinh( x) cosh( x)

1 a
x  x a
2

f)
h)
cosh 2 x  sinh 2 x  cosh(2 x)
1 1 x
tanh 1 x  ln
x 1
2 1 x
1  1  x2 

sec h 1 x  ln 
0  x 1


x


n
nx
cosh x  sinh x  e
18 - Find y  in a simple form ( if possible ):
y  sinh( x 3  7 x  9)  e x tanh(5x  1)
a)
c)
e)
g)
ln(sinh y)  tanh 1 x  sin 1 y
b)
d)
f)
h)
i)
tanh 1 y  tan 1 x  y 2
j)
y  sinh

1
x  sin x
1
 sinh x 
1
1

y  cosh ( x ) / e , x  1
4
x
k) y  csc h 1 ( x)  csc h( x)  csc( x)
m) y  sinh 1 (e x sin x)
l)
n)


y  sec h( x 2  3x  4 ) / x
1
y  x sinh (3x)
2
y  ln(tanh 1 x) , 0  x  1


y  coth sin x  sin 1 x
 cosh x  1 
y  sec 1 

 cosh x  1 
y  tan 1 x coshx 
x  e t cos t ,

y  e t sin t
19 - Find the domain of each of the following functions:
a)
f ( x)  3sin 1 ( x  5)
g)
tanh 1 (2 x  5)
x 1
f ( g ( x)) and g ( f ( x))
h)
f ( g ( x)) and g ( f ( x))
c)
f ( x) 
b)
f ( x)  ln(e x  1)
d)
f ( x) 
f)
f ( x)  cosh 1 (2 x  3)
if
tan 1 ( x 3  5)
ln( x 2  9)
e)
f
(
x
)

x2  9
x 2  16
f ( x)  ln( x  2) and g ( x)  cos x
if
f ( x)  sin 1 ( x)
and g ( x)  e x
Assignments ( Tomas' Calculus )
Assignment # 3
1(e, f),
2(c, f), 3( c, d),
4(g→i),
5(c),
17(g, h) , 18(k→n), 19(f, e)
- 9 / 11 -
MSA
Faculty of Engineering
Sheet ( 3 )
Applications & Partial Differentiation
Eng. Math. I
( MAT 151 )
[ 1 ] Use L' Hopital rule to evaluate the following limits:
(i)
lim
x 3
(iii)
 x2  x  6 
 2

x

4
x

21


(ii)
lim
y
 sin(13 t ) 
lim  tan(91t ) 
 32 y 5  1 


3
8
y

1


1
2
 3 w  5 tan(3 w) 
lim  11w  sin(5 w) 
(iv)
t 0
w0
[ 2 ] Evaluate the following limits:
(i)
lim
x 1
(iii)
 2  2x

x
 33



 1 x2 


lim
x 1  sin  x 
 1  sin( x)  1  sin( x) 

(iv) lim 

x
x 0 

 1  cos x 

(vi) lim 

x2
x 0 

1
 sin ( 2 x ) 

(viii) lim  1
x  0  tan ( 3 x ) 
(ii)
 tan x  sec x  1 
lim  tan x  sec x  1 
x 0
(v)
lim
x 0
(vii)
lim
x 0
 1  cos x 
 4

2 
 x 2x 
 1  3e 4 x  4 e3 x

 x  sin( x)



[ 3 ] Evaluate the following limits:
(i)
lim
x 
(iii)
lim
x  0
 3 x2  4 
 2

 5 x  x 1
 ln(tan 2 x) 


 ln(tan 3x) 
 tan x 


lim
 1  sec x 

(ii)
x / 2
(iv)
lim
x 
[ 4 ] Evaluate the following limits:
(i)
lim  x

ex
2
x 
(iii)
lim x  e
1/ x
(v)

x 0

(iv)
ln x

lim
lim  sin x  x 
(vi)

( 2 x   ) sec( x)
 sec x  tan x 
lim

(viii)

lim  e
x 0
x 0
(vii)
2



x  (  / 2)
1
1
lim  x
(ii)
1
x 
 x3
 2 x
e
x
1
1
 
1 x 
lim  ln( x)  ln(sin x) 
x  0
x / 2
[ 5 ] Evaluate the following limits:
(i)
lim
x 
(iv)
x
 5
1  
 x
(ii)
cot x
x 0
 tan x 
lim

cos x
x / 2
lim 1  3 x 
(v)
1
sin x
lim  x 
x 0
- 10 / 11 -
(iii)
lim x 
sin x
x 0
(vi)
lim cos x 
1/ x
(vii)
x 0
(ix)
lim  x 
lim 1  x 
ln x
(viii)
(x)
x 
lim  ln x 
x
1
x 0
x 0
1 / ln x
lim  e

x
1/ x
x 
[ 6 ] Find the first 3 nonzero terms in the Taylor series at x  a in the following cases:
a)
,
f ( x)  sin x
a  /2
2
f ( x)  cosh 3 x ,
b)
a 1
x
c)
,
f ( x)  x e
a  1/ 2
x
d)
f ( x)  e ln x ,
a  1.
[ 7 ] Find the first 4 nonzero terms in the Maclaurin series in the following cases:
a)
b)
f ( x)  cos x
f ( x)  sinh 1 x
c)
d)
f ( x)  e x /(1  x)
f ( x)  ( x  1) ln( x  2)
[ 8 ] Find the first 3 nonzero terms in the Taylor series at x  a in the following cases:
a)
,
b)
f ( x)  sin( x   / 4) ,
f ( x)  cos x
a 0.
a  1.
x
1
c)
d)
f ( x)  ( x  1) tan x ,
f ( x)  e ln( x  2) ,
a  1/ 2 .
a  0.
[ 9 ] Find the first 3 nonzero terms in the Taylor series at x  a in the following cases:
a)
,
b)
f ( x)  (tan 1 x) 2
f ( x)  cos x  sin x ,
a 0.
a  3.
c)
f ( x)  e x 
1
1 x
,
a  1/ 2 .
d)
f ( x) 
ln(1  x)
1 x
[ 10 ] Find the first partial derivatives the following cases:
f ( x, y)  tan 1 ( y / x )
f ( x, y)  e x y ln( y )
(i)
(ii)
(iii) f ( x, y)  x e x  y sin x
(iv) f ( x, y, z)  tanh( x  2 y  3 z )
[ 11 ] Find all the second partial derivatives the following cases:
(i)
(ii)
g ( x, y)  ln( x  y)
f ( x, y)  x e y  y  5
[ 12 ] Verify that:
f ( x, y)  e x  x ln( y )  y ln( x )
(i)
if
fx y  fyx
(ii)
[ 13 ] Find
(i)
(ii)
(iii)
gxz  gz x
if
g ( x, y, z ) 
x2  y2  z 2
w / u and w / v in the following cases:
y  sin u ,
w  ln( x 2  y 2  z 2 ) , x  cos u ,
w  4e ln y ,
x
x  ln(u cos v) ,
w  x y  y z  z x,
x uv,
z4 u
y  u sin v
y  u v ,
z  uv
Assignment # 4 ( Sheet 3 )
1( i, iv ), 2( i, iv, vii, viii ), 3( iii, iv ), 4( iii, vi, vii ), 5( i, iii, v, vii, ix ), 9
10( i, ii ), 11( ii ), 12( i ), 13( i, iii )
- 11 / 11 -
,
a  0.
MSA
Faculty of Engineering
Sheet ( 4 )
Find the Maclaurin series for
a)
b)
c)
d)
𝑓(𝑥 ) = ln(𝑥 + 1)
𝑓(𝑥) = sin(2𝑥)
𝑓(𝑥 ) = 𝑒 3𝑥
𝑓(𝑥 ) = 𝑥𝑒 2𝑥
2-Find the Taylor's series for
𝑓(𝑥 ) = sin(𝜋𝑥) and 𝑎 = 1
3- Find the absolute maximum and minimum values of
a) 𝑓(𝑥 ) = 2𝑥 2 − 6𝑥 over the internal [0,4]
b) 𝑓(𝑥 ) = 3𝑥(𝑥 − 4)1/3 over the internal [-5,4]
- 12 / 11 -
Eng. Math. I
( MAT 151 )