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School: WVU
School: WVU
MATH 373 Additional Exercises Chapter 1: Basics of Number Theory (A1.1) (i) Let p be a prime and let a and b are integers such that ab 0 (mod p). Show that either a 0 (mod p) or b 0 (mod p). (ii) Find an example to show that there exist integer
School: WVU
Homework 3, Solutions (2.3)(a) Let g be a primitive root for Zp . (a) Suppose that x = a and x = b are both integer solutions to the congruence g x h (mod p). Prove that a b (mod p 1). Explain why this implies that the map logg : Z Z/(p 1)Z = Zp
School: WVU
Homework 1, Solutions (1.6) Let a, b, c Z. Use the denition of divisibility to directly prove the following. (a) If a|b and b|c, then a|c. (b) If a|b and b|a, then a = b. (c) If a|b and a|c, then a|(b c). Proof: (a) Claim 1 2 3 4 5 Statement a|b &
School: WVU
Homework 9, Solutions (5.5) Find all the points on the elliptic curve. (b) E : y 2 = x3 + 2x + 7 over Z11 . Solution: (b) We first generate the square table (computing y 2 ). > for y = [0:10], [y; powermod(y, 2, 11)]', end ans = 0 0 ans = 1 1 ans = 2
School: WVU
Course: Non-engineering Calculus
, - k 2 L ; + ; 1 - , : L ) x v , l. eo x - le JT 1 t rc ) 1 w : . L = , & : 1 U x z x - = Je - 2 . (2 po in t s . Si . . " Nl " : - - Y ;: q (2 p ,I " w )A - - 1" " n l o f a r ea * . " 4 v r . F 4 A 4 J . - b - , 3 00 L : : sq . Are fe, t is t fee t 1.
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 4 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Transform the following system of equations into a sin
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 4 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Transform the following system of equations into a sin
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 3 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Find the general solution for the following dierential
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Find the general solution for each of the following di
School: WVU
Course: Elementary Differential Equations
MATH 261.005 Instr. K. Ciesielski Fall 2009 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Find the general solution for each of the following dier
School: WVU
School: WVU
Course: Non-engineering Calculus
1 . (3 Po i n t s e a c h ) Fm d t r e d e r i v a t i, e o f a B i. E n f u n ct i o n i 31 t 3 (x ) g ( h (x j (x ) ) = ) = = s n (W = _ + co . 6x z ) x ) ( 2x _ 4 3 , - D o N O T S IM PLIFY 2 .J J e - (5 x (4 . . ) U ) 7 . ) N. . e M AT H 15 5 a. " 5 (
School: WVU
Course: Non-engineering Calculus
: M . . " " ) St m 2 + b " " " : " " " b . ( " " , 11" / ? h t i. e S1 " . " . N n . n o " d , li n i t i o . . M t l," !" I . " " . L " " . " k . t; \n 9 D 4 . ( I. h ) rr y . - f. tr. . thr" j. rrf is Ti r y . H y " I " " . is Fi . - by , od d , d f
School: WVU
Course: Non-engineering Calculus
Qu i z 3 A lgeb r aic L im its _ \ \ = J - ; L A J _ , ( : I ; J > j \
School: WVU
Course: Non-engineering Calculus
M AT H 1 5 S Q u Da 1 [u t e : - Na m 1 Sk e t c h . a . b c d . . . iz 4 2 Iim O N E F U N CT IO N w it h t h e f o llk w in g p r o p er t ies x ) = 4 . - 0 Ii n i . Ji m . Iim - 0 - - . 4 4 f( f [ J: J " j (x ) (x ) - 2 = _ - Co C o . @ MAT H 155 " 4
School: WVU
Course: Non-engineering Calculus
e 1 1 e 0 e ;A 0 e t 7 . (2 ) - )L ? : s " Ci " " ) 01 3 0 ) . - O
School: WVU
Course: Non-engineering Calculus
M at h 4 Sec t io n : 5 5 H o m ew o r k D u e : F in D o d W 7 ( 2 5 p o e d n e sd a y , 2 n t s) q , Ft h e th e d e r iv a t iv e s n o t s im p li fy 2 0 1 3 y ou fo llo w i n g a n sw er s fu n c t io n s . 3 L . (4 w l. ) g (= ) = 3 s i n a - 3= 4
School: WVU
Course: Non-engineering Calculus
Do M . Ilo Lh n : . 1s 5 " 0 . k 1 (2 0 p . I. t . ) 15 L 0 (= ) : B = (b ) (5 ) R w > 0( ri 1 . +; B /a h 1a t e t l , e f . ) : q . S; ll. " I. g li. .It. . 2 . ( ) Sk , ci o n i " t 1l , t l . " ,1 f w I. ( : o f " l 0 0 m ) " " f " io " " t h . u o
School: WVU
Course: Non-engineering Calculus
* X k D . Eb , T , (25 p o in t s Mm ) au of t he p r o b t , m s . I 1o - Enal uat e t h e 1 1 Se c 2l 5 y o u r W o , k t ion : g N am o : M1 w . lim i t s 0 i w r \t th e \ hL t v ) or ol n p K o\ w or k n ot t cfw_ e Q r n " - \ , o a 6 . L _ I n 1
School: WVU
Course: Non-engineering Calculus
Ma t h k 155 il . 5 2 : . ) 0 " : . :i : ! ( " u ) Na m e: Math 1 5 S Ho m e w o r k Du e Nam e 5 T , ! . (4 ) , " " t " " " , o r , I" 2: " : g" : I. U1l y " . F1, " 1l . $+al . f " " " " " ( . , ] 9 - 3 3 . (4 ) Let f ( ) + 1. - u . T 1. l m " " 4 i.
School: WVU
Course: Non-engineering Calculus
* : X X :. D . e Ew . T l. . . (20 p J. Rsd o i . Ts AO ) 1. A r y l . At e t h e f ol i o. . R l i m i t s g I T . , S , im : : 2 0 1 3 Sho ). " " ,k . 1 6 Nm n . / qw n i n A: B - 0 H t . ? - t jx 4 < X cA &
School: WVU
Course: Numerical Analysis
Math 420 Spring, 2013 Syllabus Room 125 Brooks MWF 11:30-12:20 Course web page: http:/www.math.wvu.edu/~diamond/Math420S13 Instructor: Professor Diamond Office: 410J Armstrong Phone: 304-293-9082 email: diamond@math.wvu.edu Office hours: MWF 2:30-3:20 . Y
School: WVU
MATH 156, FALL 2012 Instructor: Office: Email: Phone: Office Hours: Text: Dr Hong-Jian Lai 320A Armstrong Hall hjlai@math.wvu.edu 293-2011 x2331 Tuesdays, and Thursdays 9:40 11:30am Essential Calculus: Early Transcendentals, by James Stewart Exams: There
School: WVU
School: WVU
MATH 373 Additional Exercises Chapter 1: Basics of Number Theory (A1.1) (i) Let p be a prime and let a and b are integers such that ab 0 (mod p). Show that either a 0 (mod p) or b 0 (mod p). (ii) Find an example to show that there exist integer
School: WVU
Homework 3, Solutions (2.3)(a) Let g be a primitive root for Zp . (a) Suppose that x = a and x = b are both integer solutions to the congruence g x h (mod p). Prove that a b (mod p 1). Explain why this implies that the map logg : Z Z/(p 1)Z = Zp
School: WVU
Homework 1, Solutions (1.6) Let a, b, c Z. Use the denition of divisibility to directly prove the following. (a) If a|b and b|c, then a|c. (b) If a|b and b|a, then a = b. (c) If a|b and a|c, then a|(b c). Proof: (a) Claim 1 2 3 4 5 Statement a|b &
School: WVU
Homework 9, Solutions (5.5) Find all the points on the elliptic curve. (b) E : y 2 = x3 + 2x + 7 over Z11 . Solution: (b) We first generate the square table (computing y 2 ). > for y = [0:10], [y; powermod(y, 2, 11)]', end ans = 0 0 ans = 1 1 ans = 2
School: WVU
Homework 10, Solutions (5.16) The MV-Elgamal system: System and its parameters: A large prime p, an elliptic curve E = E(Zp ), and a point P E. All are in public domain. Enciphering Key: Alice wants Bob to send her a message. She chooses a secret in
School: WVU
Homework Assignment 4 Homework 4. Due day: 10/25/06 (1.3) A ring R such that a R, a2 = a is called a Boolean ring. Prove that every Boolean ring is commutative and a + a = 0, a R. (1.6) A finite ring with more than one element and no zero divisors
School: WVU
Solution of Some Homework Problems Section 2.2 Permutations (2.1) Find sgn() and -1 , where = 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 . Solution: Since = (19)(28)(37)(46)(5), sgn()=(-1)9-5 =1 and that -1 = (5)(46)(37)(28)(19). (2.2) If Sn , prov
School: WVU
Homework 4, Solutions (2.6) Alice and Bob agree to use the prime p = 1373 and the base g = 2for a Diffie-Hellman key exchange. Alice sends Bob the base A = 974 . Bob asks your assistance, so you tell him to use the secret exponent b = 871. what value
School: WVU
Homework 7, Solutions (3.1) Solve the following congruences (a) x19 36 (mod 97) (c) x73 614 (mod 1159). Solution: (a) Here n = 97. Compute (n) = 96. Set e = 10. > n = 97 n = 97 > phi=eulerphi(n) phi = 96 > e = 19 e = 19 We rst nd d such that de 1
School: WVU
Math 156 10-18-2012 Worksheet Name: 1: Express the number 0.73 = 0.73737373. as a ratio of integers. Solution. We express the decimal as a geometry series to find the answer. 0.73 = 0.73737373. = 0.73 + 0.0073 + 0.000073 + . 1 1 1 1 = 73 + + . = 73 + 2 3
School: WVU
Math 156 10-22-2012 Worksheet (both sides) Name: Instruction: Use either the integral test of one of the comparison tests to determine whether each of the following series is convergent. Since this worksheet will train us how to answer these problems corr
School: WVU
Course: Non-engineering Calculus
1 . (3 Po i n t s e a c h ) Fm d t r e d e r i v a t i, e o f a B i. E n f u n ct i o n i 31 t 3 (x ) g ( h (x j (x ) ) = ) = = s n (W = _ + co . 6x z ) x ) ( 2x _ 4 3 , - D o N O T S IM PLIFY 2 .J J e - (5 x (4 . . ) U ) 7 . ) N. . e M AT H 15 5 a. " 5 (
School: WVU
Course: Non-engineering Calculus
: M . . " " ) St m 2 + b " " " : " " " b . ( " " , 11" / ? h t i. e S1 " . " . N n . n o " d , li n i t i o . . M t l," !" I . " " . L " " . " k . t; \n 9 D 4 . ( I. h ) rr y . - f. tr. . thr" j. rrf is Ti r y . H y " I " " . is Fi . - by , od d , d f
School: WVU
Course: Non-engineering Calculus
Qu i z 3 A lgeb r aic L im its _ \ \ = J - ; L A J _ , ( : I ; J > j \
School: WVU
Course: Calculus 3
Math 251 Quiz 8 v 2 , with u s cos st , v s sin st y y use the chain rule to find x , x , , . Leave your answer in terms of u, v, s, t. s t s t 1. If x x s x t y s y t uv , y x u x u y u y u u s u t u s u t 2. Let f x, y, z u2 x v x v y v y v v s v t v s
School: WVU
Course: Calculus 3
Math 251 Exam 4 1 (10) Given the iterated integral 2 2x 1 x0 y1 x y dydx a) Carry out the integration and find the value of the iterated integral. 2 2x 1 x0 y1 2 0 x 2 y dydx 4x 2 0 y2 2 xy 2x 1 2 dx 0 1 x 2x 1 2x 1 2 x 2 1 2 dx 44 3 2x dx b) Associate th
School: WVU
Course: Non-engineering Calculus
M AT H 1 5 S Q u Da 1 [u t e : - Na m 1 Sk e t c h . a . b c d . . . iz 4 2 Iim O N E F U N CT IO N w it h t h e f o llk w in g p r o p er t ies x ) = 4 . - 0 Ii n i . Ji m . Iim - 0 - - . 4 4 f( f [ J: J " j (x ) (x ) - 2 = _ - Co C o . @ MAT H 155 " 4
School: WVU
Course: Numerical Analysis
Math 420 Hw5 1. Develop an approximation formula f 0 w 1 f 1 w 2 f 2 w 3 f 4 in several ways as directed below: a) Interpolate f at x 1, 2, 4 with a polynomial p 2 x in Lagrange form, and calculate f 0 Our function p 2 x that interpolates the values f x i
School: WVU
Course: Numerical Analysis
Math 420 HW1 h f x0 with h Display the data in column form with columns that are 1. Use MATLAB to approximate f 1 for f x x0 1. and values h f x0 1 2k 40 e x , using f x 0 f x0 k1 h f x0 f x0 h E h where E is the computed error (exact value minus 2 h appr
School: WVU
Course: Numerical Analysis
Math 420 HW1 h f x0 with h Display the data in column form with columns that are 1. Use MATLAB to approximate f 1 for f x x0 1. and values h f x0 1 2k 40 e x , using f x 0 f x0 k1 h f x0 f x0 h E h where E is the computed error (exact value minus 2 h appr
School: WVU
Course: Numerical Analysis
Math 420 HW1 x0 1 2k 1. and values h f x0 40 f x0 h f x0 with h Display the data in column form with columns that are 1. Use MATLAB to approximate f 1 for f x e x , using f x 0 k1 f x0 h f x0 h E h where E is the computed error (exact value minus 2 h appr
School: WVU
Course: Numerical Analysis
Math 420 HW2 1. Here is a computational method of sorts for calculating log x accurately using Taylor series. 1 1 1 2 1 1 3 . . . using the a) Show that log 2 log 1/2 log 1 1/2 2 22 32 Talyor series for log 1 x . If we want the error to be less than 10 16
School: WVU
Course: Numerical Analysis
Math 420 Hw2 1 11 ln 1 ln 1 1 2 2 2 22 1 1 k the error is If we stop at the term k2 k1 1 1 1 1 k 2 . 12 n n k12 k22 1. a) ln 2 nk1 2 1 3 1 2 1 k 1 3 2 . 1 n nk1 k 2 11 k1 1/2 1 k 1 2 k so that the error is roughly the size of (and no bigger than) the last
School: WVU
Course: Numerical Analysis
Math 420 HW3 1. If x represents a vector with n components then we have for constants C, c independent of x, cx x 2 C x . Find, in terms of n, the best possible constants C, c : the smallest possible such C and largest possible such c. Explain. 1/2 1/2 (E
School: WVU
Course: Numerical Analysis
Math 420 HW3 1. If x represents a vector with n components then we have for constants C, c independent of x, cx x 2 C x . Find, in terms of n, the best possible constants C, c : the smallest possible such C and largest possible such c. Explain. 1/2 1/2 (E
School: WVU
Course: Numerical Analysis
Math 420 Hw5 1. Develop an approximation formula f 0 directed below: a) Interpolate f at x w1f 1 w2f 2 w 3 f 4 in several ways as 1, 2, 4 with a polynomial p 2 x in Lagrange form, and calculate f 0 1 w 1 f x 0 h w 2 f x 0 2h w 3 f x 0 4h b) Expand the sca
School: WVU
Course: Numerical Analysis
Homework Due Monday March 18 1. Calculate by hand the divided difference table corresponding to the data x201 -3 and then write the polynomial in Newton form that interpolates the data. y 5 1 -1 3 (Use fractions, not decimals) 2. Write out by hand the lin
School: WVU
Course: Numerical Analysis
Math 420 HW6 1. Derive quadrature formulas i) ii) h 0 h 0 f x dx h w1f h w2f 0 f x dx h w1f 0 w2f h w3f h w 3 f 2h 1 a) Use polynomial interpolation as applied to the unscaled formula for f x dx and scale the 0 result 1 b) Use exactness for f x 1, x, x 2
School: WVU
Course: Numerical Analysis
Homework Due Monday March 18 1. Calculate by hand the divided difference table corresponding to the data x201 -3 and then write the polynomial in Newton form that interpolates the data. y 5 1 -1 3 (Use fractions, not decimals) x f[x(i)] 2 5 2 0 1 4 -2 1 -
School: WVU
Course: Non-engineering Calculus
, - k 2 L ; + ; 1 - , : L ) x v , l. eo x - le JT 1 t rc ) 1 w : . L = , & : 1 U x z x - = Je - 2 . (2 po in t s . Si . . " Nl " : - - Y ;: q (2 p ,I " w )A - - 1" " n l o f a r ea * . " 4 v r . F 4 A 4 J . - b - , 3 00 L : : sq . Are fe, t is t fee t 1.
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 4 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Transform the following system of equations into a sin
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 4 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Transform the following system of equations into a sin
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 3 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Find the general solution for the following dierential
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Find the general solution for each of the following di
School: WVU
Course: Elementary Differential Equations
MATH 261.005 Instr. K. Ciesielski Fall 2009 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Find the general solution for each of the following dier
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 Solutions (without Ex. 2) to the SAMPLE TEST # 1 Ex. 1(a) y = ex ex , 3+4y y (0) = 1. Labeling This is separable equation, since dy dx = ex ex 3+4y is equivalent to (3 + 4y ) dy = (ex ex ) dx. Solution 3y + 2y
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 SAMPLE TEST # 1 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Each of the following dierential equations is of one of the followin
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE FINAL TEST This is an excerpt from the previous Sample Tests. The actual Final Test will be considerably shorter. Test #1 material Ex. 1. Each of the following dierential equations is of o
School: WVU
Course: Elementary Differential Equations
MATH 261.005 Instr. K. Ciesielski Fall 2011 Partial Dierential Equations, PDE, handout Crucial Boundary Value problem: Find all non-zero solutions of the BVP, where > 0: x (t) + x(t) = 0, x(0) = 0 and x(L) = 0. (1) Solution: Characteristic polynomial r2 +
School: WVU
Course: Elementary Differential Equations
MATH 261.007 Instr. K. Ciesielski Spring 2010 Bernouli Equations handout Format: y + p(t)y = g (t)y n . Solution: (a) For n = 0 and n = 1 this is linear equation. For other n use substitution v = y 1n . 1 (b) Since y = v 1n , taking the derivative we get
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Spring 2014 NAME (print): SAMPLE TEST # 1 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Evaluate 0 8 23 3 4 5 5 4 2 = 11 1 5 1 (a) 121 1 1 1 6 3
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Spring 2014 SAMPLE TEST # 3 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Show that the following limit does not exist x3 y (x,y )(0,0) x4 + 7y 4
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Spring 2014 NAME (print): SAMPLE TEST # 2 with SOLUTIONS Solve the following exercises. Show your work. Ex. 1. Find a vector equation of the line that passes through the point P (11, 13, 7) and is perpendicular to the plane w
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Spring 2014 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. Ex. 1. Find a vector equation of the line that passes through the point P (11, 13, 7) and is perpendicular to the plane with the equatio
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Spring 2014 NAME (print): Solutions for SAMPLE TEST # 1 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Evaluate (a) 0 8 23 3 4 5 5 4 2 = 11 1 5 1 6
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Fall 2013 SAMPLE TEST # 3 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1. Show that the following limit does not exist x3 y (x,y )(0,0) x4 + 7y 4 li
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Fall 2013 SAMPLE TEST # 4 Solve the following exercises. Show your work. Ex. 1. Set up the integral formulas, including the limits of the integrations, for the following problems. Do not evaluate the integrals! Where appropri
School: WVU
Course: Multivariable Calculus
Topology, Math 581, Fall 2013 last updated: December 4, 2013 1 Topology 1, Math 581, Fall 2013: Notes and homework Krzysztof Chris Ciesielski Class of August 20: Course and syllabus overview. Topology is an abstract geometry, sometimes referred to as Rubb
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Fall 2013 SAMPLE FINAL TEST (longer than the actual Final Test) Solve the following exercises. Show your work. Ex. 1. ST #1 Ex 3: Find the determinant of the matrix. Each time you expand the the matrix, you must expand it ove
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Fall 2013 SAMPLE FINAL TEST (longer than the actual Final Test) Solve the following exercises. Show your work. Ex. 1. ST #1 Ex 3: Find the determinant of the matrix. Each time you expand the the matrix, you must expand it ove
School: WVU
Course: Multivariable Calculus
MATH 251 Instr. K. Ciesielski Fall 2013 SAMPLE TEST # 4 Solve the following exercises. Show your work. Ex. 1. Set up the integral formulas, including the limits of the integrations, for the following problems. Do not evaluate the integrals! Where appropri
School: WVU
Course: College Algebra
Math126 ExponentialLaw o A=Asub0e^kt A Newamount Asub0 Originalamount Growth/decay K o K>0 Growth o K<0 T decay Time NewtonsLawofCooling o U(t)=Tplus(Usub0T)e^kt LogisticsModel o P(t)=c/1+ae^bt RowOperations o Interchange2rows o Multiplyarowbyanonzeroscal
School: WVU
Course: College Algebra
Math126 Algorithm o Arecipeoradescriptionofamechanicalsetofstepsforperformingsome task Binaryoperation o Abinaryoperationisanoperationthatcombinestwoobjectsofonetype toformanotherobjectofthesametype o Addition,subtraction,multiplication,division,andexpo
School: WVU
Course: College Algebra
Math126 Domain o Makethedenominatorequalto0 Verticalasymptote o Setthedenominatornotequalto0 Horizontalasymptote o Usetheleadingcoefficients Obliqueasymptote o Therecannotbeanobliqueasymptoteifthereisahorizontalasymptote Howdoyouknowifyouhaveholes o Anyth
School: WVU
Course: College Algebra
Math126 Orderofoperations o Parenthesis o Exponents o Multiplication o Division o Addition o Subtraction Properties o Commutative addition a+b=b+a multiplication ab=ba o Associative Addition a+(b+c)=(a+b)+c Multiplication A(bc)=(ab)c o Identity Addition A
School: WVU
Course: College Algebra
Math126 DivisionAlgorithm o Iff(x),g(x)arepolynomialswherethedegreeofgisgreaterthan0.There existsq(x)andr(x),whichareuniquepolynomialsstatethatf(x)/g(x)=q(x) +r(x)/g(x)orf(x)=q(x)g(x)+r(x)withr(x)notequalingzeroorthedegree ofrislessthanthedegreeofg Q(x
School: WVU
Course: College Algebra
Math126 AbsoluteValue o Howfarsomethingisfromzeroonthenumberline Makeanumbernotnegative *zeroisnotapositivenumber Lawofexponents o X^2=xtimesx o X^3=xtimesxtimesx o X^4=xtimesxtimesxtimesx o X^n=xtothentimes X=base N=exponent Squareroots o Squarerootofa=b
School: WVU
Course: College Algebra
Orderofoperations o Parenthesis o Exponents o Multiplication o Division o Addition o Subtraction Properties o Commutative addition a+b=b+a multiplication ab=ba o Associative Addition a+(b+c)=(a+b)+c Multiplication A(bc)=(ab)c o Identity Addition A+0=0+a=
School: WVU
Course: College Algebra
NewtonsLawofCooling o U(t)=Tplus(Usub0T)e^kt LogisticsModel o P(t)=c/1+ae^bt RowOperations o Interchange2rows o Multiplyarowbyanonzeroscalarandreplacethatrowwithresult o Replacerowwithsumofthatrowplusanotherrow o Therearecaseswheretheonesskipacolumn Divi
School: WVU
Course: College Algebra
Factors o Thefactorsofawholenumberarethenumberthatdivideevenlyintoit Solutionset o Thesolutionsettoanequationisthesetofallvaluesforthevariablesin theequationthatmaketheequationtrue Asymptote o Alinethatacurveapproachesasyoumovefurtherawayfromzero Tangen
School: WVU
Course: College Algebra
Orderofoperations o Parenthesis o Exponents o Multiplication o Division o Addition o Subtraction Properties o Commutative addition a+b=b+a multiplication ab=ba o Associative Addition a+(b+c)=(a+b)+c Multiplication A(bc)=(ab)c o Identity Addition A+0=0+a=
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Untitled.notebook November11,2013 1 Untitled.notebook November11,2013 2
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Untitled.notebook October21,2013 1 Untitled.notebook October21,2013 2
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Untitled.notebook October07,2013 1 Untitled.notebook October07,2013 2 Untitled.notebook October07,2013 3
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Untitled.notebook September30,2013 1 Untitled.notebook September30,2013 2
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Untitled.notebook October04,2013 1 Untitled.notebook October04,2013 2 Untitled.notebook October04,2013 3 Untitled.notebook October04,2013 4
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Simulation Simulation deals with systems which evolve in a probabilistic manner. In these systems, probabilities are specified for certain outcomes (or each of a sequence of outcomes) and it is desired to calculate mathematically, or estimate experimental
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Visualizing data in two and three dimensions/Graphing functions of 2 variables/Curves and Surfaces in 3-dimensions We discuss the following MATLAB commands: plot3(x,y,z) pcolor(A) or pcolor(X,Y,Z) contour(Z) or contour(X,Y,Z) surf(Z) or surf(X,Y,Z) surfc(
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Cellular automata We discuss cellular automata as a simple application of MATLAB programming and as an accessible scientific topic of recent interest. You can find a lot of information on the internet. In particular, you can visit the website of Wolfram's
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
A little bit about colors in MATLAB: RGB specification of color: There is a standard system for specifying color, based on the fact that every color can be considered a combination of red, green, and blue. In this system each color is specified by the ord
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Recursive functions: Functions that are defined in terms of themselves are called recursive functions, or recursively defined functions. This is not as strange as it sounds - a number of well-known functions can be defined in recursive form, and we are al
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Notes on iteration and Newton's method Iteration Iteration means doing something over and over. In our context, an iteration is a sequence of numbers, vectors, functions, etc. generated by an iteration rule of the type x n1 f x n where f is some fixed fun
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
(Note: When we talk about a vector below, we are generally thinking of a column vector, and in MATLAB, as a column array) If x represents a vector (in MATLAB it would just be a one dimensional array) then the sum of the squares of the components of x is x
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
General formulation of interpolation: We are given a set of datapoints x i , y i , i 1, . . , n and want to find a function x that interpolates the data, i.e. we want x i y i , i 1, . . , n. We can an interpolant. We generally look for an interpolant from
School: WVU
Course: Calculus 3
Notes on Linear Algebra Vector spaces: 1) The vector space R n is the set of ordered n-tuples of real numbers, together with the natural rules for addition and scalar multiplication (scalar multiplication is when we multiply a vector by a real number). Th
School: WVU
Course: Calculus 3
Review topics for Exam 4 Multiple integrals: fdA , fdV What is a multiple integral D D In 2 dimensions setting up a double integral as an iterated integral - different types of regions, breaking up a region into pieces if necessary, choosing an integrati
School: WVU
Course: Calculus 3
Review topics for final: * Basic matrix algebra Solutions of linear systems: Gaussian elimination: determine whether unique solution, no solution, infinitely many solutions Gaussian elimination to reduced row echelon form Writing the general solution of a
School: WVU
Course: Calculus 3
Math 251 Topics for Exam 1 Basic vector operations of addition and scalar multiplication linear combination - what is it Geometric vectors and vector operations Matrix operations: addition, scalar multiplication, matrix multiplication Viewpoints for matri
School: WVU
Course: Calculus 3
Review topics for exam 2 3-dimensional space Plotting points, vectors Vectors: geometric vectors, sums, differences, scalar multiples, position vector of a point, vector from P 1 to P 2 Dot product, interpretation in terms of angle, finding angle using do
School: WVU
Course: Calculus 3
Review topics for exam 3 1. Level curves and surfaces E.g. f x, y x 2 y 2 (just gotta know what x 2 y2 k looks like) log y f x, y xy, (could solve for y, xy k , y k/x ) , f x, y x f x, y, z x 2 y 2 z 2 (families of quadric surfaces) xy xy f x, y, z (z k ,
School: WVU
Course: Applied Linear Algebra
The four subspaces as orthogonal complements Calculating basis for an orthogonal complement - How? Calculating orthonormal basis for an orthogonal complement - How? Ax b has a solution, b is in the column space of A, if and only if Hb 0 , b is in the null
School: WVU
Course: Applied Linear Algebra
Math 441 Exam 4 Eigenvalues/eigenvectors: 1) I give you a matrix and you find the eigenvalues and eigenvectors 2) I tell you an eigenvalue and ask you to find the corresponding eigenvectors (basis of null space of A I ) Usually there is only one eigenvect
School: WVU
Course: Applied Linear Algebra
MATH 441.001 Instr. K. Ciesielski Spring 2011 NAME (print): FINAL TEST Review Final Test will start with: Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Remember, That Final Test is co
School: WVU
Course: Applied Linear Algebra
Math 441 Review topics for first exam Vectors, linear combinations of vectors Ax as a linear combination of the columns of A, as the dot product of x with the rows of A Solve Ax b Express b as a linear combination of vectors (the columns of A ) Matrix mul
School: WVU
Course: Applied Linear Algebra
General ideas: Linear independence, span, basis dimension Linear independence: c 1 v 1 . . . c k v k 0 only when c 1 0 , . . . , c k 0 Equivalent statements (if v s are column vectors): Ax 0 only if x 0 (A contains v s as columns) rankA k Span: (two usage
School: WVU
Course: Numerical And Symbolic Methods
Final Exam: MATLAB Arrays, operations (arithmetic, relational) programming structures: for/while loops, if.elseif.else.end standard mathematical functions in MATLAB, scientific notation creating MATLAB functions, for example functions that sum a series, f
School: WVU
Course: Numerical And Symbolic Methods
Review topics for exam 3 Interpolation Polynomial interpolation, interpolation with other specified functions (e.g. even or odd powers) Interpolating from data: Plotting the interpolant and the data Interpolating from functions: Plot the interpolant, the
School: WVU
Course: Numerical And Symbolic Methods
Topics for exam 2 Color in MATLAB The image and imagesc commands. Reading values off an image with colorbar. Colors in the rgb system. Colormaps. How to set up your own color map, the white/black colormap. Cellular automata - the basic idea: a series of c
School: WVU
Course: Numerical And Symbolic Methods
Topics for exam 1 The test will be mostly using MATLAB to carry out specified tasks, but some explanations are occasionally requested (why did that happen, what does this mean, why does this (not) work, what would happen if, etc) You should be familiar wi
School: WVU
School: WVU
Course: Non-engineering Calculus
1 . (3 Po i n t s e a c h ) Fm d t r e d e r i v a t i, e o f a B i. E n f u n ct i o n i 31 t 3 (x ) g ( h (x j (x ) ) = ) = = s n (W = _ + co . 6x z ) x ) ( 2x _ 4 3 , - D o N O T S IM PLIFY 2 .J J e - (5 x (4 . . ) U ) 7 . ) N. . e M AT H 15 5 a. " 5 (
School: WVU
Course: Non-engineering Calculus
: M . . " " ) St m 2 + b " " " : " " " b . ( " " , 11" / ? h t i. e S1 " . " . N n . n o " d , li n i t i o . . M t l," !" I . " " . L " " . " k . t; \n 9 D 4 . ( I. h ) rr y . - f. tr. . thr" j. rrf is Ti r y . H y " I " " . is Fi . - by , od d , d f
School: WVU
Course: Non-engineering Calculus
Qu i z 3 A lgeb r aic L im its _ \ \ = J - ; L A J _ , ( : I ; J > j \
School: WVU
Course: Non-engineering Calculus
M AT H 1 5 S Q u Da 1 [u t e : - Na m 1 Sk e t c h . a . b c d . . . iz 4 2 Iim O N E F U N CT IO N w it h t h e f o llk w in g p r o p er t ies x ) = 4 . - 0 Ii n i . Ji m . Iim - 0 - - . 4 4 f( f [ J: J " j (x ) (x ) - 2 = _ - Co C o . @ MAT H 155 " 4
School: WVU
Course: Calculus 3
Math 251 Exam 4 1 (10) Given the iterated integral 2 2x 1 x0 y1 x y dydx a) Carry out the integration and find the value of the iterated integral. 2 2x 1 x0 y1 2 0 x 2 y dydx 4x 2 0 y2 2 xy 2x 1 2 dx 0 1 x 2x 1 2x 1 2 x 2 1 2 dx 44 3 2x dx b) Associate th
School: WVU
Course: Calculus 3
Math 251 Quiz 8 v 2 , with u s cos st , v s sin st y y use the chain rule to find x , x , , . Leave your answer in terms of u, v, s, t. s t s t 1. If x x s x t y s y t uv , y x u x u y u y u u s u t u s u t 2. Let f x, y, z u2 x v x v y v y v v s v t v s
School: WVU
Course: Applied Linear Algebra
Math 441 Exam 3 1. (8 pts) Subspaces of R n are generally described either as the column space of some given matrix, or the null space of some given matrix. a) If we define a subspace V as the nullspace of the matrix A 1 1 2 1 1 11 2 express V instead as
School: WVU
Course: Applied Linear Algebra
Math 441 Exam 4 5 6 3 1. (12 pts) Given A 4 a) Find the eigenvalues and eigenvectors of A 5 6 3 det 4 2 2 0 , 1, 2 1: A 6 6 3 3 3 6 3 I ,v 6 1 ,v 2 1 2: A I 1 b) Write A in diagonalized form, A SDS 1 where D is a diagonal matrix. Then find a formula
School: WVU
Course: Applied Linear Algebra
Math 441 Exam 2 1. (2.5 pts each, total 15) Answer the following simple questions, and provide a brief explanation. a) If the vectors in the set S v 1 , . . , v k are linearly independent, what is a basis for their span, span S ? What is the dimension of
School: WVU
Course: Applied Linear Algebra
Math 441 Exam 1 x1 x2 1. (8 pts) a) Given a vector x x2 x3 find A so that Ax 0 1 1 0 0 0 x2 with A the indicated n 1 n matrix. xn xn 1 x1 x2 1 as a linear combination xn x3 xn x1 x3 b) Express the vector x2 11 x2 xn x2 0 0 . (Describe the x1 0 0 We have
School: WVU
Course: Business Calculus
Business Calc Local Extrema At the maxima and minima, we do not know the x or y values, but we do know that the derivative is zero (since the slope is zero) Projectile o In the function with a projectile, the maximum is the only point where the derivative
School: WVU
Course: Mathematical Methods
Math 261 Exam 1 1. ( 24 / 8 pts each) Solve these equations as indicated: y y e 2t e 2t , y 0 1 Integrating factor e t e t y e t e 3t d t e t 1 e 3t C , 3 1 C, C 1 y0 11 3 3 y e 2t 1 e 2t 1 e t 3 3 * y y t , y1 2 t e ty et e 3t , y e 2t 1e 3 2t Ce t 1/
School: WVU
Course: Mathematical Methods
Math 261 Exam 3 1. (8 points each) Find the general solution: y 4 y e 2t r 4 1 0 , r 2 1 r 2 1 0 , r 1 r 1 r 2 1 0 , r 1, 1, i y h c 1 e t c 2 e t c 3 cos t c 4 sin t Y Ae 2t , LY Y 4 Y 16Ae 2t Ae 2t 15Ae 2t e 2t so 15A 1 , A 1 , Y 1 e 2t 15 15 y c 1 e t
School: WVU
Course: Mathematical Methods
Math 261 Exam 4 1. (8 pts) Calculate the Fourier sine series, period 4, of the function 1 for 0 x 1 fx 0 for 1 x 2 a) Write the series using summation notation (you do not have to simplify the formula for the coefficients) The series has the form n1 b n
School: WVU
Course: Mathematical Methods
Math 261 Final Exam May, 2009 1. Solve for the general solution or solve the initial value problem as appropriate to the problem. Express y explicitly as a function of t in each case. a) y cos t y 0 b) y ty 2 , y 0 2 c) y y et , y 0 1 1 2. Find the genera
School: WVU
Course: Mathematical Methods
Math 261 Quiz 1 1. (4 pts) For the following differential equation, determine the equilibria (constant solutions) and sketch the general behavior of the solution curves: y yy 1 The equilibria are at y 0, 1. Now y 0 for y 1 , y 0 for 0 y 1 and y 0 for y 0.
School: WVU
Course: Mathematical Methods
Math 261 Quiz 2 1. (3 pts) Solve for y in terms of x : existence of the solution? dy x 1 , y 1 1. What is the interval of y dx ydy x 1 dx , 1 y 2 1 x 2 x C or y 2 x 2 2x C (different C ) 2 2 Initial condition: Plug in x, y 1, 1 : 1 2 12C, C 2 so y 2 x 2
School: WVU
Course: Mathematical Methods
Math 261 Quiz 3 1. (3 pts) Determine the largest open interval (in the independent variable) for which the solution of the given initial value problem is guaranteed to exist: ty 2y cos t , y 1 2 2 y cos t We observe a discontinuity in p, g at Put in stand
School: WVU
Course: Mathematical Methods
Math 261 Quiz 4 1. (8.5 pts) Find the general solution, or the solution of the given initial value problem, as appropriate: y y 6y 0 , y 0 1 , y 0 2 Characteristic equation: r 2 r 6 0 r 3 r 2 0 , r 3, 2 y c 1 e 3t c 2 e 2t Initial conditions: y 0 1 c1 c2
School: WVU
Course: Mathematical Methods
Math 261 Quiz 5 1. Solve the initial value problem: y 2y 5y e t , y 0 1 , y 0 1 First the homogeneous solution: r 2 2r 5 0 , r 1 2i , y 1 e t cos 2t , y 2 e t sin 2t Now a particular solution: Y Ae t is the form LY Y 2Y 5Y 1 2 5 Ae t 4Ae t e t so 4A 1 , A
School: WVU
Course: Mathematical Methods
Math 261 Quiz 6 Write the general solution: (2 pts) r3 r3 r y a) y 3y 2y 0 3r 2 0 We observe that r 1 is a root, so we can factor 3r 2 r 1 r 2 r 2 r 1 r 2 r 1 1, 1, 2 c 1 e t c 2 te t c 3 e 2t (3 pts) b) y iv 2y y e 2t (Hint: perfect square) Homogeneous s
School: WVU
Course: Mathematical Methods
Math 261 Quiz 7 1) (3 pts) Determine whether the members of the given set of vectors are linearly independent. If they are linearly dependent, find a linear relation among them. 1 2 , 3 2 1 1 , 5 1 1 1 We are solving c 1 2 c2 3 1 c3 2 1 1 12 1 c2 32 5 c
School: WVU
Course: Mathematical Methods
Math 261 Quiz 8 u cos au du 1 u sin au 12 cos au a a u sin au du 1 u cos au 12 sin au a a 1. (4 pts) Calculate the Fourier series, period 4, of the function f x x , 2 x 2. Write out the series using summation notation. Write out the first three nonzero
School: WVU
Course: Mathematical Methods
Math 261 Quiz 9 1. Use separation of variables to find a family of solutions: u xx ut u tt 0 Try u x, t X x T t and plug in: X xTt XxT t XxT t 0 X x T t X x T t X x T t and separate: T t T t Xx , separation constant Xx Tt X X 0 T T T 0 Lets assume for sp
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Final Do any 9 problems. Do not use any "fancy" MATLAB functions such as max,min,prod,sum,cumsum,sort, etc. 1. Given a matrix A, explain (using MATLAB commands) how to do the following a) Find the number of rows and columns in A b) Create a matri
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Final Exam 1. Solve this system using MATLAB (yeah its easy): 2x y 3z 1 xy z 2 3x 4z 4 2. Create these arrays with a single expression using the colon operator 1 1 1 . 1 a) 99 2 12 32 52 1 . 1 16 2 10 b) 1 2 1 4 1 8 c) 1 2 3 4 5 . 6 99 100 3. If
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 3 1. We are interested in the function f x |x 1 | on the interval 0, 2 . a) Interpolate f x on 0, 2 using a polynomial p x of degree 6 with equally spaced interpolation points that cover the interval. b) Plot the graph of p x , the datapoint
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 1 Directions: Create a single M-file with your answers, with file name smithexam1.m (where you substitute your last name for smith). When you are done, drag the file onto the Math222 icon on the desktop. NOTE: There are two sides to the exam
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 1 Instructions: All of your work should be submitted in ONE Matlab M-file. The name of the file should be of the form brownexam1.m if your last name is brown. (I dont want 25 files all named brownexam1.m please.) You will need to email me th
School: WVU
Course: Differential Equations
1. linear, ( , 32 ), non-linear 2 2. y = 4, unstable, y = -1 Asy. stable 1 3. y = cosx-2 4. y = 1 (et + c) t 5. x2 y2 + 2xy = c t t dQ Q - 50 50 ), limiting 6. dt + 50 = 1, Q(0) = 0, IF is e , Solution is Q(t) = 50(1 - e amount is 50 lbs. 7. y2 - 4t 2 >
School: WVU
Course: Differential Equations
Exam I Febuary 1, 2005, NAME: 1. Given the following differential equations. (12 pts) 2 dy + y(tant) = et , dt y(2) = 1 -x dy = , dx y - 1 answer the following y(0) = 0 a) Determine which equations are linear and non-linear. b) If any of the equations are
School: WVU
Course: Differential Equations
Exam II March 1, 2005, NAME: 1. Determine the largest interval in which the given initial value problem is certain to have an unique solution. (6 pts) ty + (tant)y - y = et , y(2) = 0, y (2) = 1 2. Consider a vibrating mechanical system in which an object
School: WVU
Course: Differential Equations
EXAM III March 31, 2005, NAME: 1. Find the inverse Laplace transform of the following (10 pts) 1-2s s2 +4s+5 2. Write the complex number -1 + 3i in the form rei . (7 pts) 3. Find the recurrence relation only by means of a power series about the point x0
School: WVU
Course: Differential Equations
Solutions for Practice Exam II 3 1. 2 3 2. The natural frequency is 4, for beats the frequency can be anything but 4, for resonance the frequency must be 4. 3. y 3 2 e2t 2 1 e t and must be equal to -2. 3 4. Y1 At 2 Bt C Y2 t 2 At B e2t Y3 At B cost Ct D
School: WVU
Course: Differential Equations
Solutions for Practice Exam II 1. 5e 2t sint 2e 2t cost 2 2. 2e 3 i n 3. an 2 an n a2 1 2 2 x2 4c1 2c2 e 1 4 2t y 5e x1 c2 e 2t F, T, F, T, F c6 e 7. 8. 9. 3 t sin 1 3t 5e 3c1 t 2 2t ! 6. y c1 cost c2 sint c3 e 3 2 t 4. 5. y y c1 c2 et c3 e2t a 0
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 3 1. We wish to find a polynomial c 0 1, 2 , 1, 3 , 2, 1 . c1x c 2 x 2 to interpolate the x, y values a) What system of equations is satisfied by c 0 , c 1 , c 2 - write it explicitly below. b) Solve the system using MATLAB and write the pol
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 2 1. Consider the positive fixed points of the function f x 5 log x. a) Find the approximation location of the fixed point(s) by graphing using MATLAB. (Youll need to go out a ways) b) Which fixed point(s) are stable and which are unstable?
School: WVU
Course: Numerical And Symbolic Methods
Practice problems for exam 2 1. Make a 20x20 random array containing integers 1,2,3 (and dont you dare use randi). Display the array in an image, with 1 Red, 2 Green, 3 Blue. 2. Write in recursive form a matlab function s mysumr(x) that will find the sum
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 2 1. Create a random 20x20 matrix of integers from 1 to 6 and display the matrix and colorbar using the following color map: 1 2 3 4 5 6 Red Red/Green (using an even mixture of red and green, with no blue) Green Blue/Green (using an even mix
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 2 1. Consider the positive fixed points of the function f x 5 log x. a) Find the approximation location of the fixed point(s) by graphing using MATLAB. (Youll need to go out a ways) b) Which fixed point(s) are stable and which are unstable?
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 2 1. (4 pts) Consider the following MATLAB statements [m,n] size(colormap); image(m:-1:1) colorbar Explain what just happened, what you are looking at, and why. There are two images in the figure - they look exactly the same except for their
School: WVU
Course: Numerical And Symbolic Methods
Math 222 Exam 1 Put your answers in a single MATLAB .m file, with the filename jonesexam1.m (if your last name is Jones). Email it to me at diamond@math.wvu.edu. Efficiency counts. The usual ground rules apply (no fancy MATLAB functions such as sum, flipl
School: WVU
Course: Abstract Algebra
Algebra Exam 1 Fall 2006 Instructions: In this exam, unless otherwise stated, G will denote a group, Z denotes the additive group of integers, and Zm denotes the additive group of integers modulo m. Work on each of the problems. 1. Let G be an abelian gro
School: WVU
Course: Discrete Structures
Math374 Final Exam 2011 May 3 Name: Do not turn the page until instructed. Directions: 1. Write your name on this page. 2. Show your work unless instructed otherwise. Answers without supporting work are incomplete and are marked accordingly. 3. You may us
School: WVU
Course: Non-engineering Calculus
e 1 1 e 0 e ;A 0 e t 7 . (2 ) - )L ? : s " Ci " " ) 01 3 0 ) . - O
School: WVU
Course: Non-engineering Calculus
M at h 4 Sec t io n : 5 5 H o m ew o r k D u e : F in D o d W 7 ( 2 5 p o e d n e sd a y , 2 n t s) q , Ft h e th e d e r iv a t iv e s n o t s im p li fy 2 0 1 3 y ou fo llo w i n g a n sw er s fu n c t io n s . 3 L . (4 w l. ) g (= ) = 3 s i n a - 3= 4
School: WVU
Course: Non-engineering Calculus
Do M . Ilo Lh n : . 1s 5 " 0 . k 1 (2 0 p . I. t . ) 15 L 0 (= ) : B = (b ) (5 ) R w > 0( ri 1 . +; B /a h 1a t e t l , e f . ) : q . S; ll. " I. g li. .It. . 2 . ( ) Sk , ci o n i " t 1l , t l . " ,1 f w I. ( : o f " l 0 0 m ) " " f " io " " t h . u o
School: WVU
Course: Non-engineering Calculus
* X k D . Eb , T , (25 p o in t s Mm ) au of t he p r o b t , m s . I 1o - Enal uat e t h e 1 1 Se c 2l 5 y o u r W o , k t ion : g N am o : M1 w . lim i t s 0 i w r \t th e \ hL t v ) or ol n p K o\ w or k n ot t cfw_ e Q r n " - \ , o a 6 . L _ I n 1
School: WVU
Course: Non-engineering Calculus
Ma t h k 155 il . 5 2 : . ) 0 " : . :i : ! ( " u ) Na m e: Math 1 5 S Ho m e w o r k Du e Nam e 5 T , ! . (4 ) , " " t " " " , o r , I" 2: " : g" : I. U1l y " . F1, " 1l . $+al . f " " " " " ( . , ] 9 - 3 3 . (4 ) Let f ( ) + 1. - u . T 1. l m " " 4 i.
School: WVU
Course: Non-engineering Calculus
* : X X :. D . e Ew . T l. . . (20 p J. Rsd o i . Ts AO ) 1. A r y l . At e t h e f ol i o. . R l i m i t s g I T . , S , im : : 2 0 1 3 Sho ). " " ,k . 1 6 Nm n . / qw n i n A: B - 0 H t . ? - t jx 4 < X cA &
School: WVU
Course: Numerical And Symbolic Methods In MATH/STAT
Math 222 Hw3 Part 2 1) Let f x 6x x 2 2 a) Locate the fixed points of f graphically. Without running any iterations or solving for the fixed points exactly, determine whether the fixed points are stable or unstable (you may use their approximate location
School: WVU
Course: Introduction To Cryptography
Math373 Homework 6 Due Fri, Feb. 21, 2014 Directions: Solve the following problems. All written work must be your own. See the course syllabus for detailed rules. 1. [JJJ 1.36] Compute the value of 2(p1)/2 (mod p) for every prime 3 p < 20. (You do not nee
School: WVU
Course: Introduction To Cryptography
Math373 Homework 2 Due Fri, Jan. 24, 2014 Directions: Solve the following problems. All written work must be your own. See the course syllabus for detailed rules. 1. Let a, a , b, b , and m be integers such that a a (mod m) and b b (mod m). Prove that a +
School: WVU
Course: Introduction To Cryptography
Math373 Homework 3 Due Fri, Jan. 31, 2014 Directions: Solve the following problems. All written work must be your own. See the course syllabus for detailed rules. 1. Modular Arithmetic Tables (a) Make addition and multiplication tables for Z3 . (b) Make a
School: WVU
Course: Introduction To Cryptography
Math373 Homework 4 Due Fri, Feb. 7, 2014 Directions: Solve the following problems. All written work must be your own. See the course syllabus for detailed rules. 1. [JJJ 1.28] Compute the following values of the order function. (a) ord2 (2816) (b) ord7 (2
School: WVU
Course: Introduction To Cryptography
Math373 Homework 5 Due Fri, Feb. 14, 2014 Directions: Solve the following problems. All written work must be your own. See the course syllabus for detailed rules. 1. Modular exponentiation in F7 . (a) Fill in the table so that row a and column k contains
School: WVU
Course: Introduction To Cryptography
Math373 Homework 1 Due Fri, Jan. 17, 2014 Directions: Solve the following problems. All written work must be your own. See the course syllabus for detailed rules. 1. The Caeser cipher. (a) Encrypt the message exchange all assets using a Caesar cipher with
School: WVU
Course: Applied Linear Algebra
1. a) Notice that the second vector is three times the first, so all linear combinations of the two vectors can be expressed as a scalar multiple of the first vector. So the result is the line t 1, 2, 3 b) The result is a plane that can be described as x,
School: WVU
Course: Applied Linear Algebra
Sec. 1.2 3,4,5,6,8,9,11,12,16,17,21,23 v 1 3, 4 and w 1 8, 6 give unit vectors in the directions of v, w 5 10 v w v w v w 1 3, 4 1 8, 6 48 24 , and Then cos 5 10 50 25 vw v w 1 24 cos 0. 283 79 (in radians) 25 Given w 8, 6 the vectors 4, 3 , 3, 4 , 4,
School: WVU
Course: Abstract Algebra
Homework Assignment 5 Homework 5. Due day: 11/6/06 (5A) Do each of the following. (i) Compute the multiplication: (12)(16) in Z24 . (ii) Determine the set of units in Z5 . Can we extend our conclusion on Z5 to Zp , for an arbitrary prime integer p? (iii)
School: WVU
Course: Abstract Algebra
Hungerford: Algebra IV.1. Modules Note: R is a ring. 1. If A is an abelian group and n > 0 an integer such that na = 0, a A, then A is a unitary Zn -module, with the action of Zn on A given by ka = ka, where k Z and k k Zn under the canonical projection Z
School: WVU
Course: Abstract Algebra
Hungerford: Algebra III.6. Factorization in Polynomial Rings 1. (a) If D is an integral domain and c is an irreducible element in D, then D[x] is not a principal ideal domain. (b) Z[x] is not a principal ideal domain. (c) If F is a eld and n 2, then F [x1
School: WVU
Course: Abstract Algebra
Hungerford: Algebra III.2. Ideals 1. The set of all nilpotent elements in a commutative ring forms an ideal. Proof: Let R be a commutative ring and let N denote the set of all nilpotent elements in R. Then I = cfw_a R : for some n Z, rn = 0. As 0 I , I =
School: WVU
Course: Calculus III
Note: Do NOT use the diagonal method to evaluate 3x3 determinants that you may have learned in high school. You will get zero points. 1) Evaluate using row/column operations and Lapalce expansion so as to reduce to a single 2x2 determinant and then evalua
School: WVU
Course: Calculus III
1) Solve the system: x 2y xy x 3y 2x y zw3 2z w 1 4z w 1 z 2w 0 a) Reduce the augmented matrix to row echelon form and use back substitution. 1 2 1 1 3 1 12 1 1 1 3 1 r2 1 2 1 4 1 2 r3 0 r4 0 4 0 000 2 2 1 1 2 1 3 0 r1 1 3 0 2 1 3 2 2 0 2r1 121 r3 3 1 4 6
School: WVU
Course: Calculus III
Exercises: 1) Consider the subspace of R 4 spanned by the vectors 1 1 1 1 1 1 1 , 1 , 1 1 1 1 1 , 1 1 . 1 a) Do these vectors span all of R 4 ? How do you know? If the vectors are not linearly independent, identify a subset of these vectors that is linear
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 1 1 (Divisibility) Let a, b, c Z. Use the denition of divisibility to directly prove the following. (Instruction: This is part of Theorem 1.2.3 on Page 3. Solutions for (a), (b) are given as examples. You only need to do (c) an
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 2, Solution 1 (Congruences) Compute the following: (No credit for using matlab or any other computer programs. The purpose is to train us to understand the arithmetic modulo a number). (a) 12 + 63 (mod 17). (b) 32 74 (mod 7) (c
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 3, Solutions Instruction: Unless otherwise stated, no credit for solutions using matlab or any other computer programs. The purpose is to train us to understand the related materials via manual solutions. 1 (Exercise 2.23.10) S
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 4, Solutions Instruction: Unless otherwise stated, no credit for solutions using matlab or any other computer programs. The purpose is to train us to understand the related materials via manual solutions. 1 Application of Euler
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 6, Solutions Instruction: In doing this set of problems, you can use any electronic devise to help your computation. But you need to present conclusions based on the analysis of your computation. 1. Convert the base-26 numbers
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 7, Solutions Instruction: In doing this set of problems, you can use any electronic devise to help your computation. But you need to present conclusions based on the analysis of your computation. The English Alphabet: Unless ot
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 8, Solutions Instruction: In doing this set of problems, you can use any electronic devise to help your computation. But you need to present conclusions based on the analysis of your computation. 1. (Exercise 8.7.4) Alice encry
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 10, Solutions Instruction: In doing this set of problems, you can use any electronic devise to help your computation. But you need to present conclusions based on the analysis of your computation. 1. Use Baby-step and Giant-ste
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 11, Due Day 4/9/2012 Instruction: In doing this set of problems, you can use any electronic devise to help your computation. But you need to present conclusions based on the analysis of your computation. 1. For each of the foll
School: WVU
Course: Introduction To Cryptography
Math 373/578 Homework, Week 12, Solutions Instruction: In doing this set of problems, you can use any electronic devise to help your computation. But you need to present conclusions based on the analysis of your computation. 1. Check that the points P = (
School: WVU
Course: Introduction To The Concepts Of Mathematics
Name: Math 283 Spring 2013 Assignment 2 Solutions 1. The statement below is not always true for real numbers x and y . Give an example where it is false, and add a hypothesis (condition) on y that makes it a true statement. If x and y are nonzero real num
School: WVU
Course: Introduction To The Concepts Of Mathematics
Math 283 Spring 2013 Assignment 3 Solutions 1. Prove that A B C if and only if A B and B C . Solution. This is false. Let A = cfw_1, B = cfw_1 and C = cfw_2. Notice that everything holds, except B C . Further, if A and B are as before, but C = cfw_1, 2, t
School: WVU
Course: Introduction To The Concepts Of Mathematics
Math 283 Spring 2013 Assignment 4 Solutions 1. P (A B ) = P (A) P (B ) Solution. This is true. () () Let X P (A B ). This means X A B and so X A and X B (since x X means x A and x B ). Since X A, we have X P (A). Similarly, X B means X P (B ). Thus, we h
School: WVU
Course: Introduction To The Concepts Of Mathematics
Math 283 Spring 2013 Assignment 5 Solutions n 1. For n N prove that i(i + 1) = i=1 n(n + 1)(n + 2) . 3 Solution. Notice that for n = 1, we have 1(1 + 1) = 2 = n If i(i + 1) = i=1 1(2)(3) . 3 n(n + 1)(n + 2) , adding (n + 1)(n + 2) to the left hand side gi
School: WVU
Course: Introduction To The Concepts Of Mathematics
Math 283 Spring 2013 Assignment 6 Solutions 1. Prove that exponentiation to a positive odd power denes a strictly increasing function. For n N, nd all solutions to xn = y n . (Hint: One possibility is to consider the cases x < 0 < y , 0 < x < y and x < y
School: WVU
Course: Introduction To The Concepts Of Mathematics
Math 283 Spring 2013 Assignment 7 Solutions 1. Prove that the natural numbers, the even natural numbers, and the odd natural numbers form sets of the same cardinality. Solution. Every even natural number is obtained by doubling a unique natural number, so
School: WVU
Course: Calculus 1 For Engineers
Worksheet 9 Math 155 KEY Group 6 (Sec 2.3/2.4/2.5) The diagram shows the major components of a stationary double acting steam engine. The piston moves along the cylinder and drives the small end of the connecting rod back and forth horizontally. The big e
School: WVU
Course: Numerical Analysis
Math 420 Hw5 1. Develop an approximation formula f 0 w 1 f 1 w 2 f 2 w 3 f 4 in several ways as directed below: a) Interpolate f at x 1, 2, 4 with a polynomial p 2 x in Lagrange form, and calculate f 0 Our function p 2 x that interpolates the values f x i
School: WVU
Course: Numerical Analysis
Homework Due Monday March 18 1. Calculate by hand the divided difference table corresponding to the data x201 -3 and then write the polynomial in Newton form that interpolates the data. y 5 1 -1 3 (Use fractions, not decimals) x f[x(i)] 2 5 2 0 1 4 -2 1 -
School: WVU
Course: Numerical Analysis
Math 420 HW6 1. Derive quadrature formulas i) ii) h 0 h 0 f x dx h w1f h w2f 0 f x dx h w1f 0 w2f h w3f h w 3 f 2h 1 a) Use polynomial interpolation as applied to the unscaled formula for f x dx and scale the 0 result 1 b) Use exactness for f x 1, x, x 2
School: WVU
Course: Numerical Analysis
Homework Due Monday March 18 1. Calculate by hand the divided difference table corresponding to the data x201 -3 and then write the polynomial in Newton form that interpolates the data. y 5 1 -1 3 (Use fractions, not decimals) 2. Write out by hand the lin
School: WVU
Course: Numerical Analysis
Math 420 Spring, 2013 Syllabus Room 125 Brooks MWF 11:30-12:20 Course web page: http:/www.math.wvu.edu/~diamond/Math420S13 Instructor: Professor Diamond Office: 410J Armstrong Phone: 304-293-9082 email: diamond@math.wvu.edu Office hours: MWF 2:30-3:20 . Y
School: WVU
MATH 156, FALL 2012 Instructor: Office: Email: Phone: Office Hours: Text: Dr Hong-Jian Lai 320A Armstrong Hall hjlai@math.wvu.edu 293-2011 x2331 Tuesdays, and Thursdays 9:40 11:30am Essential Calculus: Early Transcendentals, by James Stewart Exams: There