13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da.

Size: px
Start display at page:

Download "13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da."

Transcription

1 CHAPTE 3, MULTIPLE INTEGALS Definition. 3. Double Integrl over ectngle A function f(x, y) is integrble on rectngle [, b] [c, d] if there is number I such tht for ny given ɛ>0thereisδ>0 such tht, fir ny prtition P = { = x 0,x,,x n = b} on [, b] Q = {c = y 0,y,,y m = d} on [c, d] with P, Q <δfor ny set of representtives (x ij,y ij ) [x i,x i ] [y j,y j ] i,j f(x ij,y ij ) i j I <ɛ. I is written s f(x, y)da. Theorem. Every continuous function is integrble on rectngle. Theorem. If f is continuous on,then d b f(x, y)da = [ f(x, y)dx]dy = c b d [ f(x, y)dy]dx. c Proof. Let F (y) = b f(x, y)dx, thenf is continuous function on [c, d]. Now for ny prtition {y 0,,y m } of [c, d] nd ny prtition {x 0,,x n } of [, b] d b c f(x, y)dxdy = d c F (y)dy = m yj y j F (y)dy = m F (yj ) y j, the lst equlity comes from men vlue theorem of integrl. Agin b n F (yj )= f(x, yj )dx = f(x ij,y j ) x i. For given ɛ>0thereisδ>0 such tht for ll y j <δ, x i <δwe hve f(x, y)da f(x ij,yj ) x i y j <ɛ, i,j hence Typeset by AMS-TEX

2 2 CHAPTE 3, MULTIPLE INTEGALS f(x, y)da d b Exmple. () =[, 3] [ 2, ],f(x, y) =4x 3 +6xy 2, (2) π π cos x cos ydydx, (3) π [ey +sinx]dxdy. c f(x, y)dxdy <ɛ. 3.2 Double Integrl over More Generl egion Definition. f(x, y) is integrble over region with integrl I if for ny given ɛ>0 the is δ>0 such tht for ny prtition with y j <δ, x i <δnyiemnnsumoverthe inner prtition S stisfies S I <ɛ. Here the inner prtition mens those boxes which is contined in. Theorem. Let be the region defined by {(x, y) :0 y φ(x), x b} where φ is continuous function on [, b] nd f(x, y) is continuous function. Then f is integrble over. Theorem. nd f s bove, then b φ(x) f(x, y)da = f(x, y)dydx. Properties of Double Integrls. () cf(x, y)da = c f(x, y)da, (2) [f(x, y) +g(x, y)]da = f(x, y)da + g(x, y)da, (3) If m f(x, y) M then ma() f(x, y)da MA(), (4) If = 2 then f(x, y)da = f(x, y)da + 2 f(x, y)da. 0 Exmple. () xy2 da, x 3 y x, (2) 6x +2y2 da, y 2 x 2 y, (3) 2 0 y/2 yex3 dxdy, (4) Are of the region y =2+ 4 x2,x= ±(+y 2 ), ( 2, 3), ( 2, ), ( 2, ), (2, 3), (2, ), (2, ). Exmple. () f(x, y) =+xy, [0, 2] [0, ], (2) y = x 2 2x, y = x, (3) 0 z x, x 2 + y 2 4, (4) 2y z 6,y = x 2,y =2 x Are, Volume by Double Inetrl

3 CHAPTE 3, MULTIPLE INTEGALS Double Integrsl in Polr Coordintes β φ(θ) α 0 f(r, θ)rdrdθ. Exmple. () 0 z 25 x 2 y 2, (2) r =,r =2+cosθ, (3) x 2 + y 2 + z 2 4, (x ) 2 + y 2, (4) r 2 z 8 r 2, (5) e x2 dx. 0 Center of Mss. 3.5 Appliction of Double Integrls Centroid: x = M Mss: M = Exmple. () δ =, 0 r, 0 θ π, (2) δ = kx 2,x 2 y x +2, (3) δ = kr,0 r, 0 θ π 2. First Pppus theorem. V =2πrA. δ(x, y)da, xδ(x, y)da, y = M yδ(x, y)da. Exmple.x = b + r cos θ, y = r sin θ, r c, 0 θ 2π, < c < b. Second Pppus theorem. A =2πrs. Exmple. () ( cos t, sin t), 0 t π, (2) b + cos t, sin t), 0 t 2π. Moment of Initi. I o = I x = r 2 δ(x, y)da, y 2 δ(x, y)da, I y x 2 δ(x, y)da.

4 4 CHAPTE 3, MULTIPLE INTEGALS Exmple. () I x = y 4 (2) r, δ,i o = π4 2. y 4 y 2 dxdy = 4 7,I x = /5 /5 y2 dxdy = 4 5, 3.6 Triple Integrl in ectngulr Coordintes Exmple. () =[, ] [2, 3] [0, ],f(x, y, z) =xy + yz, (2) δ = z, =< (0, 0, 0), (2, 0, 0), (0,, 0), (0, 0, 4) >, (3) x 2 + y 2 z y +2, (x,y) dzdxdy, (4) Centroid of 0 z x, x = y 2, zxy, yzx, xzy, (5) x 2 + y 2 z y +2, dxdzdy. 3.7 Triple Integrls in Cylindricl nd Sphericl Coordintes Cylindricl Coordintes. f(r, θ, z)rdrdθdz. Exmple. () Centroid of x 2 + y 2 + z 2, 0 x, y, z, (2) b(x 2 + y 2 ) z h. Sphericl Coordintes. f(ρ, φ, θ)ρ 2 sin φdρdφdθ. Exmple. () Volume nd I z of x 2 + y 2 + z 2, (2) z of φ π,ρ 2 cos φ, 6 (3) Exercise 47,48. (L ρ cos φ)ρ 2 sin φ (L ρ cos φ)ρ 2 sin φ dθdφdρ =2π dφdρ. () (L 2 + ρ 2 2Lρ cos φ) 3/2 (L 2 + ρ 2 2Lρ cos φ) 3/2 Using u = ρ cos φ nd integrte the inner integrl right hnd side of () will get 2π ( ρ) L [ L ρ cos φ (L 2 + ρ 2 2Lρ cos φ) /2 L (L2 + ρ 2 2Lρ cos φ) /2 ] π 0, =2π ( ρ) L [( L + ρ L + ρ L + ρ L ) ( L ρ L ρ L ρ L )]. In cse L>ρ,itis4π ρ2 L 2 L<,()is 4 3 πl. nd in cse L<ρit is 0. So when L>() is 4π3 3L 2 nd when

5 CHAPTE 3, MULTIPLE INTEGALS Surfce Are of Prmetric Surfce Prmetric Surfces. r(u, v) =(x(u, v), (y(u, v),z(u, v)). Exmple. () (x, y, f(x, y)), (2) (r cos θ, r sin θ, g(r, θ)), (3) (h(φ, θ)sinφcos θ, h(φ, θ)sinφsin θ, h(φ, θ)cosφ). Surfce Are. Surfce Are of Grph. A = N dudv = A = Exmple.z =2x +2y + insidex 2 + y 2. r r dudv. +f 2 x + f 2 y dxdy. Surfce Are in Polr Coordintes. A = r 2 +(rz r ) 2 + zθ 2drdθ. Exmple. () z = r 2,r, (2) z = θ, 0 θ π, 0 r. Exmple.(x b) 2 + z 2 = 2 revolute bout z-xis. Jcobin in 2-dimension. Theorem. 3.9 Chnge Vribles in Multiple Integrls (x, y) (u, v) = f(x, y)dxdy = Exmple. () f(x, y)dxdy = f(r cos θ, r sin θ)rdrdθ, (2) I o of xy 3, x 2 y 2 4, (3) Are of xy 3, xy.4 2. S. (x, y) f(x(u, v),y(u, v)) (u, v) dudv.

6 6 CHAPTE 3, MULTIPLE INTEGALS Jcobin in 3-dimension. (x, y, z) (u, v, w) = w w z z z w. Theorem. f(x, y, z)dxdydz = S (x, y, z) f(x(u, v, w, y(u, v, w),z(u, v, w)) (u, v, w) dudvdw. Exmple. () Spericl coordintes, (2) evolution of (x b) 2 + z 2 = 2 bout z-xis.

Chapter 12 Vectors and the Geometry of Space 12.1 Three-dimensional Coordinate systems

Chapter 12 Vectors and the Geometry of Space 12.1 Three-dimensional Coordinate systems hpter 12 Vectors nd the Geometry of Spce 12.1 Three-dimensionl oordinte systems A. Three dimensionl Rectngulr oordinte Sydstem: The rtesin product where (x, y, z) isclled ordered triple. B. istnce: R 3

More information

STUDY GUIDE, CALCULUS III, 2017 SPRING

STUDY GUIDE, CALCULUS III, 2017 SPRING TUY GUIE, ALULU III, 2017 PING ontents hpter 13. Functions of severl vribles 1 13.1. Plnes nd surfces 2 13.2. Grphs nd level curves 2 13.3. Limit of function of two vribles 2 13.4. Prtil derivtives 2 13.5.

More information

WI1402-LR Calculus II Delft University of Technology

WI1402-LR Calculus II Delft University of Technology WI402-LR lculus II elft University of Technology Yer 203 204 Michele Fcchinelli Version.0 Lst modified on Februry, 207 Prefce This summry ws written for the course WI402-LR lculus II, tught t the elft

More information

b = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 )

b = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 ) ***************** Disclimer ***************** This represents very brief outline of most of the topics covered MA261 *************************************************** I. Vectors, Lines nd Plnes 1. Vector

More information

Example. Check that the Jacobian of the transformation to spherical coordinates is

Example. Check that the Jacobian of the transformation to spherical coordinates is lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.

More information

Vector Calculus. 1 Line Integrals

Vector Calculus. 1 Line Integrals Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q

More information

Section 16.3 Double Integrals over General Regions

Section 16.3 Double Integrals over General Regions Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is

More information

Polar Coordinates. July 30, 2014

Polar Coordinates. July 30, 2014 Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xy-plne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes

More information

Math 116 Calculus II

Math 116 Calculus II Mth 6 Clculus II Contents 7 Additionl topics in Integrtion 7. Integrtion by prts..................................... 7.4 Numericl Integrtion.................................... 7 7.5 Improper Integrl......................................

More information

Fubini for continuous functions over intervals

Fubini for continuous functions over intervals Fuini for ontinuous funtions over intervls We first prove the following theorem for ontinuous funtions. Theorem. Let f(x) e ontinuous on ompt intervl =[, [,. Then [, [, [ [ f(x, y)(x, y) = f(x, y)y x =

More information

Independent of path Green s Theorem Surface Integrals. MATH203 Calculus. Dr. Bandar Al-Mohsin. School of Mathematics, KSU 20/4/14

Independent of path Green s Theorem Surface Integrals. MATH203 Calculus. Dr. Bandar Al-Mohsin. School of Mathematics, KSU 20/4/14 School of Mathematics, KSU 20/4/14 Independent of path Theorem 1 If F (x, y) = M(x, y)i + N(x, y)j is continuous on an open connected region D, then the line integral F dr is independent of path if and

More information

10.4 AREAS AND LENGTHS IN POLAR COORDINATES

10.4 AREAS AND LENGTHS IN POLAR COORDINATES 65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the

More information

Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Leture 16 Double integrls Dn Nihols nihols@mth.umss.edu MATH 233, Spring 218 University of Msshusetts Mrh 27, 218 (2) iemnn sums for funtions of one vrible Let f(x) on [, b]. We n estimte the re under

More information

Study Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms

Study Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms Study Guide # 1 MA 26100 - Fll 2018 1. Vectors in R 2 nd R 3 () v =, b, c = i + b j + c k; vector ddition nd subtrction geometriclly using prllelogrms spnned by u nd v; length or mgnitude of v =, b, c,

More information

Geometric quantities for polar curves

Geometric quantities for polar curves Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions

More information

Calculus IV Math 2443 Review for Exam 2 on Mon Oct 24, 2016 Exam 2 will cover This is only a sample. Try all the homework problems.

Calculus IV Math 2443 Review for Exam 2 on Mon Oct 24, 2016 Exam 2 will cover This is only a sample. Try all the homework problems. Calculus IV Math 443 eview for xam on Mon Oct 4, 6 xam will cover 5. 5.. This is only a sample. Try all the homework problems. () o not evaluated the integral. Write as iterated integrals: (x + y )dv,

More information

Section 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.

Section 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17. Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line

More information

Estimating Areas. is reminiscent of a Riemann Sum and, amazingly enough, will be called a Riemann Sum. Double Integrals

Estimating Areas. is reminiscent of a Riemann Sum and, amazingly enough, will be called a Riemann Sum. Double Integrals Estimating Areas Consider the challenge of estimating the volume of a solid {(x, y, z) 0 z f(x, y), (x, y) }, where is a region in the xy-plane. This may be thought of as the solid under the graph of z

More information

Double Integrals over Rectangles

Double Integrals over Rectangles Jim Lmbers MAT 8 Spring Semester 9- Leture Notes These notes orrespond to Setion. in Stewrt nd Setion 5. in Mrsden nd Tromb. Double Integrls over etngles In single-vrible lulus, the definite integrl of

More information

LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY

LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd

More information

Multivariable integration. Multivariable integration. Iterated integration

Multivariable integration. Multivariable integration. Iterated integration Multivrible integrtion Multivrible integrtion Integrtion is ment to nswer the question how muh, depending on the problem nd how we set up the integrl we n be finding how muh volume, how muh surfe re, how

More information

9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations

9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations 54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show

More information

Differentiable functions (Sec. 14.4)

Differentiable functions (Sec. 14.4) Math 20C Multivariable Calculus Lecture 3 Differentiable functions (Sec. 4.4) Review: Partial derivatives. Slide Partial derivatives and continuity. Equation of the tangent plane. Differentiable functions.

More information

Multiple Integrals. Advanced Calculus. Lecture 1 Dr. Lahcen Laayouni. Department of Mathematics and Statistics McGill University.

Multiple Integrals. Advanced Calculus. Lecture 1 Dr. Lahcen Laayouni. Department of Mathematics and Statistics McGill University. Lecture epartment of Mathematics and Statistics McGill University January 4, 27 ouble integrals Iteration of double integrals ouble integrals Consider a function f(x, y), defined over a rectangle = [a,

More information

Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is

More information

VectorPlot[{y^2-2x*y,3x*y-6*x^2},{x,-5,5},{y,-5,5}]

VectorPlot[{y^2-2x*y,3x*y-6*x^2},{x,-5,5},{y,-5,5}] hapter 16 16.1. 6. Notice that F(x, y) has length 1 and that it is perpendicular to the position vector (x, y) for all x and y (except at the origin). Think about drawing the vectors based on concentric

More information

VECTOR CALCULUS Julian.O 2016

VECTOR CALCULUS Julian.O 2016 VETO ALULUS Julian.O 2016 Vector alculus Lecture 3: Double Integrals Green s Theorem Divergence of a Vector Field Double Integrals: Double integrals are used to integrate two-variable functions f(x, y)

More information

Section 10.2 Graphing Polar Equations

Section 10.2 Graphing Polar Equations Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The

More information

Polar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a

Polar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.

More information

Mock final exam Math fall 2007

Mock final exam Math fall 2007 Mock final exam Math - fall 7 Fernando Guevara Vasquez December 5 7. Consider the curve r(t) = ti + tj + 5 t t k, t. (a) Show that the curve lies on a sphere centered at the origin. (b) Where does the

More information

Conditional Distributions

Conditional Distributions Conditional Distributions X, Y discrete: the conditional pmf of X given Y y is defined to be p X Y (x y) P(X x, Y y) P(Y y) p(x, y) p Y (y), p Y (y) > 0. Given Y y, the randomness of X is described by

More information

Definitions and claims functions of several variables

Definitions and claims functions of several variables Definitions and claims functions of several variables In the Euclidian space I n of all real n-dimensional vectors x = (x 1, x,..., x n ) the following are defined: x + y = (x 1 + y 1, x + y,..., x n +

More information

Math Final Exam - 6/11/2015

Math Final Exam - 6/11/2015 Math 200 - Final Exam - 6/11/2015 Name: Section: Section Class/Times Instructor Section Class/Times Instructor 1 9:00%AM ( 9:50%AM Papadopoulos,%Dimitrios 11 1:00%PM ( 1:50%PM Swartz,%Kenneth 2 11:00%AM

More information

Domination and Independence on Square Chessboard

Domination and Independence on Square Chessboard Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq pure.hmed.omrn@uobby lon.edu.iq Domintion

More information

Mixture of Discrete and Continuous Random Variables

Mixture of Discrete and Continuous Random Variables Mixture of Discrete and Continuous Random Variables What does the CDF F X (x) look like when X is discrete vs when it s continuous? A r.v. could have a continuous component and a discrete component. Ex

More information

Triangles and parallelograms of equal area in an ellipse

Triangles and parallelograms of equal area in an ellipse 1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper

More information

University of California, Berkeley Department of Mathematics 5 th November, 2012, 12:10-12:55 pm MATH 53 - Test #2

University of California, Berkeley Department of Mathematics 5 th November, 2012, 12:10-12:55 pm MATH 53 - Test #2 University of California, Berkeley epartment of Mathematics 5 th November, 212, 12:1-12:55 pm MATH 53 - Test #2 Last Name: First Name: Student Number: iscussion Section: Name of GSI: Record your answers

More information

Functions of several variables

Functions of several variables Chapter 6 Functions of several variables 6.1 Limits and continuity Definition 6.1 (Euclidean distance). Given two points P (x 1, y 1 ) and Q(x, y ) on the plane, we define their distance by the formula

More information

NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES

NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES At Mth Univ Comenine Vol LXXXV, (06, pp 07 07 NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES G FARID Abstrt In this pper, new Ostrowski-type inequlities in two oordintes re estblished

More information

MATH 118 PROBLEM SET 6

MATH 118 PROBLEM SET 6 MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the

More information

MATH Exam 2 Solutions November 16, 2015

MATH Exam 2 Solutions November 16, 2015 MATH 1.54 Exam Solutions November 16, 15 1. Suppose f(x, y) is a differentiable function such that it and its derivatives take on the following values: (x, y) f(x, y) f x (x, y) f y (x, y) f xx (x, y)

More information

FUNCTIONS OF SEVERAL VARIABLES AND PARTIAL DIFFERENTIATION

FUNCTIONS OF SEVERAL VARIABLES AND PARTIAL DIFFERENTIATION FUNCTIONS OF SEVERAL VARIABLES AND PARTIAL DIFFERENTIATION 1. Functions of Several Variables A function of two variables is a rule that assigns a real number f(x, y) to each ordered pair of real numbers

More information

11/2/2016 Second Hourly Practice I Math 21a, Fall Name:

11/2/2016 Second Hourly Practice I Math 21a, Fall Name: 11/2/216 Second Hourly Practice I Math 21a, Fall 216 Name: MWF 9 Koji Shimizu MWF 1 Can Kozcaz MWF 1 Yifei Zhao MWF 11 Oliver Knill MWF 11 Bena Tshishiku MWF 12 Jun-Hou Fung MWF 12 Chenglong Yu TTH 1 Jameel

More information

11/1/2017 Second Hourly Practice 2 Math 21a, Fall Name:

11/1/2017 Second Hourly Practice 2 Math 21a, Fall Name: 11/1/217 Second Hourly Practice 2 Math 21a, Fall 217 Name: MWF 9 Jameel Al-Aidroos MWF 9 Dennis Tseng MWF 1 Yu-Wei Fan MWF 1 Koji Shimizu MWF 11 Oliver Knill MWF 11 Chenglong Yu MWF 12 Stepan Paul TTH

More information

2.1 Partial Derivatives

2.1 Partial Derivatives .1 Partial Derivatives.1.1 Functions of several variables Up until now, we have only met functions of single variables. From now on we will meet functions such as z = f(x, y) and w = f(x, y, z), which

More information

Vocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear.

Vocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear. Section.8 Grphs of Polr Equtions 98 9. Points:,,,,.,... The points re colliner. 9. Points:.,,.,,.,... not colliner. Section.8 Grphs of Polr Equtions When grphing polr equtions:. Test for symmetry. () )

More information

FP2 POLAR COORDINATES: PAST QUESTIONS

FP2 POLAR COORDINATES: PAST QUESTIONS FP POLAR COORDINATES: PAST QUESTIONS. The curve C hs polr eqution r = cosθ, () Sketch the curve C. () (b) Find the polr coordintes of the points where tngents to C re prllel to the initil line. (6) (c)

More information

Continuous Space Fourier Transform (CSFT)

Continuous Space Fourier Transform (CSFT) EE637 Digital Image Processing I: Purdue University VISE - February 7, Continuous Space Fourier Transform (CSFT) Forward CSFT: F (u, v) = f(x, y)e jπ(ux+vy) dxdy Inverse CSFT: f(x, y) = F (u, v)ejπ(ux+vy)

More information

Review guide for midterm 2 in Math 233 March 30, 2009

Review guide for midterm 2 in Math 233 March 30, 2009 Review guide for midterm 2 in Math 2 March, 29 Midterm 2 covers material that begins approximately with the definition of partial derivatives in Chapter 4. and ends approximately with methods for calculating

More information

Maxima and Minima. Terminology note: Do not confuse the maximum f(a, b) (a number) with the point (a, b) where the maximum occurs.

Maxima and Minima. Terminology note: Do not confuse the maximum f(a, b) (a number) with the point (a, b) where the maximum occurs. 10-11-2010 HW: 14.7: 1,5,7,13,29,33,39,51,55 Maxima and Minima In this very important chapter, we describe how to use the tools of calculus to locate the maxima and minima of a function of two variables.

More information

Test Yourself. 11. The angle in degrees between u and w. 12. A vector parallel to v, but of length 2.

Test Yourself. 11. The angle in degrees between u and w. 12. A vector parallel to v, but of length 2. Test Yourself These are problems you might see in a vector calculus course. They are general questions and are meant for practice. The key follows, but only with the answers. an you fill in the blanks

More information

10.1 Curves defined by parametric equations

10.1 Curves defined by parametric equations Outline Section 1: Parametric Equations and Polar Coordinates 1.1 Curves defined by parametric equations 1.2 Calculus with Parametric Curves 1.3 Polar Coordinates 1.4 Areas and Lengths in Polar Coordinates

More information

For each question, X indicates a correct choice. ANSWER SHEET - BLUE. Question a b c d e Do not write in this column 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X

For each question, X indicates a correct choice. ANSWER SHEET - BLUE. Question a b c d e Do not write in this column 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X For each question, X indicates a correct choice. ANSWER SHEET - BLUE X ANSWER SHEET - GREEN X ANSWER SHEET - WHITE X ANSWER SHEET - YELLOW For each question, place an X in the box of your choice. X QUESTION

More information

METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin

METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or

More information

INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS

INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite

More information

Solutions to the problems from Written assignment 2 Math 222 Winter 2015

Solutions to the problems from Written assignment 2 Math 222 Winter 2015 Solutions to the problems from Written assignment 2 Math 222 Winter 2015 1. Determine if the following limits exist, and if a limit exists, find its value. x2 y (a) The limit of f(x, y) = x 4 as (x, y)

More information

Chapter 16. Partial Derivatives

Chapter 16. Partial Derivatives Chapter 16 Partial Derivatives The use of contour lines to help understand a function whose domain is part of the plane goes back to the year 1774. A group of surveyors had collected a large number of

More information

Exam 2 Review Sheet. r(t) = x(t), y(t), z(t)

Exam 2 Review Sheet. r(t) = x(t), y(t), z(t) Exam 2 Review Sheet Joseph Breen Particle Motion Recall that a parametric curve given by: r(t) = x(t), y(t), z(t) can be interpreted as the position of a particle. Then the derivative represents the particle

More information

Topic 6: Joint Distributions

Topic 6: Joint Distributions Topic 6: Joint Distributions Course 003, 2017 Page 0 Joint distributions Social scientists are typically interested in the relationship between many random variables. They may be able to change some of

More information

Exam 1 Study Guide. Math 223 Section 12 Fall Student s Name

Exam 1 Study Guide. Math 223 Section 12 Fall Student s Name Exam 1 Study Guide Math 223 Section 12 Fall 2015 Dr. Gilbert Student s Name The following problems are designed to help you study for the first in-class exam. Problems may or may not be an accurate indicator

More information

Double Integrals over More General Regions

Double Integrals over More General Regions Jim Lambers MAT 8 Spring Semester 9-1 Lecture 11 Notes These notes correspond to Section 1. in Stewart and Sections 5.3 and 5.4 in Marsden and Tromba. ouble Integrals over More General Regions We have

More information

Math 232. Calculus III Limits and Continuity. Updated: January 13, 2016 Calculus III Section 14.2

Math 232. Calculus III Limits and Continuity. Updated: January 13, 2016 Calculus III Section 14.2 Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 14.2 Limits and Continuity In this section our goal is to evaluate its of the form f(x, y) = L Let s take a look back at its in

More information

7/26/2018 SECOND HOURLY PRACTICE I Maths 21a, O.Knill, Summer Name:

7/26/2018 SECOND HOURLY PRACTICE I Maths 21a, O.Knill, Summer Name: 7/26/218 SECOND HOURLY PRACTICE I Maths 21a, O.Knill, Summer 218 Name: Start by printing your name in the above box. Try to answer each question on the same page as the question is asked. If needed, use

More information

11.7 Maximum and Minimum Values

11.7 Maximum and Minimum Values Arkansas Tech University MATH 2934: Calculus III Dr. Marcel B Finan 11.7 Maximum and Minimum Values Just like functions of a single variable, functions of several variables can have local and global extrema,

More information

Convolutional Networks. Lecture slides for Chapter 9 of Deep Learning Ian Goodfellow

Convolutional Networks. Lecture slides for Chapter 9 of Deep Learning Ian Goodfellow Convolutionl Networks Lecture slides for Chpter 9 of Deep Lerning In Goodfellow 2016-09-12 Convolutionl Networks Scle up neurl networks to process very lrge imges / video sequences Sprse connections Prmeter

More information

47. Conservative Vector Fields

47. Conservative Vector Fields 47. onservative Vector Fields Given a function z = φ(x, y), its gradient is φ = φ x, φ y. Thus, φ is a gradient (or conservative) vector field, and the function φ is called a potential function. Suppose

More information

IMAGE PROCESSING (RRY025) THE CONTINUOUS 2D FOURIER TRANSFORM

IMAGE PROCESSING (RRY025) THE CONTINUOUS 2D FOURIER TRANSFORM IMAGE PROCESSING (RRY5) THE CONTINUOUS D FOURIER TRANSFORM INTRODUCTION A vital tool in image processing. Also a prototype of other image transforms, cosine, Wavelet etc. Applications Image Filtering -

More information

Slinky vs. guitar. W.E. Bailey, APAM/MSE EN1102

Slinky vs. guitar. W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar W.E. Bailey, APAM/MSE EN1102 Differential spring element Figure: Differential length dx of spring under tension T with curvature is not a constant. θ = θ(x) W.E. Bailey, APAM/MSE EN1102

More information

Applying the Filtered Back-Projection Method to Extract Signal at Specific Position

Applying the Filtered Back-Projection Method to Extract Signal at Specific Position Applying the Filtered Back-Projection Method to Extract Signal at Specific Position 1 Chia-Ming Chang and Chun-Hao Peng Department of Computer Science and Engineering, Tatung University, Taipei, Taiwan

More information

B) 0 C) 1 D) No limit. x2 + y2 4) A) 2 B) 0 C) 1 D) No limit. A) 1 B) 2 C) 0 D) No limit. 8xy 6) A) 1 B) 0 C) π D) -1

B) 0 C) 1 D) No limit. x2 + y2 4) A) 2 B) 0 C) 1 D) No limit. A) 1 B) 2 C) 0 D) No limit. 8xy 6) A) 1 B) 0 C) π D) -1 MTH 22 Exam Two - Review Problem Set Name Sketch the surface z = f(x,y). ) f(x, y) = - x2 ) 2) f(x, y) = 2 -x2 - y2 2) Find the indicated limit or state that it does not exist. 4x2 + 8xy + 4y2 ) lim (x,

More information

Proofs of a Trigonometric Inequality

Proofs of a Trigonometric Inequality Proofs of a Trigonometric Inequality Abstract A trigonometric inequality is introduced and proved using Hölder s inequality Cauchy-Schwarz inequality and Chebyshev s order inequality AMS Subject Classification:

More information

Review Problems. Calculus IIIA: page 1 of??

Review Problems. Calculus IIIA: page 1 of?? Review Problems The final is comprehensive exam (although the material from the last third of the course will be emphasized). You are encouraged to work carefully through this review package, and to revisit

More information

State Math Contest Junior Exam SOLUTIONS

State Math Contest Junior Exam SOLUTIONS State Math Contest Junior Exam SOLUTIONS 1. The following pictures show two views of a non standard die (however the numbers 1-6 are represented on the die). How many dots are on the bottom face of figure?

More information

Transport Capacity and Spectral Efficiency of Large Wireless CDMA Ad Hoc Networks

Transport Capacity and Spectral Efficiency of Large Wireless CDMA Ad Hoc Networks Transport Capacity and Spectral Efficiency of Large Wireless CDMA Ad Hoc Networks Yi Sun Department of Electrical Engineering The City College of City University of New York Acknowledgement: supported

More information

y = sin x 5.5 Graphing sine and cosine functions

y = sin x 5.5 Graphing sine and cosine functions Graphing sine and csine functins: Class ntes, G. Battaly 5.5 Graphing sine and csine functins GOALS: 1. Recgnize that f(x) = sin x and g(x) = cs x are functins. (Each value f x results in exactly 1 y value.)

More information

CS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates

CS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the

More information

ECE 274 Digital Logic. Digital Design. Datapath Components Shifters, Comparators, Counters, Multipliers Digital Design

ECE 274 Digital Logic. Digital Design. Datapath Components Shifters, Comparators, Counters, Multipliers Digital Design ECE 27 Digitl Logic Shifters, Comprtors, Counters, Multipliers Digitl Design..7 Digitl Design Chpter : Slides to ccompny the textbook Digitl Design, First Edition, by Frnk Vhid, John Wiley nd Sons Publishers,

More information

Section 3: Functions of several variables.

Section 3: Functions of several variables. Section 3: Functions of several variables. Compiled by Chris Tisdell S1: Motivation S2: Function of two variables S3: Visualising and sketching S4: Limits and continuity S5: Partial differentiation S6:

More information

Unit 1: Chapter 4 Roots & Powers

Unit 1: Chapter 4 Roots & Powers Unit 1: Chpter 4 Roots & Powers Big Ides Any number tht cn be written s the frction mm, nn 0, where m nd n re integers, is nn rtionl. Eponents cn be used to represent roots nd reciprocls of rtionl numbers.

More information

MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES

MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine E-mil: rt_science@hotmil.com 1 INTRODUCTION During the considertion

More information

Exam 2 Summary. 1. The domain of a function is the set of all possible inputes of the function and the range is the set of all outputs.

Exam 2 Summary. 1. The domain of a function is the set of all possible inputes of the function and the range is the set of all outputs. Exam 2 Summary Disclaimer: The exam 2 covers lectures 9-15, inclusive. This is mostly about limits, continuity and differentiation of functions of 2 and 3 variables, and some applications. The complete

More information

F13 Study Guide/Practice Exam 3

F13 Study Guide/Practice Exam 3 F13 Study Guide/Practice Exam 3 This study guide/practice exam covers only the material since exam 2. The final exam, however, is cumulative so you should be sure to thoroughly study earlier material.

More information

33. Riemann Summation over Rectangular Regions

33. Riemann Summation over Rectangular Regions . iemann Summation over ectangular egions A rectangular region in the xy-plane can be defined using compound inequalities, where x and y are each bound by constants such that a x a and b y b. Let z = f(x,

More information

Section 14.3 Partial Derivatives

Section 14.3 Partial Derivatives Section 14.3 Partial Derivatives Ruipeng Shen March 20 1 Basic Conceptions If f(x, y) is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant.

More information

Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3

Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3 Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan Review Problems for Test #3 Exercise 1 The following is one cycle of a trigonometric function. Find an equation of this graph. Exercise

More information

Practice problems from old exams for math 233

Practice problems from old exams for math 233 Practice problems from old exams for math 233 William H. Meeks III October 26, 2012 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These

More information

Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION

Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION Exercise 1-1 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You

More information

c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?

c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK? 3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem

More information

Lecture 12: Image Processing and 2D Transforms

Lecture 12: Image Processing and 2D Transforms Lecture 12: Image Processing and 2D Transforms Harvey Rhody Chester F. Carlson Center for Imaging Science Rochester Institute of Technology rhody@cis.rit.edu October 18, 2005 Abstract The Fourier transform

More information

March 13, 2009 CHAPTER 3: PARTIAL DERIVATIVES AND DIFFERENTIATION

March 13, 2009 CHAPTER 3: PARTIAL DERIVATIVES AND DIFFERENTIATION March 13, 2009 CHAPTER 3: PARTIAL DERIVATIVES AND DIFFERENTIATION 1. Parial Derivaives and Differeniable funcions In all his chaper, D will denoe an open subse of R n. Definiion 1.1. Consider a funcion

More information

[f(t)] 2 + [g(t)] 2 + [h(t)] 2 dt. [f(u)] 2 + [g(u)] 2 + [h(u)] 2 du. The Fundamental Theorem of Calculus implies that s(t) is differentiable and

[f(t)] 2 + [g(t)] 2 + [h(t)] 2 dt. [f(u)] 2 + [g(u)] 2 + [h(u)] 2 du. The Fundamental Theorem of Calculus implies that s(t) is differentiable and Midterm 2 review Math 265 Fall 2007 13.3. Arc Length and Curvature. Assume that the curve C is described by the vector-valued function r(r) = f(t), g(t), h(t), and that C is traversed exactly once as t

More information

Synchronous Machine Parameter Measurement

Synchronous Machine Parameter Measurement Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions

More information

Name: ID: Section: Math 233 Exam 2. Page 1. This exam has 17 questions:

Name: ID: Section: Math 233 Exam 2. Page 1. This exam has 17 questions: Page Name: ID: Section: This exam has 7 questions: 5 multiple choice questions worth 5 points each. 2 hand graded questions worth 25 points total. Important: No graphing calculators! Any non scientific

More information

SOLUTIONS 2. PRACTICE EXAM 2. HOURLY. Problem 1) TF questions (20 points) Circle the correct letter. No justifications are needed.

SOLUTIONS 2. PRACTICE EXAM 2. HOURLY. Problem 1) TF questions (20 points) Circle the correct letter. No justifications are needed. SOLUIONS 2. PRACICE EXAM 2. HOURLY Math 21a, S03 Problem 1) questions (20 points) Circle the correct letter. No justifications are needed. A function f(x, y) on the plane for which the absolute minimum

More information

MATH 259 FINAL EXAM. Friday, May 8, Alexandra Oleksii Reshma Stephen William Klimova Mostovyi Ramadurai Russel Boney A C D G H B F E

MATH 259 FINAL EXAM. Friday, May 8, Alexandra Oleksii Reshma Stephen William Klimova Mostovyi Ramadurai Russel Boney A C D G H B F E MATH 259 FINAL EXAM 1 Friday, May 8, 2009. NAME: Alexandra Oleksii Reshma Stephen William Klimova Mostovyi Ramadurai Russel Boney A C D G H B F E Instructions: 1. Do not separate the pages of the exam.

More information

ANSWER KEY. (a) For each of the following partials derivatives, use the contour plot to decide whether they are positive, negative, or zero.

ANSWER KEY. (a) For each of the following partials derivatives, use the contour plot to decide whether they are positive, negative, or zero. Math 2130-101 Test #2 for Section 101 October 14 th, 2009 ANSWE KEY 1. (10 points) Compute the curvature of r(t) = (t + 2, 3t + 4, 5t + 6). r (t) = (1, 3, 5) r (t) = 1 2 + 3 2 + 5 2 = 35 T(t) = 1 r (t)

More information

CHAPTER 11 PARTIAL DERIVATIVES

CHAPTER 11 PARTIAL DERIVATIVES CHAPTER 11 PARTIAL DERIVATIVES 1. FUNCTIONS OF SEVERAL VARIABLES A) Definition: A function of two variables is a rule that assigns to each ordered pair of real numbers (x,y) in a set D a unique real number

More information

MATH 105: Midterm #1 Practice Problems

MATH 105: Midterm #1 Practice Problems Name: MATH 105: Midterm #1 Practice Problems 1. TRUE or FALSE, plus explanation. Give a full-word answer TRUE or FALSE. If the statement is true, explain why, using concepts and results from class to justify

More information

A General Procedure (Solids of Revolution) Some Useful Area Formulas

A General Procedure (Solids of Revolution) Some Useful Area Formulas Goal: Given a solid described by rotating an area, compute its volume. A General Procedure (Solids of Revolution) (i) Draw a graph of the relevant functions/regions in the plane. Draw a vertical line and

More information

On Surfaces of Revolution whose Mean Curvature is Constant

On Surfaces of Revolution whose Mean Curvature is Constant On Surfaces of Revolution whose Mean Curvature is Constant Ch. Delaunay May 4, 2002 When one seeks a surface of given area enclosing a maximal volume, one finds that the equation this surface must satisfy

More information