EXAM 4 -A MATH 6: Elementary Differential Equation MATH 6 FALL 00 EXAMINATION COVER PAGE Profeor Moeley PRINT NAME ( ) Lat Name, Firt Name MI (What you wih to be called) ID # EXAM DATE Friday, Nov. 9, 00 :30 am I wear and/or affirm that all of the work preented on thi exam i my own and that I have neither given nor received any help during the exam. Date Signature SIGNATURE DATE INSTRUCTIONS: Beide thi cover page, there are page of quetion and problem on thi exam. MAKE SURE YOU HAVE ALL THE PAGES. If a page i miing, you will receive a grade of zero for that page. Page contain Laplace tranform you need not memorize. Read through the entire exam. If you cannot read anything, raie your hand and I will come to you. Place your I.D. on your dek during the exam. Your I.D., thi exam, and a traight edge are all that you may have on your dek during the exam. NO CALCULATORS! NO SCRATCH PAPER! Ue the back of the exam heet if neceary. You may remove the taple if you wih. Print your name on all heet. Page - are Fillin-the Blank/Multiple Choice or True/Fale. Expect no part credit on thee page. For each Fill-in-the Blank/Multiple Choice quetion write your anwer in the blank provided. Next find your anwer from the lit given and write the correponding letter or letter for your anwer in the blank provided. Then circle thi letter or letter. There are no free repone page. However, to inure credit, you hould explain your olution fully and carefully. Your entire olution may be graded, not jut your final anwer. SHOW YOUR WORK! Every thought you have hould be expreed in your bet mathematic on thi paper. Partial credit will be given a deemed appropriate. Proofread your olution and check your computation a time allow. GOOD LUCK!! REQUEST FOR REGRADE Pleae regrade the following problem for the reaon I have indicated: (e.g., I do not undertand what I did wrong on page.) Score page point core 5 3 4 7 5 6 6 7 8 8 4 9 0 6 7 --- 3 4 5 6 7 8 (Regrade hould be requeted within a week of the date the exam i returned. Attach additional heet a neceary to explain your reaon.) I wear and/or affirm that upon the return of thi exam I have written nothing on thi exam except on thi REGRADE FORM. (Writing or changing anything i conidered to be cheating.) 9 0
Total 00 MATH 6 EXAM 4A- Prof. Moeley Page Lat Name, Firt Name MI, What you wih to be called True-fale. Laplace tranform.. ( pt) A)True or B)Fale The definition of the Laplace tranform i {f(t)}() = provided the improper integral exit. t t0 f(t)e t dt. ( pt) A)True or B) Fale Since the Laplace tranform i defined in term of an improper integral, it involve only one limit proce. 3. ( pt) A)True or B)Fale The Laplace tranform exit for all continuou function on [0,). 4. ( pt) A)True or B)Fale The Laplace tranform doe not exit for all dicontinuou function. 5. ( pt) A)True or B)Fale The function f(t) = /(t-3) i piecewie continuou on [0,7]. 4t 6. ( pt) A)True or B)Fale The function i of exponential order. f (t) e co(t) 7. ( pt) A)True or B)Fale The Laplace tranform :TF i a linear operator. 8. ( pt) A)True or B)Fale The invere Laplace tranform :FT i a linear operator. 9. ( pt) A)True or B)Fale The Laplace tranform i a one-to-one mapping on the et of continuou function on [0,) for which the Laplace tranform exit. 0. ( pt) A)True or B)Fale There i only one continuou function in the null pace of.. ( pt) A)True or B)Fale The trategy of olving an ODE uing Laplace tranform i to tranform the problem from the time domain T to the (complex) frequency domain F, olve the tranformed problem uing algebra intead of calculu, and then tranform the olution back to the time domain T.
Total point thi page =. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page Lat Name, Firt Name MI, What you wih to be called Follow the intruction on the Exam Cover Sheet for Fill-in-the Blank/Multiple Choice quetion. Alo, circle your anwer. 8 0 t 5. (5 pt.) The Laplace tranform of the function f(t) 0 t 5 i. A B C D E Hint: Ue the definition. Be careful to handle the limit appropriately a dicued in cla. 8 8 5 8 A) B) C) e 5 8 D) e 5 8 E) ( e 5 8 ) AB) ( e 5 ) AC) 5 5 AD) AE) BC) e 8 5 BD) e 8 5 BE) ( e 8 ) CD) 5 CE) (e 8 ABC) 5 ) ( 8 e ) ABCDE) None of the above. 5 ( e 8 ) 8 (e 5 )
Total point thi page = 5. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page 3 Lat Name, Firt Name MI, What you wih to be called Follow the intruction on the Exam Cover Sheet for Fill-in-the Blank/Multiple Choice quetion. Compute the Laplace tranform of the following function. 3. (4 pt.) f(t) = t 3t (f) =. A B C D E 4. (4 pt.) f(t) = e t + 3 e 3t (f) =. A B C D E 5 (4 pt.) f(t) = in(t) + 3 co(3t) (f) =. A B C D E Poible anwer thi page 3 A) B) C) D) E) AB) AC) AD) 3 3 3 6 6 3 6 3 3 6 3 AE) 3 3 3 3 BC) BD) BE) 3 3 3 3 3 3 3 3 CD) CE) DE) ABC) 3 3 3 3 4 3 4 3 4 3 4 3 ABD) ABE) ACD) ACE) 4 9 4 9 4 9 4 9 4 3 4 3 4 3 4 3 ADE) BCD) BDE ) CDE) 4 9 4 9 4 9 4 9
3 ABCD) ABCE) 3 3 ABDE) ( ) ( 3) 3 3 ACDE) {f} exit but none of the above i {f} BCDE) {f} doe not exit. ABCDE)None of the above. Total point thi page =. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page 4 Lat Name, Firt Name MI, What you wih to be called Follow the intruction on the Exam Cover Sheet for Fill-in-the Blank/Multiple Choice quetion. DEFINITION. Let f:xy. Then f i one-to-one if x, x X we have 6.( pt.) A B C D E implie 7(pt) A B C D E THEOREM. Let T:VW be a linear operator where V and W are vector pace over the ame field K. If the null pace N T contain only the zeo vector, then T i a one-to-one mapping. Proof. We begin our proof of the theorem by firt proving the following lemma: Lemma. Let T:VW be a linear operator and N T = {0}. If T(v ) 0, then v 0. Proof of lemma: Let T:VW be a linear operator, N T = {0} and T(v ) 0. By the definition of the null pace we have that N T = { v V: 8.( pt.) A B C D E} o that T(v ) 0 implie that v N T. Since N T contain 9.( pt.) A B C D E, we have that v 0 a wa to be proved. QED for lemma. Having finihed the proof of the lemma, we now finih the proof of the theorem. To how that T i one-to-one, for v, v V we aume 0.( pt.) A B C D E and how that.( pt.). A B C D E. We ue the STATEMENT/REASON format. STATEMENT REASON T(v ) T(v ).( pt.) A B C D E 3.(pt.) A B C D E Vector algebra in W 4.( pt.) A B C D E T(v v ) 0 v v 0 v v Hence T i one-to-one a wa to be proved. 5.( pt.) A B C D E Vector algebra in V QED for the theorem. Poible anwer for thi page: A)x = 0 B)x = x C)x + x = 0 D) f(x + x ) = 0 E) f(x ) = f(x ) AB) f(x ) + f(x ) =0 AC)T( v ) = 0 AD) T(α v ) = αt( v ) AE) T( v ) = 0 BC) T(v ) T(v ) BD) Hypothei (or Given) BE) only the zero vector CD) The lemma proved above
v v v CE) Vector algebra in V DE) Vector algebra in W ABC) only the vector ABD) v = 0 ABE) T ( v v ) BCD) T(v BCE) BDE) v ) 0 v 0 CDE) T i a one-to-one mapping ABCD) T i a linear operator ABCE) Definition of T ACDE) Theorem from Calculu BCDE) Definition of f ABCDE) None of the above. Total point thi page = 7. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page 5 Lat Name, Firt Name MI, What you wih to be called Follow the intruction on the Exam Cover Sheet for Fill-in-the Blank/Multiple Choice quetion. Compute the invere Laplace tranform of the following function. 3 6. (4 pt.) F() = + {F} =. A B C D E + + 4 7. (4 pt.) F() = {F} =. A B C D E + 9 + 3 8. (4 pt.) F() = {F} =. A B C D E + Poible anwer thi page A) +3 e t B) 3e t C) +3e t D) 3e t E) + 3e t AB) 3e t AC) +3e t AD) 3e t AE) co 3t + 4 in 3t BC) co 3t 4 in 3t AD) co 3t + 4 in 3t AE) co 3t 4 in 3t BC) co t +(4/3)in 3 t BC) co 3t (4/3) in 3t
AD) co 3t + (4/3) in 3t AE) co 3t (4/3) in 3t BD) e t co t + 5e t in t BE) e t co t 5e t in t CD) e t co t + 5e t in t CE) e t co t 5e t in t BD) e t co t + e t in t BE) e t co t e t in t CD) e t co t + e t in t CE) e t co t e t in t ACDE) {f} exit but none of the above i {f} BCDE) {f} doe not exit. ABCDE)None of the above. Total point thi page =. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page 6 Lat Name, Firt Name MI, What you wih to be called Anwer quetion uing the intruction on the Exam Cover Sheet. Alo, circle your anwer. Conider the IVP: ODE y + 4 y = 0 IC : y(0) = 3, y(0) = Let Y = {y(t)}(). 9. (3 pt.) A dicued in cla (attendance i mandatory), taking the Laplace tranform of the ODE and uing the initial condition we may obtain. A B C D E Be careful, if you mi thi quetion, you will alo mi the next quetion. A) Y + + 3 + 4Y = 0 B) Y + 3 + 4Y = 0 C) Y +3 + 4Y = 0 D) Y 3 4 Y = 0 E) Y +3 + 4Y = 0 AB) Y + 3 4 Y = 0 AC) Y 3 + 4Y = 0 AD) Y 3 4Y = 0 AE) Y + 4Y = 0 BC) Y 3 + 4Y = 0 ABCDE) None of the above 30. (3 pt.) The Laplace tranform of the olution to the IVP i Y =. A B C D E 3 3 3 3 3 3 A) B) C) D) E) AB) 4 4 4 4 4 4 3 3 3 AC) AD) AE) BC) ABCDE) None of the above 4 4 4 4
Total point thi page = 6. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page7 PRINT NAME ( ) ID No. Lat Name, Firt Name MI, What you wih to be called Anwer quetion uing the intruction on the Exam Cover Sheet. Alo, circle your anwer. {v,v,...,v } 3. ( 4 pt.) Let S = n V where V i a vector pace and (*) be the vector equation c. Chooe the correct completion of the following: v +cv +...+cnv n = 0 Definition. The et S i linearly independent if. A B C D E A). (*) ha an infinite number of olution. B)(*) ha only the olution c = c = = c n = 0 C) (*) ha a olution other than the trivial olution. D)(*) ha at leat two olution. E) (*) ha no olution. AB) the aociated matrix i noningular. AC) the aociated matrix i ingular. ABCDE) None of the above 3. (4 pt.)now let S ={ [x (t), y (t), z (t)] T, [x (t), y (t), z (t)] T,..., [x n (t), y n (t), z n (t)] T } A 3 ( R,R ) and (**) be the vector equation c [x (t), y (t), z (t)] T + c [x (t), y (t), z (t)] T + + c n [x n (t), y n (t), z n (t)] T = [0, 0, 0] T t R Apply the definition above to the pace of time varying "vector" A 3 ( R,R ). That i, by the definition above the et S A 3 ( R,R ) i linearly independent if. A B C D E A)(**) ha an infinite number of olution B)(**) ha only the olution c = c = = c n = 0 C) (**) ha a olution other than the trivial olution. D) (**) ha at leat two olution. E) (**) ha no olution. AB) the aociated matrix i noningular. AC) the aociated matrix i ingular. ABCDE) None of the above
Total point thi page = 8. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page8 PRINT NAME ( ) ID No. Lat Name, Firt Name MI, What you wih to be called Anwer quetion uing the intruction on the Exam Cover Sheet. Alo, circle your anwer. 33. (4 pt.) You are to determine Directly Uing the Definition (DUD) if the following et of time varying "vector" are linearly independent. Let S = {x(t),x(t)} A ( R,R ) where t 3e x (t) t 4e and t 6e x (t). Then S t 9e i. A B C D E A) linearly independent a c x(t) + c = [0,0] T tr implie c = c = 0. x(t) B) ) linearly dependent a c x + c = [0,0] T tr implie c = c = 0. (t) x ( t) C) linearly independent a x + = [0,0] T tr. (t) x ( t) D) linearly dependent a x (t) + x ( t) = [0,0] T tr. E) linearly independent a the aociated matrix i noningular. D) linearly dependent a the aociated matrix i noningular AB) linearly independent a the aociated matrix i ingular. AC) linearly dependent a the aociated matrix i ingular. AD) neither linearly independent or linearly dependent a the definition doe not apply. ABCDE) None of the above tatement are true.
Total point thi page = 4. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page 9 Lat Name, Firt Name MI, What you wih to be called Anwer quetion uing the intruction on the Exam Cover Sheet. Alo, circle your anwer. Uing the procedure illutrated in cla (attendance i mandatory), find the eigenvalue of i 4 0 A = C x. 34. (4 pt.) A polynomial p(λ) where olving p(λ) = 0 yield the eigenvalue of A can be written a p(λ) =. A B C D E A) (i+λ)(+λ) B) (i+λ)(λ) C) (iλ)(+λ) D) (iλ)(λ) E) (i+λ)(+λ) AB) (i+λ)(λ) AC) (iλ)(+λ) AD)(iλ)(λ) AE) (i+λ)(+λ) BC)(i+λ)(λ) BD)(iλ)(+λ) BE) (iλ)(λ) CD) (i+λ)(+λ) CE)(i+λ)(λ) DE)(iλ)(+λ) ABC)(iλ)(λ) ABD) (3iλ)(+λ) ABCDE)None of the above. 35. ( pt.) The degree of p(λ) i. A B C D E A) B) C) 3 D) 4 E) 5 AB) 6 AC) 7 ABCDE) None of the above. 36. ( pt.) Counting repeated root, the number of eigenvalue of A i. A B C D E A) 0 B) C) D) 3 E) 4 AB) 5 AC) 6 AD) 7 AE) 8 ABCDE) None of the above 37. (4 pt.) The eigenvalue of A can be written a. A B C D E A) λ =, λ = i B) λ =, λ = i C) λ =, λ = i D) λ =, λ = i E) λ =, λ = i AB) λ =, λ = i AC) λ =, λ = i AD) λ =, λ = i AE) λ =, λ = i BC) λ =, λ = i BD) λ =, λ = i BE) λ =, λ = i CD) λ =, λ = i CE) λ =, λ = i DE) λ =, λ = i ABC) λ =, λ = i ABCDE) None of the above
Total point thi page =. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Prof. Moeley Page 0 Lat Name, Firt Name MI, What you wih to be called Follow the intruction on the Exam Cover Sheet for Fill-in-the Blank/Multiple Choice quetion. Alo, circle your anwer. 3 Note that λ = i an eigenvalue of the matrix A = 4 38. (5 pt.) Uing the convention dicued in cla (attendance i mandatory), a bai B for the eigenpace aociated with λ i B =. A B C D E A) {[,] T, [4,4] T } B) {[,] T } C) {[,] T } D) {[,] T, [4,8] T } E) {[,] T } AB) {[,3] T } AC) {[,4] T } AD) {[4,] T } AE) {[3,] T } BC) {[,] T, [4,4] T } BD) {[,] T } BE) {[,] T } CD) {[,] T, [4,8] T } CE) {[,] T } DE) {[,3] T } ABC) {[,4] T } ABD) {[4,] T } ABE) {[3,] T } ACD) λ = i not an eigenvalue of the matrix A ACE) λ = i not an eigenvalue of the matrix A ADE) λ = 3 i not an eigenvalue of the matrix A ABCDE) None of the above i correct. 39. (pt.) Although there are an infinite number of eigenvector aociated with any eigenvalue, the eigenpace aociated with λ i often one dimenional. Hence convention for electing eigenvector() aociated with λ have been developed (by engineer). We ay that the eigenvector() aociated with λ i (are). A B C D E A) [,] T, [4,4] T B) [,] T C) {[,] T } D) [,] T, [4,8] T E) [,] T AB) [,3] T AC) [,4] T AD) [4,] T AE) [3,] T BC) [,] T, [4,4] T BD) [,] T BE) [,] T CD) [,] T, [4,8] T CE) [,] T DE) [,3] T ABC) [,4] T ABD) [4,] T ABE) [3,] T ACD) λ = i not an eigenvalue of the matrix A ACE) λ = i not an eigenvalue of the matrix A ADE) λ = 3 i not an eigenvalue of the matrix A ABCDE) None of the above i correct.
Total point thi page = 6. TOTAL POINTS EARNED THIS PAGE MATH 6 EXAM 4A- Fall 009 Profeor Moeley Page Lat Name, Firt Name MI, What you wih to be called Anwer quetion uing the intruction on the Exam Cover Sheet. Alo, circle your anwer. TABLE Let the x matrix A have the eigenvalue table Eigenvalue Eigenvector Let L:A (R,R )A (R,R ) be defined by L[x] x Ax r = and let the null pace of L be N L r = 40. ( pt). The dimenion of N L i. A B C D E A) 0 B) C) D) 3 E) 4 AB) 5 AC) 6 ABCDE) None of the above. 4. (3 pt.) A bai for the null pace of L i. A B C D E t t t t t t A) B e, e B) B e, e C) B e, e D) t t t t t t E) B e, e AB) B e, e AC) B e, e AD) t t B e, e t t t t t t AE) B e, e BC) B e, e BD) B e, e BE) ABCDE) None of the above t t B e, e t t B e, e 4. ( pt.) The general olution of x Ax i. A B C D E t t t t t t x(t) c e c e t t A) x(t) c e c e B) x(t) c e c e C) x(t) c e c e D) t t E) x(t) c e c e AB) x(t) c e c e AC) x( t) c e c e AD) t t t t t t x(t) c e c e t x(t) c e c e t t t t x(t) c e c e t t AE) x(t) c e c e BC) x(t) c e c e BD) t BE) ABCDE)None of the above
Total point thi page = 7. TOTAL POINTS EARNED THIS PAGE