Umm Al-Qura University Final exam
Faculty of Applied Science First semester 1439/1440 H
Department of Mathematical Science Math 4041101-4
Saturday: 8 / 4 / 1440 H Time Limit: 2 hours
Differentiation and integration (1)
Student information:
Name (Print): Group (Print):
University ID (Print): Serial No. (Print):
This exam contains 7 pages (including this cover page) and 5 problems. Check to see if any pages are
missing. Enter all requested information on the top of this page, and write your name on the top of
every page, in case the pages become separated.
You may not use your books, notes, or any calculator on this exam.
You are required to show your work on each problem on this exam. The following rules apply:
• If you use a “fundamental theorem” you
Problem Points Score
must indicate this and explain why the theorem
may be applied.
1 10
• Organize your work, in a reasonably neat and
coherent way, in the space provided. Work scat- 2 10
tered all over the page without a clear ordering
will receive very little credit. 3 10
• Mysterious or unsupported answers will not 4 10
receive full credit. A correct answer, unsup-
ported by calculations, explanation, or algebraic 5 10
work will receive no credit; an incorrect answer
supported by substantially correct calculations and Total: 50
explanations might still receive partial credit.
Do NOT write in the table to the right.
Math 4041101-4 Final exam - Page 2 of 7 Saturday: 8 / 4 / 1440 H
dy
1. (10 points) Find for each of the following:
dx
(a) (2 points) y = 2x3 + 3x2 + 5x
(b) (2 points) y = (x2 + 2)(x3 + 1)
3x−5
(c) (2 points) y = x2 +7
(d) (2 points) y = (x2 − x + 1)5
(e) (2 points) y = sin(3x2 )
Math 4041101-4 Final exam - Page 3 of 7 Saturday: 8 / 4 / 1440 H
2. (10 points)
√ 1
(a) (6 points) Let f (x) = x3 + 3x, g(x) = x+3 and h(x) = .
4x − 1
(i) Find the following:
• the natural domain of f
• the natural domain of g
• the natural domain of h
• (f + g)(1)
• (f ◦ g)(1)
(ii) Show that the function f is odd.
(b) (4 points) Find each of the following limits
2x x3
(i) lim (iii) lim
x→1 − x −1 x→−∞ 2x3 − 5x
x2 − 25 x2 − 4
(ii) lim (iv) lim
x→5 x − 5 x→2 x2 + 4
Math 4041101-4 Final exam - Page 4 of 7 Saturday: 8 / 4 / 1440 H
3. (10 points) Find each of the following:
Z 4 √
(a) (2 points) (3t2 + t) dt
0
Z 1
(b) (2 points) (x3 + 3 x) dx
−1
Z π/4
(c) (2 points) (cos x − sin x) dx
0
Z
1 3
(d) (2 points) (x 3 + x 4 ) dx
Z
(e) (2 points) (x3 + 6x)5 (3x2 + 6) dx
Math 4041101-4 Final exam - Page 5 of 7 Saturday: 8 / 4 / 1440 H
4. (10 points) For the curve y = x3 − 12x + 1, find the following:
(a) (2 points) the critical points,
(b) (2 points) the increasing and decreasing intervals,
(c) (2 points) the local Maxima and Minima,
(d) (2 points) the concavity intervals
(e) (2 points) the inflection point,
Math 4041101-4 Final exam - Page 6 of 7 Saturday: 8 / 4 / 1440 H
5. (10 points)
(a) (5 points) The function g is defined by
7 − 3x if x ≤ 3,
g(x) = √
1 − 3x if x > 3.
(i) (2 points) Find lim g(x) =
x→3−
(ii) (2 points) Find lim g(x) =
x→3+
(iii) (1 point) Is the function g continuous or discontinuous, at x = 3 ?
(b) (2 points) Given that f (x) = sin2 (x), find
(i) (1 point) f 0 (x) (ii) (1 point) f 00 (0)
(c) (3 points) For the curve x2 + y 2 = 25, find the following:
dy
(i) (2 points)
dx
d2 y
(ii) (1 point)
dx2
Math 4041101-4 Final exam - Page 7 of 7 Saturday: 8 / 4 / 1440 H
Best regards.
