Almajma’ah University
Engineering College
2nd Level
Time allowed: 150 min.
15/2/1431
Final Exam (Math 105)
Question #1: [4 marks]
Find the set of solutions (represented as intervals) for each of the following
inequalities. Then explain your answer graphically.
a) −5 ≤
4−3x
2
b) |2𝑥 − 5| ≥ |𝑥 − 4|
<1
Question #2: [6 marks]
a) Proof that the following points (-2,1), (-2,4), (3,4), (3,1) are representing
vertices of a rectangle.
b) Find the equation of the straight line that passes with the point (1,-1)
Perpendicular to the line 2x -3y - 8 = 0
Question #3: [6 marks]
a) Find the Domain of each of the following functions:
i. 𝑓(𝑥) =
b) If f(x) =
1
ii. 𝑓(𝑥) = 𝑥
𝑥 2 −1
x
2
and
(x+1)
1
−√𝑥
g(x) = √x 3 − 2 Find:
i. (f+g)(3)
ii. (f o g)( 3)
c) Examine the function ℎ(𝑥) =
𝑥 2 +4
if it is even or odd?
𝑥 3 −𝑥
Question #4: [6 marks]
Find the following limits:
a) Lim
3x2 −5x+8
𝑏)
x→∞ 9x−4x2 +13
𝑐)
𝐿𝑖𝑚
𝑥→2
(𝑥 2 +𝑎𝑥+𝑏)
𝑥 2 −4
=
7
2
,
𝑙𝑖𝑚
5𝑥+𝑠𝑖𝑛3𝑥
𝑥→0 2𝑥−𝑡𝑎𝑛 𝑥
find a and b
and
Differential Calculus: Math 105, Final exam
Question #5: Answer only two of the following questions: [4 marks]
a) If 𝑔(𝑡) = 𝑎𝑡 2 + 𝑏𝑡 + 𝑐 and 𝑔(1) = 5 , 𝑔/ (1) = 3 , 𝑔// (1) = −4 Find
a, b and c
b) Find a and b so that the following function will be defined at all points
𝑥+1
𝑓(𝑥) = {𝑎𝑥 + 𝑏
3𝑥
c) If y = cos 2x , Find
dy
dx
,
d3 y
d10 y
dx
dx10
,
3
𝑓𝑜𝑟 𝑥 < 1
𝑓𝑜𝑟 1 ≤ 𝑥 < 2
𝑓𝑜𝑟 𝑥 ≥ 2
Question #6: Answer the following questions: [5 marks]
a) If 4𝑥 2 𝑦 − 3𝑦 = 𝑥 3 − 1 Find
b) If
𝒅𝒚
𝒅𝒙
𝑥 = 2𝑡 + 6 , and 𝑦 = 6𝑡 3 + 4
c) Find 𝐷𝑥 (𝑥3 + tan 𝑥)
Find
𝑑𝑦
𝑑𝑥
at t =4
6
Question #7: [9 marks]
a) If 𝑓(𝑥) = 𝑥 4 + 1 Find the real number c that satisfy “The Mean Value
Theory” in the interval [-2,4].
b) Find where the following function is increasing, decreasing, concaving
upward and concaving downward?
1 3
𝑥 − 𝑥 2 − 3𝑥 + 4
3
c) Find all the infliction points for the following function:
𝑓(𝑥) = 𝑥 4 − 2𝑥 2 − 12
𝑓(𝑥) =
Best wishes…
Dr. SaMeH Ahmed
2 P.T.O