University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
One hour exam. No communication with the outside world allowed.
Name:______________________________________________________(please write clearly)
75 Points Total: Choose 3 of the first 4.
Only three of the first four will be graded.
Clearly indicate below which ones you DO want graded.
Open notebooks are allowed.
SHOW ALL WORK!
Calculators are allowed.
SHOW ALL WORK!
Neatness counts!
Points will be
deducted for
excessive
sloppiness.
Grading (25 points each):
#1 : _______________
Please check the box of
the problems you want
to have graded
#2 : _______________
#3 : _______________
A
#4 : _______________
Bonus : _______________
Total points (out of 75): ______________
Grade (percent):
0
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
16.364
1. (25 pts)
Phone: (978) 934-3300
Fax:
(978) 934-3027
Evaluation of Functions of Complex Numbers
For each of the following functions, find ( ) in rectangular form for
If the function is undefined at , state that fact. (5 pts each).
Please enter answers clearly at the bottom of the page.
(a)
( )
(b)
( )
(c)
( )
(d) ( )
(e)
A
̅
(
(
)
(answer as a decimal number)
) ̅
(
| |
)
( )
[
(
)]
[
(
̅̅̅̅̅̅
( )
)]
(a)__________________
(b)__________________
(c)__________________
(d)__________________
(e)__________________
1
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
16.364
2. Derivative of a function of complex numbers (25 pts):
(Please enter answers for (a), (b) and (c) clearly at the bottom of the page.)
A
For ( )
(a) For what values of
(b) Find
(c) Find
( )
( )
( ) undefined? (5 pts)
in lowest terms. (10 pts).
evaluated at
(10pts)
(a)__________________
(b)__________________
(c)__________________
2
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
3
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
3. Cauchy-Riemann and Laplacian analysis. (25 pts)
)
Consider (
,
(a)
(b)
(c)
(d)
(a)
Find
. (10 pts).
Explain why (
) is harmonic (5 pts)
Find (
) that is a harmonic conjugate of (
) (10 pts).
Find ( ) in simplest terms of . ([Bonus!] 5 pts).
_____________________________
A
________________________________________
(b)________________________________________________________________________________
(c)________________________________________________________________________________
(d)___________________
4
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
5
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
4. Complex Trigonometry (25 pts)
(
)
Use the following identities to prove that ̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(
(
( )
( )
)
( )
)
( )
( )
( )
(̅ )
( )
( )
( ) ( )
( )
( )
A
Show all work clearly (do not use exponentials).
6
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
Bonus (7.5 pts): Parts (a) and (b) ( ) and general complex power (2.5 pts each). First
evaluate to “calculator ready” form and put a rectangle around that answer. Then find the
numerical answer (this is where you use the calculator). Put the numerical answers to (a) and (b)
to five significant figures of precision at the bottom of the page. If you don’t do the first part you
will get no credit for the second part. Show all Work!
Part (c) Sets (2.5 pts).
(a) Evaluate
(b) Evaluate | (
(
A
).
)
|
[ ( )]
(c) Find the set of all such that
(| |).
Sketch the set (use the back of this sheet). Is it a domain?
(a)_________________________
(b)_________________________
(c)__________________________
7
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
One hour exam. No communication with the outside world allowed.
Name:______________________________________________________(please write clearly)
75 Points Total: Choose 3 of the first 4.
Only three of the first four will be graded.
Clearly indicate below which ones you DO want graded.
Open notebooks are allowed.
SHOW ALL WORK!
Calculators are allowed.
SHOW ALL WORK!
Neatness counts!
Points will be
deducted for
excessive
sloppiness.
Grading (25 points each):
#1 : _______________
Please check the box of
the problems you want
to have graded
#2 : _______________
#3 : _______________
B
#4 : _______________
Bonus : _______________
Total points (out of 75): ______________
Grade (percent):
8
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
16.364
1. (25 pts)
Phone: (978) 934-3300
Fax:
(978) 934-3027
Evaluation of Functions of Complex Numbers
For each of the following functions, find ( ) in rectangular form for
If the function is undefined at , state that fact. (5 pts each).
Please enter answers clearly at the bottom of the page.
(a)
( )
(b)
( )
(c)
( )
(d) ( )
(e)
B
̅
(
(
)
(answer as a decimal number)
) ̅
(
| |
)
( )
[
(
)]
[
(
̅̅̅̅̅̅
( )
)]
(a)__________________
(b)__________________
(c)__________________
(d)__________________
(e)__________________
9
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
16.364
2. Derivative of a function of complex numbers (25 pts):
(Please enter answers for (a), (b) and (c) clearly at the bottom of the page.)
For ( )
(d) For what value(s) of
(e) Find
(f) Find
( )
( )
( ) undefined? (5 pts)
B
in lowest terms. (10 pts).
evaluated at
(10pts)
(a)__________________
(b)__________________
(c)__________________
10
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
3. Cauchy-Riemann and Laplacian analysis. (25 pts)
)
Consider (
,
(a)
(b)
(c)
(d)
(a)
Find
(10 pts).
Explain why (
) is harmonic (5 pts)
Find (
) that is a harmonic conjugate of (
) (10 pts).
Find ( ) in simplest terms of . ([Bonus!] 5 pts).
_____________________________
B
________________________________________
(b)________________________________________________________________________________
(c)________________________________________________________________________________
(d)___________________
11
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
12
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
4. Complex Trigonometry (25 pts)
(
)
Use the following identities to prove that ̅̅̅̅̅̅̅̅̅̅̅̅̅
(
(
( )
( )
)
( )
)
( )
( )
( )
(̅ )
( )
( )
( ) ( )
( )
( )
B
Show all work clearly (do not use exponentials).
13
University of Massachusetts, Lowell
301 Ball Hall
One University Avenue
Lowell, Massachusetts 01854
16.364
Phone: (978) 934-3300
Fax:
(978) 934-3027
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
Engineering Math
Exam #1
25 Feb. 2015
Bonus (7.5 pts): Parts (a) and (b) ( ) and general complex power (2.5 pts each). First
evaluate to “calculator ready” form and put a rectangle around that answer. Then find the
numerical answer (this is where you use the calculator). Put the numerical answers to (a) and (b)
to five significant figures of precision at the bottom of the page. If you don’t do the first part you
will get no credit for the second part. Show all Work!
Part (c) Sets (2.5 pts).
(a) Evaluate
(b) Evaluate | (
(
).
)
B
|
[ ( )]
(c) Find the set of all such that
(|
|).
Sketch the set (use the back of this sheet). Is it a domain?
(a)_________________________
(b)_________________________
(c)__________________________
14