American University of Technology
F i n a l
E x a m i n a t i o n
Fall Semester 2022-23
MAT 221 – Calculus and Applied Math for Business
Instructor:
Joseph Keirouz
Duration:
2 hours
Use of Scientific Calculators is allowed
Programmable Calculators are not allowed
Closed book Exam
Mobile phones are forbidden in Exam Room
ID#:
Last Name:
First Name:
Middle Name:
…………………………
………………………..
……………………….
……………………….
Course Code:
Section:
Room:
Seat #:
MAT 221
A
Grade ……/100
**********************
Good Luck!
MAT 221 – Final Exam-fall 2022-23
1/5
American University of Technology
Formulas
2
The quadratic Function: ax + bx + c = 0
 = b 2 − 4ac
x=
−b 
2a
Total Revenue: TR = (aQ + b)Q = aQ 2 + bQ
Total Cost
Profit
TC = FC + (VC ) Q
Average Cost
AC =
TC
Q
 = TR − TC
M = b n  log b M = n
Logarithms
log( x  y ) = log( x) + log( y )
log( x / y ) = log( x ) − log( y )
log( x m ) = m log( x)
Derivatives
f ( x) = x n  f ( x) = nx n −1
f ( x) = e g ( x ) = eU  f ( x) = g ( x)e g ( x ) = U ' eU
g ( x) U '
f ( x) = ln ( g ( x )) = ln U  f ( x) =
=
g ( x) U
y = U  V  y ' = U 'V '
U
U 'V − V 'U
y =  y' =
V
V2
y = U  V  y ' = U 'V + V 'U
y = U n  y ' = n U n −1  U '
Optimization of Economic Functions
To find and classify stationary points:
1. Solve the equation f’(x)=0 and find stationary points x=a.
2. Find f”(x) and if:
a. f”(a)>0 then the function has a minimum at x=a.
b. f”(a)<0 then the function has a maximum at x=a.
c. f”(a)=0 then x=a cannot be classified using the available information.
Integration
If F’(x) = f(x) then F ( x) =  f ( x) dx
x
n
dx =
1 n +1
x +c
n +1
1
 x dx = ln( x) + c
1
 e dx = m e + c
 [af ( x)  bg ( x)] dx = a  f ( x) dx  b g ( x) dx + c
TC =  MC dQ Where c is obtained from finding when Q = 0, TC = FC
TR =  MR dQ Where c = 0 since when Q = 0, TR = 0
mx
mx
MAT 221 – Final Exam-fall 2022-23
2/5
American University of Technology
b
 f ( x) dx = [ F ( x)]
b
a
= F (b) − F (a )
a
 Determinant of a 3x3 matrix:
➔ To find the determinant of a 3x3 matrix, rewrite the first
and the second columns next to the matrix and multiply
the diagonals as follows:
Matrix multiplication
row 1
a
A =  d
 g
b
e
h
➔ How to multiply:
column 1
 1 3 1

 1 3 − 5 

  − 1 − 1 0 
2
1
3


  2 0 1 
c a b
f  d e
i  g h
element (1,1)

 − 12
= 




| A |= det( A) = (aei + bfg + cdh) − (ceg + afh + bdi )
MAT 221 – Final Exam-fall 2022-23
3/5
American University of Technology
1. (10 points) The demand function of a good is P = 30 − Q and the total cost function is
1
TC = Q 2 + 6Q + 7
2
a. (5 points) Find an expression of the profit in term of the output Q.
b. (5 points) Find the level of output that maximizes the profit found in part a) above.
2. (10 points) Find the derivative of the following functions:
a. y = x 4 e 2 x
b.
y = ln x − x
3. (10 points) We have the following average cost function: A C =
25
− 2 +Q .
Q
a- (5 points) Find the value of the total cost (TC) at Q=2.
b- (5 points) Find the value of the marginal cost (MC) at Q=2.
4. (10 points) We have the following three-equation system:
x1 + 2 x2 − 2 x3 = 1
2 x 2 + x3 = 4
x 1 + x3 = 8
Find the value of (x3) using Cramer's rule.
5- (10 points) Given the following matrices:
 1 2 3
 2 4 −1
A=
, B =  −2 0 4 

 3 −2 5 
 2 3 1 
a. Find, A+B if possible. (3 points).
b. Find AB , if possible. (7 points).
6-(10 points) We have the following matrices:
10 0 10
A =  0 20 0  ,
10 0 15
5
B = 10 ,
15
Find the value of D where D =
C = 4 2 4
det( A)
.
CB
MAT 221 – Final Exam-fall 2022-23
4/5
American University of Technology
7- (10 points) Find the value of X in the following matrix if det(A) = 25.
 5 3 − 1
A =  X 2 2 
 5 2 1 
8- (20 points) The marginal revenue function for a commodity is given by
MR = −2Q 2 + 10 .
And the total cost function for the commodity is
TC = Q 2 + 4Q + 2
a) Find the Total Revenue function. (8 points)
b) Determine the maximum profit. (12 points)

9- (10 points) Find the value of  in the following expression:  (2Q + 5) dQ = 8
1
MAT 221 – Final Exam-fall 2022-23
5/5