Days and Time: - 16-364-FPT
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
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