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Mathematics Exam 11/13/2024

⏱ 60’

1) LIMITS Calculate the following limit

lim log $ √𝑥

!→#

%

o 0
o −∞
o

!
"

o +∞

2) VECTORS Determine the polar coordinates (𝜌, 𝛼) of 𝑢
(⃗ with A (0, –4)
#

o 𝜌 = 4 ; 𝛼 = $

%

o 𝜌 = −4 ; 𝛼 = − $ 𝜋
o they are not defined
%

o 𝜌 = 4 ; 𝛼 = $ 𝜋

3) VECTORS a- Determine 𝑘 so that 𝑢 (2, 2𝑘) and 𝑣 4√3, 18 are orthogonal
b- Determine the angle 𝛼 between 𝑢 and the versor −𝚥
(⃗(0, – 1)
o 𝑢 = 4√3, 18 ; 𝛼 =

#
%

o 𝑢 = 41, −√38 ; 𝛼 =

#
&

o 𝑢 = 42, −2√38 ; 𝛼 =

#
&
#

o 𝑢 = 42, 2√38 ; 𝛼 = − &

4) DERIVATIVE Given the function 𝑓 (𝑥 ) = −10(𝑥 $ − 2), determine if there
is a maximum or a minimum in 𝑥 = 0
o the function has a minimum in 𝑥 = 0

o the function has a maximum in 𝑥 = 0
o the function neither has a maximum nor a minimum in 𝑥 = 0
o the function is increasing in 𝑥 = 0

2 3
5) ALGEBRA Determine the inverse matrix 𝐴'! of A ?
@
−1 1
1
?
( 1
5
o ?
5
! 5
o (?
0
1
o ?
1
o

!

−3
@
2
−15
@
10
0
@
5
−3
@
2

6) ALGEBRA Determine the real eigenvalues and eventual orthogonal
−3 0
eigenvectors of A ?
@
0 3
o no, there are no real eigenvalues
o yes, there are two real eigenvalues, but no eigenvectors
o yes, there are two real eigenvalues with their respective eigenvectors,
but they are not necessarily orthogonal
o yes, there are two eigenvalues with their respective orthogonal
eigenvectors

7) ALGEBRA Given the following equations, determine 𝑚∥ and 𝑚* so that
the lines can be respectively parallel and orthogonal
2𝑥 + 𝑦 = 7
C
𝑚𝑥 − 𝑦 = −1

!

o 𝑚∥ = 2 ; 𝑚* = $
o 𝑚∥ = 2 ; 𝑚* = 1

o 𝑚∥ = −1 ; 𝑚* = 2
!

o 𝑚∥ = −2 ; 𝑚* = $

8) DERIVATIVE Calculate the first derivative of 𝑓(𝑥) = √2𝑥 + 1
o 𝑓 +(-) =
o 𝑓

+(-)

!
√$-0!

= 2𝑥 + 3
"

o 𝑓

!
+(-)1$($-0!) #

o 𝑓 +(-) =

$
√$-0!

1
9) ALGEBRA Given 𝑣⃗ (0, −2, 0) and 𝐴 F2H, calculate 𝑣⃗ ∙ 𝐴
6
o
o
o
o

−12
−4
0
it’s not defined

10) PARTIALD Given the function 𝑓(𝑥, 𝑦) = 5𝑥 + cos 𝑦, determine the
second derivative 𝑓-2 (0, 0)
o
o
o
o

1
5
0
−∞

11) FUNCTIONS Determine the centre C and the radius R of the following
function
𝑥 $ + 𝑦 $ = 6y

o
o
o
o

𝐶 (0, 0) ; 𝑅 = P6𝑦
𝐶 (0, 6) ; 𝑅 = 9
𝐶 (0, 3) ; 𝑅 = 3
𝐶 = 0 ; 𝑅 = 3

12) PARTIALD Given the function 𝑓(𝑥, 𝑦) = 5𝑥 + ln 𝑦, determine the second
derivative 𝑓-2 (0, 0)
o 5
o −∞
o

!
(

o 0

13) LIMITS Calculate the following limit

lim log$' 𝑒 !

!→&#

o
o
o
o

−∞
+∞
it’s not defined
0

14) SERIES To which function the following series corresponds
6

𝑓(𝑥) = T

($3)#$

417 ($4)!

o
o
o
o

𝑓(𝑥) = cos 𝑥
𝑓(𝑥) = sin 𝑥
𝑓(𝑥) = cos 2𝑥
𝑓(𝑥) = e$3

(((⃗ (−1, 1) of 𝑓(𝑥, 𝑦) = 𝑥 $ 𝑦
15) PARTIALD Determine ∇f

o
o
o
o

(−2, 1)
(2, 1)
−2
1

16) SERIES To which function the following series corresponds
6

𝑓 (𝑥 ) = T

417

(%3)$
4!

o 𝑓(𝑥) = cos 3𝑥
!

o 𝑓(𝑥) = (!'-)"
o 𝑓(𝑥) = e%3
!

o 𝑓(𝑥) = !0-"
17) DERIVATIVE Determine the tangent line to 𝑓(𝑥) = 3𝑥 $ in the point 𝑥7 =
1
o
o
o
o

𝑦 = 3(2𝑥 − 1)
𝑦=0
𝑦=6
𝑦 = 6𝑥

18) INTEGRALS Determine if the following integral is proper, improper,
eventually calculable
!

Y
7

o
o
o
o

it’s improper but calculable
it’s improper and not calculable
it’s proper and calculable
it’s not calculable

1
√𝑥

𝑑𝑥

2 3
−2 −5
19) ALGEBRA Given the following matrices A ?
@ and B ?
@,
−1 1
1
2
calculate C = AB
−4 −1
o C?
@
7
3
0 −2
o C?
@
0 3
−1 −4
o C?
@
3
7
−4 −15
o C?
@
−1
2

20) FUNCTIONS Determine if [
o
o
o
o

0 𝑠𝑒 𝑥 ≤ 0
is continuous in 𝑥 = 0
0,1 𝑠𝑒 𝑥 > 0

𝑓(𝑥) is defined and continuous in 𝑥 = 0
𝑓(𝑥) is discontinuous in 𝑥 = 0
𝑓(𝑥) is not defined in 𝑥 = 0
𝑓(𝑥) is continuous but singular in 𝑥 = 0

21) FUNCTIONS Determine the inverse function of 𝑦 = sin 𝑥 on the entire
real line
o
o
o
o

arcsin 𝑥
the function does not have the inverse on the entire real line
cos 𝑥
sin'! 𝑦

22) DIFFEQ Determine the functional development in 𝑦(𝑡) of
o
o
o
o

quadratic in t
constant in t
exponential in t
linear in t

82
89

= 𝑦𝑡

23) FUNCTIONS Determine the inverse function of 10o
o
o
o

#

(𝑥 $ )!7
log $ 10an inverse function does not exist
2 log!7 𝑥

24) FUNCTIONS Given the following functions
#

𝑓! (𝑥) = 𝑒 '- 𝑓$ (𝑥) = 𝑒 '!7- 𝑓% (𝑥) = 𝑒 '- 𝑓" (𝑥) = 𝑒 '√- ,
determine the higher-order infinitesimal 𝑥 → +∞
o
o
o
o

𝑓"
𝑓!
𝑓%
𝑓$

25) LIMITS Calculate the following limit

sin 2𝑥
!→' 2𝑥
lim

o
o
o
o

1
it’s not defined
0
2𝜋

Solutions
1) −∞
%

2) 𝜌 = 4 ; 𝛼 = $ 𝜋
3) 𝑢 = 42, −2√38 ; 𝛼 =

#
&

4) the function has a maximum in 𝑥 = 0
! 1 −3
5) ( ?
@
1 2

6) yes, there are two eigenvalues with their respective orthogonal
eigenvectors
!

7) 𝑚∥ = −2 ; 𝑚* = $
8) 𝑓 +(-) =

!
√$-0!

9) −4
10) 0
11) 𝐶 (0, 3) ; 𝑅 = 3
12) 0
13) −∞
14) 𝑓(𝑥) = cos 2𝑥
15) (−2, 1)
16) 𝑓(𝑥) = e%3
17) 𝑦 = 3(2𝑥 − 1)
18) it’s improper but calculable

−1 −4
19) C?
@
3
7
20) 𝑓(𝑥) is discontinuous in 𝑥 = 0
21) the function does not have the inverse on the entire real line
22) constant in t
23) 2 log!7 𝑥
24) 𝑓$ (𝑥) = 𝑒 '!725) 1



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