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Course: MAT 312, Fall 2008
School: SUNY Stony Brook
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Word Count: 192

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Algebra, Applied MAT312/AMS351 Practice Problems for Midterm 1 1. Find the greatest common divisor of 12n + 1 and 30n + 2. 2. Prove that for every natural number n, the number 32n+2 + 8n 9 is divisible by 16. 3. Recall that the Fibonacci sequence is dened as F1 = 1, F2 = 1, and then for every n &gt; 2, Fn = Fn1 + Fn2 . Prove that for every n, F1 + F3 + + F2n1 = F2n . 4. Find all n &gt; 2 such that n3...

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Algebra, Applied MAT312/AMS351 Practice Problems for Midterm 1 1. Find the greatest common divisor of 12n + 1 and 30n + 2. 2. Prove that for every natural number n, the number 32n+2 + 8n 9 is divisible by 16. 3. Recall that the Fibonacci sequence is dened as F1 = 1, F2 = 1, and then for every n > 2, Fn = Fn1 + Fn2 . Prove that for every n, F1 + F3 + + F2n1 = F2n . 4. Find all n > 2 such that n3 3 is divisible by n 1. 5. When questioned by the police, the suspect claimed that he did not remember his home address but could denitely recall that the house number is less than 1000 and is divisible 7, by 11, and 13. Is the suspect telling the truth? And what if he said that the number was divisible by 7, 11, and 14? 6. Let a, b, and c be positive integers such that a2 + b2 = c2 . Prove that at least one of them is divisible by 3. (Hint: reduce mod 3.) 7. Solve the following linear congruences: (a) 5x 7 mod 31; (b) 2x 19 mod 2006; (c) 19x + 3 4...

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SUNY Stony Brook - MAT - 312
Applied Algebra, MAT312/AMS351 Practice Problems for Midterm II (1) Let R = {(a, b) | a b mod 5} be a subset of Z Z. Prove or disprove that aRb is an equivalence relation on Z. 1234567 1234567 (2) Let = and = . 4627315 3651427 Compute , 1 . Det
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Practice Final Exam MAT 125May 8, 2006 Name: Recitation number (e.g., R01): (for evening lecture, use ELC 4)Lecture 1 R01 R02 R03 R04 R05 R31 Lecture 2 R06 R07 R08 R09 R10 Lecture 3 R11 R12 R13 R14 R15 R32 Evening Lec 4 MWF 9:3510:30 M 11:45am12:40
SUNY Stony Brook - MAT - 125
MATH 125Name:Second MidtermMarch 20, 2007ID:Rec:Question: Points: Score:1 122 123 84 85 166 147 108 10Total 90There are 8 problems in this exam, printed on 6 pages (not including this cover sheet). Make sure that you hav
SUNY Stony Brook - MAT - 125
MAT125.R91: QUIZ 7SOLUTIONSCompute derivatives of the following functions: sin x (a) f (x) = 1 + sin x (sin x) (1 + sin x) sin x(1 + sin x) cos x(1 + sin x) sin x cos x f (x) = = = (1 + sin x)2 (1 + sin x)2 cos x cos x + cos x sin x sin x cos x
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MAT125.R92: QUIZ 5SOLUTIONSFind the equation of the tangent line to the graph of the function 4 f (x) = 2 3x2 x at x = 2. The tangent line will pass through the point (2, f (2). 4 Since f (2) = 2 3(2)2 = 1 12 = 11, the point is (2, 11). 2 The s
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Calculus IV with Applications MAT303 Solutions to Practice Problems for Midterm 1 1.3, 17. f (x, y ) = x1 and Dy f (x, y ) = xy21 are continuous in the neighbory hood of (0, 1), thus the solution near x = 0 exists and is unique. 1.3, 18. f (x, y )
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Calculus IV with Applications MAT303 Solutions to Practice Problems for Midterm II 3.1, 26. If dependent, then f = cg for a constant c, i.e. 2 cos x + 3 sin x = c(3 cos x 2 sin x). Then comparing coecients at cos x and sin x, we get 2 = 3c and 3 = 2
SUNY Stony Brook - MAT - 125
MAT125.R91: QUIZ 8SOLUTIONSNo score will be assigned for this quiz. For the total of 10 points, answer the following questions. Do not skip any steps. Show your computations, if any. No calculators or notes are allowed. You have 10 minutes. The le
SUNY Stony Brook - MAT - 125
MAT125.R92: QUIZ 8SOLUTIONSFind the absolute maximum and absolute minimum values of f (x) = (x2 1)7 on the interval [1, 1]. f (x) = 7(x2 1)6 (2x) = 14x(x2 1)6 (we use the chain rule with u = x2 1) Critical points: The derivative exists everywhere
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MAT125.R91: QUIZ 1SOLUTIONSLet f (x) =x and g (x) = x2 + 1.(a) Find the function f g . (f g )(x) = x2 + 1 (b) What are the domain and range of f g ? Domain: all x such that x2 + 1 0 (to make the square root dened). Since x2 + 1 is alway
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MAT125.R91: QUIZ 0SOLUTIONSNo score was assigned for this quiz. The graph of the function f (x) is given below:11(a) Determine the domain of f (x). 4 &lt; x &lt; 2 and 3 x 4 (or, in other notations, (4, 2) and [3, 4]). We exclude 4 and 2 because
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MAT125.R92: QUIZ 7SOLUTIONSTwo people start walking from the same point. One goes north at 3 mi/h and the other goes west at 4 mi/h. How fast is the distance between them increasing when the rst person is 6 miles away from their starting point? Le
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CS-034 Getting Started in CThomas Doeppner Pascal Van Hentenryck2/2/2005CS-034: Lecture 1 (twd/pvh: 2005)1A C Programint main() { int main() { return 0; return 0; } }C Programs always starts in a main function main always return an error
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CS034CS-034 Continuing with CThomas Doeppner Pascal Van Hentenryck2/9/2005CS-034: Lecture 2 (twd/pvh)1Memory0 1 2. . .1,073,741,8232/9/2005CS-034: Lecture 2 (twd/pvh)2CS0341CS034Basic Data Typesint-2,147,483,648 2,14
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CS-034 Memory Allocation/deallocationThomas Doeppner Pascal Van Hentenryck2/16/2005CS-034: Lecture 3 (twd/pvh: 2005)1Naming Typestypedef char *String; typedef char *String; typedef int *IntPtr; typedef int *IntPtr;typedef allows you to n
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CS-034 From C to C+Thomas Doeppner Pascal Van Hentenryck3/2/2005CS-034: Lecture 5 (twd/pvh: 2005)1From C to C+3/2/2005CS-034: Lecture 5 (twd/pvh: 2005)2From C to C+3/2/2005CS-034: Lecture 5 (twd/pvh: 2005)3Memory Allocation
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CS34 Lecture 63/9/2005CS-034 From C+ toC+1Thomas Doeppner Pascal Van Hentenryck3/9/2005CS-034: Lecture 6 (twd/pvh)Stacks class Stack { public: Stack(); ~Stack(); void push(int); int pop(); bool empty(); private: Node *top; };3/9/2005
Sanford-Brown Institute - CS - 034
CS-034 To be or not to be (efficient)T. Shakespeare Doeppner P. Hugo Van Hentenryck3/16/2005CS-034: Lecture 7 (twd/pvh: 2005)1C/C+ versus JavaWhere does the time go?arrays arrays virtual methods virtual stack machine: JVM stack garbage col
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CS34 Lecture 8CS-034 Its Magic Thomas Doeppner Pascal Van Hentenryck3/23/2005CS-034: Lecture 8 (twd/pvh: 2005)1Storagestack Local Variablesdynamic data textmalloc and new everything else Code3/23/2005CS-034: Lecture 8 (twd/pvh: 2
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CS034Intro to Systems ProgrammingDoeppner &amp; Van HentenryckLab 1.1Out: February 2, 2005 What youll learn.In this lab, you will write and compile your rst C program, all from scratch. Well go through step by step what is required to write a com
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Sanford-Brown Institute - CS - 034
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CS034Intro to Systems ProgrammingDoeppner &amp; Van HentenryckLab 2.1Out: 9 February 2005 What youll learn.In the rst part of this lab, you will practice using bitwise operators. In the second part, you will open an image and read formatted data
Sanford-Brown Institute - CS - 034
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SUNY Stony Brook - MAT - 125
MAT125.R92: QUIZ 6SOLUTIONSName: Using linear approximation, estimate3 8.1. (Hint: 3 8.1 = f (8.1), where f (x) = 3 x.) f (x) f (a) + f (a)(x a) for a close to x. Specically, f (8.1) = f (8) + f (8)(8.1 8) (we choose a = 8 because its close t
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SUNY Stony Brook - MAT - 125
MAT125.R92: QUIZ 2SOLUTIONS2x + 5 . Find the inverse function of f (x). x3 2x + 5 The function f (x) is given by the equation y = . We need to solve x3 for x: 2x + 5 y= x3 (x 3)y = 2x + 5 xy 3y = 2x + 5 xy 2x = 5 + 3y (y 2)x = 3y + 5 3y + 5 x=
Sanford-Brown Institute - CS - 034
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SUNY Stony Brook - MAT - 125
MAT125.R91: QUIZ 5SOLUTIONS3x + 1 . Evaluate the following limits: x3 3x + 1 (a) lim f (x) = lim . x3+ x3+ x 3 When x approaches 3 from the right, 3x + 1 is close to 3(3) + 1 = 10 and x 3 is a small positive number. 3x + 1 is 10 divided by a sma
Sanford-Brown Institute - CS - 034
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SUNY Stony Brook - MAT - 125
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SUNY Stony Brook - MAT - 125
MAT125.R91: QUIZ 2SOLUTIONS Let f (x) = 2 x2 + 5, x 0. Find the inverse function of f (x). The function f (x) is given by the equation y = 2 x2 + 5. We need to solve for x: y = 2 x2 + 5 y 2 = x +5 2 y2 = x2 + 5 2 y2 5 = x2 2 y2 5 x= 2 y2 Hence
SUNY Stony Brook - MAT - 125
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Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
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Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
Team Number: Week: Program Size (SLOC) Base Added Deleted Modified Reused # of COTS Total New SLOC Effort (Hours) Project Mgmt. Requirements COTS Assessment Design Life Cycle Planning Configuration Mgmt. Feasibility Analysis Code COTS Tailoring COTS
USC - CSCI - 577
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USC - CSCI - 577
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ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32Task Name Inception First Team Interaction Members' time preferences Assigning Roles Project Detail discussion Client meeting preperation First Client Meeting
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Answers to the MAT127 Homework No.12 Chapter 7 Section 4 Problem 2-4,7,10 &amp; Section 5 Problem 2,5,7,8,11,13,14,15 Section 7.4 2.(a) Let P (t) be the population of bacteria w.r.t. time t in the unit of hour, and let k be the relative growth rate in th