ω = radians per sec, t = 3 sec
|
|
- Ιάσων Μιχαηλίδης
- 7 χρόνια πριν
- Προβολές:
Transcript
1 Secion. Linear and Angular Speed 7. From exercise, =. A= r A = ( 00 ) (. ) = 7,00 in 7. Since 7 is in quadran IV, he reference angle is = =. In quadran IV, he cosine is posiive. Thus, 7 cos = cos =. Convering 7 o degrees, we have 7 7 = ( 80 ) =. The reference angle is 80 =. Thus, 7 cos = cos = cos =. is coerminal wih 7 + = + =. Since 7 is in quadran III, he reference angle is 7 7 = =. In quadran III, he sine is negaive. Thus, 7 sin = sin = sin =. Convering 7 o degrees, we have 7 7 = ( 80 ) = 0. The reference angle is 0 80 = 0. Thus, 7 sin = sin = sin 0 = sin 0 =. is coerminal wih =. So an = an = 0 Convering o degrees, we have 80 = 0. The reference angle is 0º 0º = 80º. Thus an = an 0 = an80 = 0. 0., ; sins = Recall ha sin = and in quadran II, sin s is posiive. Therefore, sin = sin =, and hus, s =. Secion. Linear and Angular Speed. The circumference of he uni circle is. radian per, = = =. The circumference of he uni circle is. v = uni per, s = unis s v = = =. r = 0 cm, (a) radian per, = = = (b) s = r s = 0 = 0 cm r 0 0 (c) v = v = = = cm per. r = 0 cm,. (a) (b) (c) radian per, = 0 = = = 0 0 s = r s = 0 = cm r 0 v = v = = = cm per per, = = = Copyrigh 0 Pearson Educaion, Inc.
2 8 Chaper Radian Measure and he Uni Circle per min, = min = = =, = 8 = = = radian per 8 8 =, = 0 = = radian per 0 0 = radian, radian per min 7 = = 7 7 = = = min =, radian per min 8 8 = = = 7 = = min 8. =.87, = radian per.7. =., = per radian per min, =.88 min 0.07 =.88 = per min, =.7 min. =.7 = r = m, per v = = 8 m per. r = 8 m, per 7 v = 8 = cm per 7. v = m per, r = m = ω per v = 8 f per, r = f 8 = ω per. v = 07.7 m per, r = 7 m 07.7 = 7ω per 7 0. r =. cm, 0.7 radian per v = cm per. r = cm, per, = s = ( ) = 8 cm. r = yd, per, = s = ( ) = yd. s = cm, r = cm, radian per = = = =. s = m, r = m, per = = = = Copyrigh 0 Pearson Educaion, Inc.
3 Secion. Linear and Angular Speed. s = km, r = km, = = ω = 8ω = radian per 8. s = 8 m, r = m, = 8 = ω 8 8 = ω = radian per 8 7. The hour hand of a clock moves hrough an angle of (one complee revoluion) in hours, so = = radian per hr. The ond hand makes one revoluion per minue. Each revoluion is, so we = per min. There have are 0 onds in min, so = 0 0 radian per.. The minue hand makes one revoluion per hour. Each revoluion is, so we = per hr. There have are 0 minues in hour, so = 0 0 radian per min. 0. The line makes 00 revoluions per minue. Each revoluion is, so we have 00 = 00 per min. The minue hand of a clock moves hrough an angle of in 0 min, and a he ip of he minue hand, r = 7 cm, so we have r 7 ( ) 7 v = v = = cm per min 0 0. The ond hand makes one revoluion per minue. Each revoluion is, and a he ip of he ond hand, r = 8 mm, so we have v = 8 = mm per min. There are 0 onds in min, so v = = mm per. 0. The flywheel making roaions per min = 8 urns hrough an angle in minue wih r = m. So, r 8 ( ) v = v = = 8 m per min. The poin on he read of he ire is roaing imes per min. Each roaion is. Thus, we have = 70 per min. Since, we have v = 8 70 = 0 cm per min.. A 00 roaions per min, he propeller urns hrough an angle of = 00( ) = 000 in min wih r = =. m, we have r.( 000 ) v = v = = 00 m per min.. The poin on he edge of he gyroscope is roaing 80 imes per min. Each roaion is. 80 = 0 per min, so v = 8 0 =,880 cm per min 7. A revoluions per minue, he bicycle ire is moving ( ) = 0 per min. This is he angular velociy ω. The linear velociy of he bicycle is =.0 0 = 0 in. per min. Conver his o miles per hour: 0 in. 0 min f mi v = min hr in. 80 f. mph Mars will make one full roaion (of ) during he course of one day. Thus, hr. hr 0. radian. (a) (b) = = radian radian per day = radian per hr = radian per hr 80 Copyrigh 0 Pearson Educaion, Inc.
4 0 Chaper Radian Measure and he Uni Circle (c) v = (,000,000) 7,000 mph (a) The earh complees one revoluion per day, so i urns hrough = in ime = day = hr. So, we have: = per day = radian per hr = radian per hr (b) A he poles, r = 0, so = 0. (c) A he equaor, r = 00 km. So, v = 00 =,800 km per day,800 = km per hr km per hr (d) Salem roaes abou he axis in a circle of radius r a an angular velociy per day. r sin = 00 r = 00sin = 00 = 00 km v = 00 ( ) 00 km per day 8,000 km per day 00 v = km per hr 77 km per hr 00 km per hr. (a) Since s = cm of bel go around in = 8, he linear velociy is s 8 v = v = =. cm per 8 (b) Since he -cm bel goes around in 8, we have = (.) ω 8 8 = (.) ω 8 0. radian per.. The larger pulley roaes imes in or imes per. Thus, is angular velociy is = per. The 8 linear velociy of he bel is v = = cm per 8 To find he angular velociy of he smaller pulley, use v = cm per and r = 8 cm. = 8ω = per 8 8. = 0 per min 0 = per 0 7 = per 7. = r r =..7 cm 7. Le s = he lengh of he rack on he arc. Firs, convering 0 o, we have 0 = 0 radian = radian. 80 The lengh of rack, s, is he arc lengh when = and r = 800 f. s = r s = 800 = 00 f (coninued on nex page) Copyrigh 0 Pearson Educaion, Inc.
5 Chaper Review Exercises (coninued) Expressing he velociy v = 0 mph in f per, we have 0 v = 0 mph = mi per 00 ( 0) 80 = f per = f per 00 s 00 v = = = = =. In one minue, he propeller makes 000 revoluions. Each revoluion is, so we have 000( ) = 0,000 per min. There are 0 in a minue, so 0, =. per 0. r = f; per Chaper v = = f per Review Exercises. A cenral angle of a circle ha inerceps an arc of lengh imes he radius of he circle has a measure of.. (a) Since.7 and., we have < <. Thus, he erminal side is in quadran II. (b) Since. and.7, we have < <. Thus, he erminal side is in quadran III. (c) Since.7 and., we have > >. Thus, he erminal side is in quadran III. (d) Since.8 and 7. we have < 7 <. Thus, he erminal side is in quadran I.. To find a coerminal angle, add or subrac muliples of. Three of he many possible answers are +, +, and +.. To find a coerminal angle, add or subrac muliples of. Since n represens any ineger, he expression + n generaes all coerminal angles wih an angle of radian.. = radian = radian = 0 radian = 80 7 = 7 radian = 80 0 = 0 radian = = 800 radian = = 00 radian = 80 = = = = = = 80 = = = = = = 7 7. Since = roaion, we have 0 = ( ) =. Thus, s = r s = = in. Copyrigh 0 Pearson Educaion, Inc.
16. 17. r t te 2t i t 1. 18 19 Find the derivative of the vector function. 19. r t e t cos t i e t sin t j ln t k. 31 33 Evaluate the integral.
SECTION.7 VECTOR FUNCTIONS AND SPACE CURVES.7 VECTOR FUNCTIONS AND SPACE CURVES A Click here for answers. S Click here for soluions. Copyrigh Cengage Learning. All righs reserved.. Find he domain of he
Διαβάστε περισσότερα10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations
//.: Measures of Angles and Rotations I. Vocabulary A A. Angle the union of two rays with a common endpoint B. BA and BC C. B is the vertex. B C D. You can think of BA as the rotation of (clockwise) with
Διαβάστε περισσότεραis like multiplying by the conversion factor of. Dividing by 2π gives you the
Chapter Graphs of Trigonometric Functions Answer Ke. Radian Measure Answers. π. π. π. π. 7π. π 7. 70 8. 9. 0 0. 0. 00. 80. Multipling b π π is like multipling b the conversion factor of. Dividing b 0 gives
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραSection 8.2 Graphs of Polar Equations
Section 8. Graphs of Polar Equations Graphing Polar Equations The graph of a polar equation r = f(θ), or more generally F(r,θ) = 0, consists of all points P that have at least one polar representation
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραCHAPTER 12: PERIMETER, AREA, CIRCUMFERENCE, AND 12.1 INTRODUCTION TO GEOMETRIC 12.2 PERIMETER: SQUARES, RECTANGLES,
CHAPTER : PERIMETER, AREA, CIRCUMFERENCE, AND SIGNED FRACTIONS. INTRODUCTION TO GEOMETRIC MEASUREMENTS p. -3. PERIMETER: SQUARES, RECTANGLES, TRIANGLES p. 4-5.3 AREA: SQUARES, RECTANGLES, TRIANGLES p.
Διαβάστε περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραVolume of a Cuboid. Volume = length x breadth x height. V = l x b x h. The formula for the volume of a cuboid is
Volume of a Cuboid The formula for the volume of a cuboid is Volume = length x breadth x height V = l x b x h Example Work out the volume of this cuboid 10 cm 15 cm V = l x b x h V = 15 x 6 x 10 V = 900cm³
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραThe Euler Equations! λ 1. λ 2. λ 3. ρ ρu. E = e + u 2 /2. E + p ρ. = de /dt. = dh / dt; h = h( T ); c p. / c v. ; γ = c p. p = ( γ 1)ρe. c v.
hp://www.nd.ed/~gryggva/cfd-corse/ The Eler Eqaions The Eler Eqaions The Eler eqaions for D flow: + + p = x E E + p where Define E = e + / H = h + /; h = e + p/ Gréar Tryggvason Spring 3 Ideal Gas: p =
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραLecture 12 Modulation and Sampling
EE 2 spring 2-22 Handou #25 Lecure 2 Modulaion and Sampling The Fourier ransform of he produc of wo signals Modulaion of a signal wih a sinusoid Sampling wih an impulse rain The sampling heorem 2 Convoluion
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότερα( ) ( ) ( ) Fourier series. ; m is an integer. r(t) is periodic (T>0), r(t+t) = r(t), t Fundamental period T 0 = smallest T. Fundamental frequency ω
Fourier series e jm when m d when m ; m is an ineger. jm jm jm jm e d e e e jm jm jm jm r( is periodi (>, r(+ r(, Fundamenal period smalles Fundamenal frequeny r ( + r ( is periodi hen M M e j M, e j,
Διαβάστε περισσότεραAREAS AND LENGTHS IN POLAR COORDINATES. 25. Find the area inside the larger loop and outside the smaller loop
SECTIN 9. AREAS AND LENGTHS IN PLAR CRDINATES 9. AREAS AND LENGTHS IN PLAR CRDINATES A Click here for answers. S Click here for solutions. 8 Find the area of the region that is bounded by the given curve
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραPrinciples of Mathematics 12 Answer Key, Contents 185
Principles of Mathematics Answer Ke, Contents 85 Module : Section Trigonometr Trigonometric Functions Lesson The Trigonometric Values for θ, 0 θ 60 86 Lesson Solving Trigonometric Equations for 0 θ 60
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραAnswers to practice exercises
Answers to practice exercises Chapter Exercise (Page 5). 9 kg 2. 479 mm. 66 4. 565 5. 225 6. 26 7. 07,70 8. 4 9. 487 0. 70872. $5, Exercise 2 (Page 6). (a) 468 (b) 868 2. (a) 827 (b) 458. (a) 86 kg (b)
Διαβάστε περισσότεραSampling Basics (1B) Young Won Lim 9/21/13
Sampling Basics (1B) Copyright (c) 2009-2013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any
Διαβάστε περισσότερα2 2 2 The correct formula for the cosine of the sum of two angles is given by the following theorem.
5 TRIGONOMETRIC FORMULAS FOR SUMS AND DIFFERENCES The fundamental trignmetric identities cnsidered earlier express relatinships amng trignmetric functins f a single variable In this sectin we develp trignmetric
Διαβάστε περισσότεραChapter 7 Analytic Trigonometry
Chapter 7 Analytic Trigonometry Section 7.. Domain: { is any real number} ; Range: { y y }. { } or { }. [, ). True. ;. ; 7. sin y 8. 0 9. 0. False. The domain of. True. True.. y sin is. sin 0 We are finding
Διαβάστε περισσότεραTRIGONOMETRIC FUNCTIONS
Chapter TRIGONOMETRIC FUNCTIONS. Overview.. The word trigonometry is derived from the Greek words trigon and metron which means measuring the sides of a triangle. An angle is the amount of rotation of
Διαβάστε περισσότεραChapter 5. Exercise 5A. Chapter minor arc AB = θ = 90 π = major arc AB = minor arc AB =
Chapter 5 Chapter 5 Exercise 5. minor arc = 50 60.4 0.8cm. major arc = 5 60 4.7 60.cm. minor arc = 60 90 60 6.7 8.cm 4. major arc = 60 0 60 8 = 6 = cm 5. minor arc = 50 5 60 0 = cm 6. major arc = 80 8
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότερα4.5 SUMMARY OF CURVE SKETCHING. Click here for answers. Click here for solutions. y cos x sin x. x 2 x 3 4. x 1 x y x 3 x
SECTION.5 SUMMARY OF CURVE SKETCHING.5 SUMMARY OF CURVE SKETCHING A Click here for answers. S Click here for solutions. 9. 8 Use the guidelines of this section to sketch the curve. cos sin. 5. 6 8 7. cot..
Διαβάστε περισσότεραLeaving Certificate Applied Maths Higher Level Answers
0 Leavin Certificate Applied Maths Hiher Level Answers ) (a) (b) (i) r (ii) d (iii) m ) (a) 0 m s - 9 N of E ) (b) (i) km h - 0 S of E (ii) (iii) 90 km ) (a) (i) 0 6 (ii) h 0h s s ) (a) (i) 8 m N (ii)
Διαβάστε περισσότεραMock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
Διαβάστε περισσότεραTrigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραSpherical Coordinates
Spherical Coordinates MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Spherical Coordinates Another means of locating points in three-dimensional space is known as the spherical
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότερα26 28 Find an equation of the tangent line to the curve at the given point Discuss the curve under the guidelines of Section
SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5. THE NATURAL LOGARITHMIC FUNCTION A Click here for answers. S Click here for solutions. 4 Use the Laws of Logarithms to epand the quantit.. ln ab. ln c. ln
Διαβάστε περισσότεραTrigonometry 1.TRIGONOMETRIC RATIOS
Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραSection 7.7 Product-to-Sum and Sum-to-Product Formulas
Section 7.7 Product-to-Sum and Sum-to-Product Fmulas Objective 1: Express Products as Sums To derive the Product-to-Sum Fmulas will begin by writing down the difference and sum fmulas of the cosine function:
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραSolution to Review Problems for Midterm III
Solution to Review Problems for Mierm III Mierm III: Friday, November 19 in class Topics:.8-.11, 4.1,4. 1. Find the derivative of the following functions and simplify your answers. (a) x(ln(4x)) +ln(5
Διαβάστε περισσότερα= e 6t. = t 1 = t. 5 t 8L 1[ 1 = 3L 1 [ 1. L 1 [ π. = 3 π. = L 1 3s = L. = 3L 1 s t. = 3 cos(5t) sin(5t).
Worked Soluion 95 Chaper 25: The Invere Laplace Tranform 25 a From he able: L ] e 6 6 25 c L 2 ] ] L! + 25 e L 5 2 + 25] ] L 5 2 + 5 2 in(5) 252 a L 6 + 2] L 6 ( 2)] 6L ( 2)] 6e 2 252 c L 3 8 4] 3L ] 8L
Διαβάστε περισσότερα6.003: Signals and Systems
6.3: Signals and Sysems Modulaion December 6, 2 Communicaions Sysems Signals are no always well mached o he media hrough which we wish o ransmi hem. signal audio video inerne applicaions elephone, radio,
Διαβάστε περισσότεραPg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Διαβάστε περισσότεραAppendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)
Aenix Aenix A: The equaion o he sock rice. The soluion egins wih Eq..5 rom he ex, which we reea here or convenience as Eq.A.: [ [ E E X, A. c α where X u ε, α γ, an c α y AR. Take execaions o Eq. A. as
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότερα7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple
A/ Ονόματα και ένα παράδειγμα 1 Present Simple 7 Present PERFECT Simple 2 Present Continuous 8 Present PERFECT Continuous 3 Past Simple (+ used to) 9 Past PERFECT Simple she eats she is eating she ate
Διαβάστε περισσότεραMATH 150 Pre-Calculus
MATH 150 Pre-Calculus Fall, 014, WEEK 11 JoungDong Kim Week 11: 8A, 8B, 8C, 8D Chapter 8. Trigonometry Chapter 8A. Angles and Circles The size of an angle may be measured in revolutions (rev), in degree
Διαβάστε περισσότεραUNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level * 0 5 1 0 3 4 8 7 8 5 * MATHEMATICS (SYLLABUS D) 4024/21 Paper 2 May/June 2013 Candidates answer on the
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραCalculus and Differential Equations page 1 of 17 CALCULUS and DIFFERENTIAL EQUATIONS
alculus and Diffrnial Equaions pag of 7 ALULUS and DIFFERENTIAL EQUATIONS Th following 55 qusions concrn calculus and diffrnial quaions. In his vrsion of h am, h firs choic is always h corrc on. In h acual
Διαβάστε περισσότεραUniversity of Washington Department of Chemistry Chemistry 553 Spring Quarter 2010 Homework Assignment 3 Due 04/26/10
Universiy of Washingon Deparmen of Chemisry Chemisry 553 Spring Quarer 1 Homework Assignmen 3 Due 4/6/1 v e v e A s ds: a) Show ha for large 1 and, (i.e. 1 >> and >>) he velociy auocorrelaion funcion 1)
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραWritten Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
Διαβάστε περισσότερα3.4. Click here for solutions. Click here for answers. CURVE SKETCHING. y cos x sin x. x 1 x 2. x 2 x 3 4 y 1 x 2. x 5 2
SECTION. CURVE SKETCHING. CURVE SKETCHING A Click here for answers. S Click here for solutions. 9. Use the guidelines of this section to sketch the curve. cos sin. 5. 6 8 7 0. cot, 0.. 9. cos sin. sin
Διαβάστε περισσότεραAdvanced Subsidiary Unit 1: Understanding and Written Response
Write your name here Surname Other names Edexcel GE entre Number andidate Number Greek dvanced Subsidiary Unit 1: Understanding and Written Response Thursday 16 May 2013 Morning Time: 2 hours 45 minutes
Διαβάστε περισσότερα2 2 2 The correct formula for the cosine of the sum of two angles is given by the following theorem.
5 TRIGONOMETRIC FORMULAS FOR SUMS AND DIFFERENCES The fundamental trignmetric identities cnsidered earlier express relatinships amng trignmetric functins f a single variable In this sectin we develp trignmetric
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραDerivations of Useful Trigonometric Identities
Derivations of Useful Trigonometric Identities Pythagorean Identity This is a basic and very useful relationship which comes directly from the definition of the trigonometric ratios of sine and cosine
Διαβάστε περισσότερα1. For each of the following power series, find the interval of convergence and the radius of convergence:
Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.
Διαβάστε περισσότεραMathCity.org Merging man and maths
MathCity.org Merging man and maths Exercise 10. (s) Page Textbook of Algebra and Trigonometry for Class XI Available online @, Version:.0 Question # 1 Find the values of sin, and tan when: 1 π (i) (ii)
Διαβάστε περισσότεραUNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level * 6 3 1 7 7 7 6 4 0 6 * MATHEMATICS (SYLLABUS D) 4024/21 Paper 2 October/November 2013 Candidates answer
Διαβάστε περισσότεραSimilarly, we may define hyperbolic functions cosh α and sinh α from the unit hyperbola
Universit of Hperbolic Functions The trigonometric functions cos α an cos α are efine using the unit circle + b measuring the istance α in the counter-clockwise irection along the circumference of the
Διαβάστε περισσότεραthe total number of electrons passing through the lamp.
1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΕΤΑΙΡΕΙΑ IΔ ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ 2013 21 ΑΠΡΙΛΙΟΥ 2013 Β & Γ ΛΥΚΕΙΟΥ. www.cms.org.cy
ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΕΤΑΙΡΕΙΑ IΔ ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ 2013 21 ΑΠΡΙΛΙΟΥ 2013 Β & Γ ΛΥΚΕΙΟΥ www.cms.org.cy ΘΕΜΑΤΑ ΣΤΑ ΕΛΛΗΝΙΚΑ ΚΑΙ ΑΓΓΛΙΚΑ PAPERS IN BOTH GREEK AND ENGLISH ΚΥΠΡΙΑΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραwave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:
3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved. 3.0 - Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition,
Διαβάστε περισσότεραAnti-aliasing Prefilter (6B) Young Won Lim 6/8/12
ni-aliasing Prefiler (6B) Copyrigh (c) Young W. Lim. Permission is graned o copy, disribue and/or modify his documen under he erms of he GNU Free Documenaion License, Version. or any laer version published
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραCambridge International Examinations Cambridge International General Certificate of Secondary Education
Cambridge International Examinations Cambridge International General Certificate of Secondary Education GREEK 0543/04 Paper 4 Writing For Examination from 2015 SPECIMEN PAPER Candidates answer on the Question
Διαβάστε περισσότεραFINAL TEST B TERM-JUNIOR B STARTING STEPS IN GRAMMAR UNITS 8-17
FINAL TEST B TERM-JUNIOR B STARTING STEPS IN GRAMMAR UNITS 8-17 Name: Surname: Date: Class: 1. Write these words in the correct order. /Γράψε αυτέσ τισ λέξεισ ςτη ςωςτή ςειρά. 1) playing / his / not /
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραPaper Reference. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced. Thursday 11 June 2009 Morning Time: 1 hour 30 minutes
Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Thursday 11 June 2009 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae
Διαβάστε περισσότερα6.003: Signals and Systems. Modulation
6.3: Signals and Sysems Modulaion December 6, 2 Subjec Evaluaions Your feedback is imporan o us! Please give feedback o he saff and fuure 6.3 sudens: hp://web.mi.edu/subjecevaluaion Evaluaions are open
Διαβάστε περισσότεραReview Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -
Διαβάστε περισσότεραDifferential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραGeodesic Equations for the Wormhole Metric
Geodesic Equations for the Wormhole Metric Dr R Herman Physics & Physical Oceanography, UNCW February 14, 2018 The Wormhole Metric Morris and Thorne wormhole metric: [M S Morris, K S Thorne, Wormholes
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότερα*2354431106* GREEK 0543/02 Paper 2 Reading and Directed Writing May/June 2009
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *2354431106* GREEK 0543/02 Paper 2 Reading and Directed Writing May/June 2009 1 hour 30 minutes
Διαβάστε περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότερα6.4 Superposition of Linear Plane Progressive Waves
.0 - Marine Hydrodynamics, Spring 005 Lecture.0 - Marine Hydrodynamics Lecture 6.4 Superposition of Linear Plane Progressive Waves. Oblique Plane Waves z v k k k z v k = ( k, k z ) θ (Looking up the y-ais
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότερα