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21 Pages

### Lect_Notes_2_Leahy11

Course: EE 483, Fall 2011
School: USC
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Word Count: 1873

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(lecture Outline #2) A (very) brief review of continuous time signals and LTI sytems time signals and LTI sytems Examples of discrete time signals Discrete time systems time systems Building blocks Examples How do we characterize these systems? 1 Copyright 2005, S. K. Mitra/ 2006-2010 R.Leahy Continuous time review Continuous time signals A real or complex function x(t) defined on the real or complex...

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(lecture Outline #2) A (very) brief review of continuous time signals and LTI sytems time signals and LTI sytems Examples of discrete time signals Discrete time systems time systems Building blocks Examples How do we characterize these systems? 1 Copyright 2005, S. K. Mitra/ 2006-2010 R.Leahy Continuous time review Continuous time signals A real or complex function x(t) defined on the real or complex function x(t) on the real line Continuous time systems A mapping from an input signal x(t) to an output signal y(t) 2 Continuous time systems Linear time invariant system Superposition: if y1 (t ) = H { x1 (t )} , y2 (t ) = H { x2 (t )} and y (t ) = H { x1 (t ) + x2 (t )} then th y (t ) = y1 (t ) + y2 (t ) Time invariance If y (t ) = H { x (t )} then H { x(t t0 )} = y (t t0 ) t0 Characterized by impulse response y (t ) = 3 x(t )h( )d = x( )h(t )d Example and the Dirac Delta Dirac Delta and the sifting property y (t ) = x(t ) ( )d = (t ) x( )d = x(t ) Simple Echo y (t ) = x(t ) + ax(t t0 ) = x(t ) { ( ) + a ( t )} d 0 y (t ) = x(t )* ( (t ) + a (t t0 ) ) 4 Eigenfunctions of LTI systems What are the eigenfunctions of an LTI system? i.e. signals for which output is a system? signals for which output is scaled version of the input x(t) H(t) y(t)=x(t) 5 Eigenfunctions and the Fourier Transform Let x(t ) = exp { jt} real. Then: y (t ) = h( ) x(t )d Then: = h( ) exp{ j(t )}d = exp{ jt} h( ) exp{ j }d = x(t ) H ( j} So x(t ) = exp { jt} is an Eigenfunction of an LTI system. What are the eigenvalues? What about x(t ) = exp { t} for complex? 6 Fourier Transform Definitions X ( j ) = x(t )e j t R dt 1 x(t ) = 2 X ( j )e j t d t R Existence: 7 Dirichlet conditions absolutely integrable, finite number of discontinuties More relaxed forms including generalized functions Examples Dirac Delta (t ) Fourier transform (by sifting property): D ( j ) = (t )e jt dt = 1 Delayed delta function: Dt0 ( j) = (t t )e 0 jt dt = e jt0 Note: you cannot find inverse of these transforms through standard integrals as the 8 examples of generalized functions. Examples Complex Sinusoid x(t ) = exp { j0t} X ( j) = 2 ( 0 ) inverse Fourier transform: 1 x(t ) = 2 X ( j )e j t 1 d = 2 ( )e 0 jt d = e j0 t 9 Examples Rect Function: .5 X ( j ) = e .5 1 x(t ) = 0 j t t 0.5 t > 0.5 e jt dt = j .5 .5 e j /2 e j /2 sin( / 2) = = j /2 10 sinc( / 2 ) Characterization of Continuous time LTI systems Impulse response: h(t) Linear differential equations Characterizes realizable LTI systems through LDE involving inputs and outputs Frequency response H(j) - Fourier transform of impulse response System function: H(S) Laplace transform of impulse response 11 Analog vs. Discrete Time Systems Impulse response: h(t) Impulse response: h(n) differential equations Linear differential equations Characterizes realizable LTI systems through LDE involving inputs and outputs Linear difference equations Linear difference equations Characterizes realizable LTI systems through LDE involving inputs and outputs Frequency response H(jW) - Fourier transform of impulse response Frequency response H (e j ) Discrete Time Fourier transform of impulse of impulse response System function: H(S) Laplace transform of impulse response System function: H(Z) Z-transform of impulse response 12 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation Signals represented as sequences of numbers called samples numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range n x[n] defined only for integer values of n and undefined for noninteger values of n Discrete-time signal represented by {x[n]} 13 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation Discrete-time signal may also be written as a sequence of numbers inside braces: sequence of numbers inside braces: {x[n]} = {, 0.2, 2.2,1.1, 0.2, 3.7, 2.9,} In the above, x[1] = 0.2, x[0] = 2.2, x[1] = 1.1, etc. The arrow is placed under the sample at time index n = 0 14 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation Graphical representation of a discrete-time signal with real signal with real-valued samples is as shown samples is as shown below: 15 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal xa (t ) at uniform intervals of time 16 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation Here, n-th sample is given by x[n] = xa (t ) t = nT = xa (nT ), n = , 2, 1,0,1, The spacing T between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval T, denoted as FT , is called the sampling frequency: 1 FT = T 17 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation Unit of sampling frequency is cycles per second, or hertz (Hz), if T is in seconds if Whether or not the sequence {x[n]} has been obtained by sampling, the quantity x[n] is called the n-th sample of the sequence sequence {x[n]} is a real sequence, if the n-th sample x[n] is real for all values of n Otherwise, {x[n]} is a complex sequence 18 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation A complex sequence {x[n]} can be written as as {x[n]} = {xre [n]} + j{xim [n]} where where xre [n] and xim [n] are the real and imaginary parts of x[n] The complex conjugate sequence of {x[n]} is given by {x * [n]} = {xre [n]} j{xim [n]} Often the braces are ignored to denote a sequence if there is no ambiguity 19 DiscreteDiscrete-Time Signals: TimeTime-Domain Representation Example - {x[n]} = {cos 0.25n} is a real sequence sequence {y[n]} = {e j 0.3n} is a complex sequence can We write {y[n]} = {cos 0.3n + j sin 0.3n} where 20 = {cos 0.3n} + j{sin 0.3n} {yre [n]} = {cos 0.3n} {yim [n]} = {sin 0.3n} DiscreteDiscrete-Time Signals: TimeTime-Domain Representation Example {w[n]} = {cos 0.3n} j{sin 0.3n} = {e j 0.3n} is the complex conjugate sequence of {y[n]} That is, {w[n]} = {y * [n]} 21 Basic Sequences 1, n = 0 Unit sample sequence - [n] = 0, n 0 1 n 4 3 2 1 0 1 2 3 4 5 6 1, n 0 0, n < 0 Unit step sequence - [ n] = 1 22 4 3 2 1 0 1 2 3 4 5 6 n Basic Basic Sequences Real sinusoidal sequence x[n] = A cos(o n + ) where A is the amplitude, o is the angular frequency, and is the phase of x[n] Example = 0.1 o 2 Amplitude 1 0 -1 -2 0 10 20 Time index n 30 40 23 Basic Sequences Exponential sequence x[n] = A n , < n < where A and are real or complex numbers j If we write = e( o + jo ) , A = A e , then we can express x[n] = A e je(o + jo ) n = xre [n] + j xim [ n], where xre [n] = A eon cos(o n + ), 24 xim [ n] = A eon sin(o n + ) Basic Basic Sequences xre [ n] and xim [n] of a complex exponential sequence are real sinusoidal sequences with constant (o = 0 ), growing (o > 0 ), and decaying (o < 0 ) amplitudes for n > 0 Imaginary part Real part 0.5 0.5 Amplitude 1 Amplitude 1 0 -0.5 0 -0.5 -1 0 10 20 Time index n 30 -1 0 40 10 20 Time index n 1 x[n] = exp( 12 + j )n 6 25 30 40 Basic Sequences Real exponential sequence x[n] = A n , < n < where A and are real numbers = 0.9 = 1.2 20 50 15 Amplitude Amplitude 40 30 20 5 10 0 0 26 10 5 10 15 20 Time index n 25 30 0 0 5 10 15 20 Time index n 25 30 Basic Basic Sequences: Relations An arbitrary sequence can be represented in the time-domain as a weighted sum of some basic sequence and its delayed sequence and its delayed (advanced) versions versions x(n) x[n] = 0.5 [n + 2] + 1.5 [n 1] [n 2] + [n 4] + 0.75 [n 6] x[n] = x(2) [n + 2] + x(1) [n 1] + x(2) [n 2] 27 Basic Sequences: Relations General relationship between sequences and unit impulse x[n] = x(m) [n m] m = Unit step function and Unit Impulse 28 m = u ( n) = m =0 u (m) (n m) = (n m) Operations Operations on Sequences A single-input, single-output discrete-time system operates on sequence called the system operates on a sequence, called the input sequence, according some prescribed rules and develops another sequence, called the output sequence, with more desirable properties x[n] Discrete-time system y[n] Output sequence Input sequence 29 Basic Operations Product (modulation) operation: x[n] Modulator y[n] y[n] = x[n] w[n] w[n] An application is in forming a finite-length sequence from an infinite sequence from an infinite-length sequence sequence by multiplying the latter with a finite-length sequence called a window sequence Process called windowing 30 Basic Basic Operations Addition operation: Adder x[n] + y[n] y[n] = x[n] + w[n] w[n] Multiplication operation Multiplier A y[n] = A x[n] y[n] x[n] 31 Basic Operations Time-shifting operation: y[n] = x[n N ] where N is an integer If N > 0, it is delaying operation Unit delay x[n] z 1 y[n] y[n] = x[n 1] If N < 0, it is an advance operation x[n] Unit advance 32 z y[n] y[n] = x[n + 1] Basic Basic Operations Time-reversal (folding) operation: y[n] = x[n] Branching operation: Used to provide multiple copies of a sequence x[n] x[n] x[n] 33 Basic Operations Example - Consider the two following sequences of length defined for sequences of length 5 defined for 0 n 4: {a[n]} = {3 4 6 9 0} {b[n]} = {2 1 4 5 3} New sequences generated from the above two sequences by applying the basic two sequences by applying the basic operations are as follows: 34 Basic Basic Operations {c[n]} = {a[n] b[n]} = {6 4 24 45 0} {d [n]} = {a[n] + b[n]} = {5 3 10 4 3} {e[n]} = 3 {a[n]} = {4.5 6 9 13.5 0} 2 As pointed out by the above example, operations on two or more sequences can be carried out if all sequences involved are of carried out if all sequences involved are of same length and defined for the same range of the time index n 35 Combinations of Basic Operations Example - y[n] = 1x[n] + 2 x[n 1] + 3 x[n 2] + 4 x[n 3] (Youll see this many times in this course its an FIR filter) 36 A Useful Example Useful M-point moving-average system M 1 1 y[ n] = M x[ n k ] k =0 1 = 2 = ... = M M=3 = 1/ M An application: Consider application: Consider x[n] = s[n] + d[n], where s[n] is the signal corrupted by a noise d[n] 37 Simple moving average filter s[n] = 2[ n(0.9) n ], d[n] - random signal 8 d[n] s[n] x[n] 6 Amplitude x(n)=s(n)+d(n) 4 2 0 -2 0 10 20 30 Time index n 40 50 7 s[n] y[n] 6 5 Amplitude y(n) = filtered/smoothed version of x(n) 4 3 2 1 38 0 0 10 20 30 Time index n 40 50 Another Another Example n Accumulator - y[n] = x[ ] n 1 = = x[ ] + x[n] = y[n 1] + x[n] = The output y[n] at time instant n is the sum of the input sample x[n] at time instant n and the previous output y[n 1] at time instant instant n 1, which is the sum of all which is the sum of all previous input sample values from to n 1 The system cumulatively adds, i.e., it accumulates all input sample values 39 Accumulator as recursive filter We can implement the accumulator as x(n) + 1 y(n)=x(n)+y(n-1) y(n) Z-1 This is a discrete-time approximation to an integrator. Its also an example of a recursive (feedback) system Can modify using a lossy integrator 40 y(n)=x(n)+y(n-1) So. Weve seen a couple of practically useful systems but does it make sense to use them? Are they stable (and what do we mean by stable) What are their properties and how do we define these properties? Can we find better ways of doing these operations (smoothing and accumulating) .. and others Well answer these in the next few weeks 41
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USC - EE - 483
Black box=importantOutline (lecture #3) Properties of Discrete Time LTI SystemsLinearityTime (shift) invarianceCausalityStability Time Domain Characterization of DiscreteTime LTI SystemsTime LTI Systems Impulse response Parallel and cascade s
USC - EE - 483
Outline (lecture #4) Finite Dimensional LTI Systems Finite order linear difference equationsFilidiff FIR systems and their impulse response IIR systems and their impulse response Multiple solutions for IIR forms of LDEs Importance of initial cond
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Outline (lecture #5) The Discrete Time Fourier Transform(DTFT)(DTFT)DefinitionPeriodicity and symmetriesInverse DTFTConvergencePropertiesBlack box=important1Copyright 2005, S. K. Mitra, 2006-2010 R.LeahyJean Baptiste Joseph Fourier 1768-183
USC - EE - 483
Outline (lecture #6) The Discrete Time Fourier Transform(DTFT)(DTFT) - continued DTFT Properties LTI Systems and the Frequency Response Magnitude and Phase ResponseBlack box=important1Copyright 2005, S. K. Mitra, 2006-2010 R.LeahyImportant Pro
USC - EE - 483
OutlineOutline (lecture #7) The Discrete Time Fourier Transform(DTFT)(DTFT) - continued The importance of phase in signals andsystem Phase response and discontinuities Linear phase filters Group delayBlack box All pass filters=important1Cop
USC - EE - 483
OutlineOutline (lecture #8) The Discrete Fourier Transform (DFT)Forward and inverse DFT, definition and proofDFTExamplesDFT in matrix formBasis function of the DFTThe FFT AlgorithmBlack box=important1Copyright 2005, S. K. Mitra; 2006-2010 R.L
USC - EE - 483
OutlineOutline (lecture #9) The DFT and Other Unitary TransformsThThe unitary transformDiscrete Cosine TransformThe Haar Transform2D Transforms Relationship between DFT and DTFTbetween DFT and DTFT (coming soon .)Black box=important1Copyrig
USC - EE - 483
OutlineOutline (lecture #10) Relationship between the Fouriertransforms the Samping Theorem andtransforms, the Samping Theorem andSpectral Analysis I 1. Relationship between DTFT and DFT Relationship between FT and DTFT (samplingtheorem) Analog s
USC - EE - 483
OutlineOutline (lecture #11) Relationship between the Fouriertransforms the Samping Theorem andtransforms, the Samping Theorem andSpectral Analysis IIBlack box=important1Copyright 2005, S. K. Mitra; 2006-2010, R.LeahyPractical Spectral Analysis
USC - EE - 483
Lecture #12 OutlineZ-transform definitiondefinitionRegions of convergenceRational Z-transforms: poles and zerosInverse Z-transformBlack box=important1Copyright 2005, S. K. Mitra; 2006-2010, R.LeahyThe z-Transform A generalization of the DTFT d
USC - EE - 483
Lecture #13 Outline LTI systems: the impulse response and thesystems: the impulse response and thesystem function. Rational z-transforms and LTI systems Causality and stability in LTI systems System functions and the frequencyfunctions and the freq
USC - EE - 483
Lecture #14 OutlineA couple of random examplesGeneralized linear phase (again)Minimum and all pass filtersDifferentiators and Hilbert transformsDesign of general purpose filters .Black box=important1Copyright 2005, S. K. Mitra/R. Leahy 2006-2010
USC - EE - 483
Lecture #15 Outline FIR Filter design develop a numericalFilter designnumericaltechnique for determing FIR filter coefficientsto optimally approximate the desired frequencyresponse. Least squares design Design using windows Equiripple filter desi
USC - EE - 483
Lecture #16 Outline An extended, complete FIR Filter designextended complete FIR Filter designexampleBlack box=important#Copyright 2005, S. K. Mitra/ RM Leahy 2008-2010FIR Design Example Well conclude our study of FIR filter design withconclude
USC - EE - 483
Lecture #17 Outline IIR filter Designfilter Design Analog prototype filters and their designBlack box=important#Copyright 2005, S. K. Mitra/ RM Leahy 2008Analog Prototypes The set of analog filters that are mostset of analog filters that are mo
USC - EE - 483
Lecture #18 Outline IIR filter Designfilter Design The Bilinear transform design method Spectral transformations IIR design by numerical optimizationBlack box=importantCopyright 2005, S. K. Mitra/ RM Leahy 2009Bilinear Transformation Method Ide
USC - EE - 483
Lecture #19An Introduction to AdaptiveFilteringMotivationThe ideal transfer functionFiltering by power minimizationSteepest descentdescentLMS algorithmApplications#Copyright 2010 R. LeahyMotivationMotivation+-G(z)#Copyright 2010 R. Leahy
USC - EE - 483
Lecture #20 Outline: Quanztizationand Digital Filter Structures Quantization: Oversampling DACs Representation of binary signals and quantization noise Sigma-delta converters. Issues in implementation of digital filters: Efficiency (minimize number
USC - EE - 483
EE483: Digital Signal ProcessingInstructor: Professor LeahySAMPLE MIDTERM, FALL 2006Time Allowed: 80 MinutesPlease answer all questions. For partial credit you must show how you reach your solutions.Make sure that any sketches you draw are labelled a
Texas A&M - MATH - 410
Math 410.500Exam 1Solutions1. (15 pts.)(a) What is the Uniform Cauchy Criterion for uniform convergence of asequence of functions fn (x) on a set E ? Answer: For every &gt; 0there exists N so that m, n N implies that |fn (x) fm (x)| &lt; holds for all x
Texas A&M - MATH - 410
Math 410.500Exam 2, version ASolutions1. (20 pts.) Determine the interval of convergence of the power seriesn=1(1)n n (x 3)3n .n8Solution: Its probably easiest to use the ratio test:(1)n+1 (x 3)3(n+1) / n + 18n+1|x 3|3lim= limnn8(1)n (x 3)
Texas A&M - MATH - 410
Math 410.500Exam 2, version BSolutions1. (20 pts.) Determine the interval of convergence of the power seriesn=1(1)n n (x 1)3n .n6Solution: Its probably easiest to use the ratio test:(1)n+1 (x 1)3(n+1) / n + 16n+1|x 1|3lim= limnnn6(1)n (x
Texas A&M - MATH - 410
Math 410.500: answers to exam 11. (a) f is analytic on (a, b) i for each x0 (a, b) there exists a power serieskak (x x0 ) and a number R &gt; 0 so that f (x) =k=0ak (x x0 )kk=0on (x0 R, x0 + R). (It's too much to ask for that f equals a single1powe
Texas A&M - MATH - 410
Math 410.500: answers to exam 21. (a) U Rn is open if and only if for every a U there exists r &gt; 0 sothat Br (a) U .(b) The boundary of U is the set of all points x Rn so that r &gt; 0,both Br (x) U = and Br (x) U c = .(c) E Rn is connected if there doe
Texas A&M - MATH - 410
Math 410.500: answers to exam 31. (a) True. (This is Theorem 11.13.)(b) False. (Problem 2, section 11.2 has rst order partials, but is notdierentiable.)(c) False. (See example 11.11.)(d) True. (This is Theorem 11.15.)(e) False. (See example 11.18.)
Texas A&M - MATH - 410
Math 410.500Exam 12/16/051. (20 pts.) Short answer:(a) Dene: f is analytic on an open interval (a, b).(b) State Dirichlet's test for uniform convergence offk (x) gk (x)k=1on a set E .(c) State the Cauchy criterion for uniform convergence of a seq
Texas A&M - MATH - 410
Math 410.500Exam 23/30/05There are problems on both sides of this sheet!1. (18 pts.) Dene the term in italics :(a) U Rn is an open set.(b) What is the boundary of a set U Rn ?(c) E Rn is a connected set. (Either the denition from the bookor the de
Texas A&M - MATH - 410
Math 410.500Exam 34/29/05There are problems on both sides of this sheet!1. (10 pts.) True or false? (no explanation is needed)(a) If f is dierentiable at a Rn then f is continuous at a.(b) If all rst order partials of f exist at a then f is dierenti
Texas A&M - MATH - 410
Solutions to homework #1 a) False. You can use a divergent p series as a counterexample, or6.1.01k=1 k+ k+1 , which I showed in class was a diverging telescoping series.b) False. As a counterexample, take any convergent series whose termsare nonzer
Texas A&M - MATH - 410
Solutions to homework # 2 6.3.2d. A routine ratio test:limk(1 3 (2k 1) (2k + 1) / (2k + 2)!)(1 3 (2k 1) / (2k )!==limk2k + 1(2k + 1) (2k + 2)0,hence the series converges absolutely and therefore converges. Since everything is obviously positi
Texas A&M - MATH - 410
Solutions # 3x 7.1.1 a) On any interval [a, b], we have |x| max (|a| , |b|), so that n 0 cfw_xmax(|a|,|b|)max(|a|,|b|)x. For any &gt; 0, if&lt; n, then n 0 &lt; , thus nnconverges uniformly to zero on [a, b]. b) There are two things to show.1First, fn
Texas A&M - MATH - 410
Solutions to assignment # 4 7.3.1b: Writing out the series as 1+ x2 +32 x4 + x6 +34 x8 + x10 + , we seethat the sequence of roots of the coecients is 1, 0, 1, 0, 31/2 , 0, 1, 0 , 31/2 ,and so on, so the limit supremum is 3, and the radius of convergenc
Texas A&M - MATH - 410
Solutions to assignment 5 8.1.1: a)x y = x z + z y x z + z y &lt; 2 + 3 = 5 b)|x y x z| = |x (y z)| x y z x y + (z) x (y + z)&lt; 2 (3 + 4) = 14,where I used the Cauchy-Schwartz inequality in the rst step, andthen the triangle inequality. c)|x (
Texas A&M - MATH - 410
Assignment # 6Solutions: 8.3.5 a) The sketch is below: the region in question is the intersection ofthe two disks.b) U is relatively open in E1 , since its the intersection of an open set withE1 . c) U is relatively closed in E2 , since its the inter
Texas A&M - MATH - 410
Solutions to suggested problems for exam 2 9.1.1a: Obviously, we suspect that the limit is (0, 1). To prove thisfrom the denition, we need to produce N () so that k N () implies(1)1k , 1 k2 (0, 1) &lt; . This is equivalent to()1111=,+ 4 &lt; .k
Texas A&M - MATH - 367
Denition 38 Triangular region is the union of a triangle and its interior. Denition 39 A polygonal region is the union of a nite number of triangular regions such that if two triangular regions intersect, their intersection is an edge or vertex of both. D
Texas A&M - MATH - 367
Homework 7 Identify the congruent triangles and show they are congruent. Note a picture can have more than one pair of triangles.1.2.3.4.5.6.7.8.12
Texas A&M - MATH - 367
An axiomatic system example. Assume that a club of two or more students is organized into committees in such a way that each of the following conditions are satised. a) Every committee is a set of one or more students. b) For each pair of students, there
Texas A&M - MATH - 367
Denition 28 Parallel Two lines are parallel in a plane if they dont intersect. Postulate 16 Parallel Postulate Through a point not on a given line there is exactly one parallel to the line. Postulate 17 Lobachevsky and Bolya Postulate Through a point not
Texas A&M - MATH - 367
Theorem 29 If A and B are equidistant from P and Q then every point between A and B has the same property. Theorem 30 If a line L contains the midpoint of P Q and contains another point which is equidistant from P and Q, then L P Q.Theorem 40 Through a g
Texas A&M - MATH - 367
Denition 33 polygon A polygon is the union of n segments in a plane, intersecting at and only at their endpoints, such that exactly two segments contain each endpoint and no two consecutive segments are on the same line. We are going to assume that any po
Texas A&M - MATH - 367
Denition 12 Segment If A and B are two points then the segment between them is the set of points A X B , for all X in S , along with A and B . It is written AB Denition 13 Ray Is the set of points C on AB with A not between B and C . It is written AB . De
Texas A&M - MATH - 367
Theorem 14 Segment construction [3.6.C-2] Given a segment AB and a ray CD, there is exactly one point E on CD such that AB CE . = Theorem 15 Segment Addition [3.6.C-3] If A B C and D E F and AB DE and = BC EF then AC DF . = = Denition 22 Convex A set G is
Texas A&M - MATH - 367
Postulate 12 Space separation postulate [4.5.ss-1] Given a plane in space. The set of all points that do not lie in the plane is the union of two sets S1 , S2 (the book uses H1 and H2 we will reserve those for half-planes) such that each of the sets is co
Texas A&M - MATH - 367
Denition 7 Logical System consists of undened terms, denitions, assumptions and theorems. The undened terms for geometry are set, point, line, plane. Denition 8 One-to-one correspondence is if two sets have the same number of elements. If two sets have a
Texas A&M - MATH - 367
Denition 1 Negation If p is a statement, the statement p is the negation of p. Example 1 Form the negation of the following statements. a The moon is rising. b ABC is a remote interior angle. c Point C is between points A and B . d m 3 = 25 Solution a The
Texas A&M - MATH - 251
Answers to exam 2, version A1. In general, the directional derivative of f at (a, b) in the direction of a unitvector u is Du f (a, b) = f (a, b) u. So,=exy + xyexy , x2 exy=f2e, eat (1, 1). The vector going from (1, 1) to (2, 3) is 1, 2 , but thi
Texas A&M - MATH - 251
Answers to exam 3, version A 1. If D is the region bounded by x and x2 , put in = ky to getx = =which integrates out to 5 . 8Dx dA dA D1 x xky dy dx 0 x2 , 1 x 2 ky dy dx 0 x2. The solid that we're integrating over has its top as part of a spherical
Texas A&M - MATH - 251
Exam 1, version A 9/19/08 1. For the function f (x, y ) = 16 4x2 + 9 y 2 , (a) (6 pts.) sketch the domain. (b) (6 pts.) nd the range. 2. (8 pts.) Find two unit vectors perpendicular to both 2, 1, 1 and 1, 2, 1 . 3. (10 pts.) Find the equation of the plane
Texas A&M - MATH - 251
Math 251.512Exam 2, version A10/15/081. (11 pts.) If f (x, y ) = xexy , nd the directional derivative of f at (1, 1) in thedirection from (1, 1) to (2, 3).2. (11 pts.) FindRxdA, where R is the rectangle 1 x 2, 0 y 4.1 + 2yx, use dierentials to
Texas A&M - MATH - 251
Exam 3, version A 11/12/08 Conversion from spherical to cartesian coordinates:Math 251.512x = sin cos y = sin sin z = cos dV =2 sin d d d1. (10 pts.) Determine x of the center of mass of a plate bounded by y = x and y = x2 , if density is proportional
Texas A&M - MATH - 251
Math 251.504Exam 1, version ASolutions1. (11 pts.) Find the equation of the plane containing the points (0, 1, 1), (2, 1, 2), and (3, 0, 1).Answer: To nd the equation of a plane, we need a point in the plane and a vector normal tothe plane. Weve alre
Texas A&M - MATH - 251
Math 251.504Exam 1, version BSolutions1. (11 pts.) Find the equation of the plane containing the points (1, 1, 1), (1, 1, 0), and (3, 0, 1).Answer: To nd the equation of a plane, we need a point in the plane and a vector normal tothe plane. Weve alre
Texas A&M - MATH - 251
Math 251.504Exam 2, version ASolutions1. (10 pts.) Evaluate D x dA, where D is the triangular region withvertices (0, 0), (1, 1), and (1, 4). Solution: In the order dy dx, thedouble integral becomes1 1 4xxy |y=4x dxx dy dx =y =x0x0 1=3x2 dx
Texas A&M - MATH - 251
Math 251.504Exam 2, version BSolutions1. (10 pts.) Evaluate D x dA, where D is the triangular region withvertices (0, 0), (1, 1), and (1, 3). Solution: In the order dy dx, thedouble integral becomes1 1 3xxy |y=3x dxx dy dx =y =x0x0 1=2x2 dx
Texas A&M - MATH - 151
11.1: VectorsDenitions:vector:additionscalar multiplicationsubtractionmagnitude:unit vector:i and j:1Examples:Given the vectors a =&lt; 3, 5 &gt; and b starts at the point (1, 1) and ends at the point (1, 3),write i + j in terms of a and b.Given a
Texas A&M - MATH - 151
11.2: Dot ProductDenitions:The dot product of the vectors a and b is given byDot Product computation formulaFrom the denition, it follows that the angle between two vectors is given bya and b are orthogonal if and only ifOrthogonal complementsScal
Texas A&M - MATH - 151
11.3: Vector Functions and Parametrized CurvesDenitions:(Recall) function:Vector Valued function:Parametrized Curve:Eliminating the ParameterVector and Parametric Equations of a Line1Examples:Given the curve parametrized by r(t) = (t2 + 1)i + (t
Texas A&M - MATH - 151
12.1/2.2: Intro to Calculus and LimitsGoal #1: To nd the slope of a line tangent to a curve at a given point.Concept of a Limit (Maplet):Innite Limits and Vertical Asymptotes:1Examples:x2 + 1x 1 x 1limOn Beyond Average: Find the vertical asympto
Texas A&M - MATH - 151
12.3: Analytic Computation of LimitsProperties of Limits: (pp 91-93. Basis for the techniques used in the following examples.)Examples:lim x3 3x2 + 1x 1limx42x + 8x2 + x 12lim r(t) where r(t) =t25t3 + 4t3i+t2 4t2j1Squeeze Theorem: If g