Differential Calculus: Math 105, Final exam
King Saud University
Almajmaa’h Engineering College
Time allowed: 150 min.
2nd Level
27/6/1430
Question 1:
Find the set of solutions (represented as intervals) for each of the following
inequalities. Then explain your answer graphically.
a) (x+1)(x-1)(x-3) ≤0
b) |2𝑥 − 5| ≥ |𝑥 + 4|
Question 2:
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:
a) Find the Domain of each of the following functions:
1
1
i. 𝑓(𝑥) = 2
ii. 𝑓(𝑥) = 𝑥
𝑥 −1
b) If f(x) =
x
(x+1)
and
i. (f+g)(3)
4
−√𝑥
g(x) = √x 3 − 2 Find:
ii. (g/f)(2)
c) Examine the function ℎ(𝑥) =
𝑥 2 +4
𝑥 3 −𝑥
if it is even or odd?
Question 4:
Find the following limits:
i. lim
x−1
x→1 √x−1
ii. 𝑙𝑖𝑚
2𝑥+𝑠𝑖𝑛3𝑥
𝑥→0 𝑠𝑖𝑛 𝑥+𝑡𝑎𝑛 4𝑥
Question 5:
If 𝑔(𝑡) = 𝑎𝑡 2 + 𝑏𝑡 + 𝑐 and 𝑔(1) = 5 , 𝑔/ (1) = 3 , 𝑔// (1) = −4
Find a, b and c
1
and
Differential Calculus: Math 105, Final exam
Question 6: Answer only two of the following questions:
a) Find a and b so that the following function will be defined at all points
𝑥+1
𝑓(𝑥) = {𝑎𝑥 + 𝑏
3𝑥
𝑓𝑜𝑟 𝑥 < 1
𝑓𝑜𝑟 1 ≤ 𝑥 < 2
𝑓𝑜𝑟 𝑥 ≥ 2
b) Redefine the following function so that it is being continuous at x =2
𝑓(𝑥) =
𝑥 2 −4
𝑥−2
,x≠2
c) Use the definition of the first derivatives to find the derivative of:
𝑓(𝑥) = 𝑥 2 + 𝑥 + 1
Question 7: Answer only two of the following questions:
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 8:
a) If 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 1 Find all the real number c that satisfy “Rolle’s
Theorem” in the interval [0,4].
b) If y = sin 2x , Find
dy
dx
,
d3 y
d8 y
dx
dx8
,
3
c) Find where the following function is increasing, decreasing, concaving
upward and concaving downward?
𝑓(𝑥) =
Best wishes…
Dr. SaMeH Ahmed
2
1 3
𝑥 − 𝑥 2 − 3𝑥 + 4
3