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2 Pages

### hw6

Course: MATH 163, Fall 2008
School: Concordia Chicago
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Word Count: 366

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6: Homework The Nature of Sin Denition 1 =2 1 1 x2 dx. 1 1. Geometrically motivate this denition (with picture(s)!). Denition 2 For 1 x 1, x 1 x2 + A(x) = 2 1 1 t2 dt. x 2. Geometrically motivate this denition. Hint: the rst term is the area of a triangle. 3. Calculate A (x) and A(1), A(1). Denition 3 The function cos : [0, ] R is dened by letting cos x be the unique number in [1, 1] such that A(cos...

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6: Homework The Nature of Sin Denition 1 =2 1 1 x2 dx. 1 1. Geometrically motivate this denition (with picture(s)!). Denition 2 For 1 x 1, x 1 x2 + A(x) = 2 1 1 t2 dt. x 2. Geometrically motivate this denition. Hint: the rst term is the area of a triangle. 3. Calculate A (x) and A(1), A(1). Denition 3 The function cos : [0, ] R is dened by letting cos x be the unique number in [1, 1] such that A(cos x) = x . The function sin : [0, ] R 2 is dened by: sin x = 1 (cos x)2 . 4. Prove that cos and sin are well dened by showing that A is a bijection from [1, 1] to [0, ]. 5. Consider a line in the x, y plane through the origin that makes an angle of with the positive x-axis, where 0 . Prove that the line intersects the unit circle (i.e. the circle centered at the origin with radius 1) at the point (cos , sin ). Denition cos, 4 sin : R R are dened as the unique functions agreeing with Denition 3 on [0, ] that satisfy: sin x = sin(x) cos x = cos(x) sin(x + 2 ) = sin x cos(x + 2 ) = cos x. 1 6. Prove that: sin x = cos x, 7. Dene f (x) = Prove that f is continuous at 0. Recall that a function f : R R is twice-dierentiable if f is dierentiable and f is dierentiable. Naturally, we write f for the derivative of f . 8. Suppose that f : R R is twice-dierentiable and satises: f +f =0 f (0) = 0 f (0) = 0. Prove that f ...

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Concordia Chicago - MATH - 163
Homework 7: Polynomial ApproximationsThe goal here is to discuss polynomial approximations of dierentiable functions. Lets begin with a warm-up. Denition 1 A function s : [a, b] R is called a step function if there exists a partition a = t0 &lt; . .
Concordia Chicago - MATH - 163
Concordia Chicago - MATH - 163
Concordia Chicago - MATH - 163
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Cornell - PRH - 7090
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Geog 191/291 Introduction to Optimization for Geographic ProblemsInitialization MethodsThere are several methods for initializing a Classical Transportation Problem (CTP). In addition to the Northwest Corner Rule, there is Vogels Approximation Me
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Penn State - SCB - 223
Serge BallifMATH 527 Homework 3September 21, 2007Problem 1. Let X be a compact Hausdorff space. Let K1 K2 . . . be a sequence of closed connected subsets of X . Let K = 1 Ki . Show that K is connected. i=We note rst that X is normal since
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Penn State - SCB - 223
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Serge BallifMATH 527 Homework 6October 12, 2007Problem 1. Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map f : S 1 S 1 must there exist (x, y) S 1 S 1 , such that f (x, y) = f (-x, -y)?R2No. We can imbed th
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Serge BallifProblem 1. Let K = {(x, y, z) group of X = 3 - K.MATH 527 Homework 7October 26, 2007RR3 | x2 + y2 = 1, z = 0}, the standard circle in R3. Compute the fundamentalX deformation rectracts onto a space X1 which is sphere centered
Penn State - SCB - 223
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Penn State - SCB - 223
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Penn State - SCB - 223
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Texas A&M - ELEN - 449
Lab # 3 ISA Prototyping and using the ABEL LanguageIntroduction In this lab, we introduce the ISA interface bus we will be using this semester. We also explore the Lattice MACH ispLEVER software to write, simulate, and debug an ABEL description of
Texas A&M - ELEN - 449
Lab # 4 Wire Wrapping and Programming the MACH 64/32Introduction In this lab, we introduce the skill (art?) of wire-wrapping as a means to prototype digital circuits. We begin by wire-wrapping the data transceiver block of the ISA interface prototy
Texas A&M - ELEN - 449
Lab # 5 - I/O Latch Interfacing and Timing ConsiderationsIntroduction In this lab, we learn to program the Lattice MACH 64/32 device with the JEDEC file we created in the last lab using the Lattice ispVMsystem tool. We learn how to use the Oscillosc
Texas A&M - ELEN - 449
Lab # 7 D/A Converter and Wait State GenerationIntroduction In this lab, we interface a DAC converter on our ISA prototype card. Care must be taken in the design process to insure the device will operate correctly. We will modify our CPLD to add a
Texas A&M - ELEN - 449
Lab # 8 A/D Converter and Interrupt ProcessingIntroduction In this lab, we interface an Analog-to-Digital Converter (ADC) to our ISA prototype card. We modify the ABEL source and program our CPLD to generate a conversion clock, chip select, and int
Texas A&M - ELEN - 449
LAB # 1 - Introduction to DOS, EDIT, and DEBUGPart I - Introduction to DOS Commands Introduction: In this part of the lab, we present several of the features supported by the disk operating system (DOS). The disk operating system is a program that c
Texas A&M - ELEN - 449
Lab # 1 Report Requirements (100 Points)Sign-off (20 Points) You must be signed-off by your TA before you leave the Lab. Students are required to Develop the required program for Part III, step 8 in the lab manual. Save your program as a .com file
Texas A&M - ELEN - 449
Lab # 2 Report Requirements (100 points)Sign-off (20 points) You must be signed-off by your TA before you leave the Lab. Students are required to Make demos for steps 9, 10, 11, 13; Correct the errors in Example 5 and make the demo for step 12, th
Texas A&M - ELEN - 449
Lab # 3 Report Requirements (150 points)Sign-off (50 points) You must be signed-off by your TA before you leave the Lab. Students are required to 1. Understand the ISA Interface Card-Block Diagram (figure 1) and the 8-Bit ISA Bus Cycle (figure 2) as
Texas A&M - ELEN - 449
Lab # 4 Sign-off and Report Requirements (150 points)Sign-off (50 points) You must be signed-off by your TA before you leave the Lab. Students are required to 1. Finish the required wire-wrapping. TA will check off your connections. (15 points) 2. F