r(t) =hf(t), g(t), h(t)i = f(t)i + g(t)j + h(t)k
|
|
- Junior Ramsey
- 5 years ago
- Views:
Transcription
1 Chapter 3 Vector Functions 3. Vector Functions and Space Curves efinition. A vector-valued function of one variable is a function whose domain is a set of real numbers and whose range is a set of vectors, usuall in R 3. If r :! R 3 is a vector-valued function then each component of r(t) is a function of t: r(t) =hf(t), g(t), h(t)i = f(t)i + g(t)j + h(t)k We call f(t), g(t) and h(t) thecomponent functions. ample. Let r(t) = t, 3sin(3t +3 /), p t 5. Find the domain of r(t), and find the range of each component function. Solution: As usual, the domain is implied to be all the real numbers that make the formulas defined. The onl component function that s not defined for some values of t is the square root. Thus, : t. We ll describe the range component b component: (, 3 apple apple 3, 5). efinition. If r(t) =hf(t), g(t), h(t)i then lim r(t) = lim f(t), lim g(t), lim h(t) t!a t!a t!a t!a ample. Let r(t) = Solution: t, 3sin(3t +3 /), p t 5 (if these limits eist).findlim t! r(t). lim r(t) = lim t, lim 3sin(3t +3 /), lim (p t 5) t! t! t! t! = h, 3, 5i 47
2 CHAPTR 3. VCTOR FUNCTIONS 48 efinition. If r(t) = hf(t), g(t), h(t)i and lim t!a r(t) =r(a), then r(t) iscontinuous at t = a efinition. The range of a vector-valued function r(t) is called a space curve. In other words, a space curve equals the set of all (,, ) such that = f(t) = g(t) = h(t) (3.) and as t varies. We call these equations the parametric equations of the curve, and call t the parameter. ample 3. Use a computer package to graph the space curve defined b r(t) = t, 3 cos(3t +3 /), p t 5. Solution: We show below the Matlab code, together with the resulting graph t= linspace(,4*pi); = t; = 3*cos(3*t +3*pi/); = sqrt(t)-5; plot3(,,); label(''),label(''), label('') ample 4. Find the parametric equations of the following space curve, and use a computer package to graph the result Solution: The parametric equations are Notice that r(t) = cos(t)i + tj +sin(t)k = cos(t), = t, =sin(t), + = cos (t)+sin (t) = What this means is that all the (, )-values lie on the clinder + =. As the -value increases, the (, )-values wind around the clinder. We show below the Matlab code, together with the resulting graph
3 CHAPTR 3. VCTOR FUNCTIONS t = linspace(,4*pi); = cos(*t); = t; = sin(*t); plot3(,,); label(''),label('' ),label('') ample 5. Find a vector function that represents the curve of intersection of the following shapes: + + =(sphere), = (plane). Solution: To intersect two surfaces we combine the two equations in the usual fashion, in this case substituting = into the other equation +( )+ = + = + = In the -plane this is an ellipse (although in the = plane the curve is a circle). The parametric equations of an ellipse start with sine and cosine, multipling one of these b a larger number to give the longer ais: = p cos(t), =sin(t), =sint, apple t apple. Shown below is the Matlab code and resulting graph t = linspace(,*pi); = sqrt()*cos(t); = sin(t); = sin(t); plot3(,,); label(''),label('' ),label('')
4 CHAPTR 3. VCTOR FUNCTIONS 5 ample 6. [#36] Find a vector function that represents the curve of intersection of the following two shapes + =4, and = This is where we ended on Wednesda, Januar 3 Solution: The shape + =4iscircleinR, and so makes a vertical clinder in R 3. The parametric equations of a circle start with sine and cosine, and multipl these b the radius. So, in the -plane we get = cos t, =sint, t [, ] Now we combine the parametric equations with the =, and simplif the result with atrigidentit: = = 4 cos(t)sin(t) So now we can write, and in parametric equations: 9 = cos(t) =sin(t) >= =4sin(t) cos(t) >; r(t) = cos ti +sintj +sin(t)k apple t apple Here is the Matlab code and the resulting graph: t = linspace(,*pi); = *cos(t); = *sin(t); = 4*sin(t).*cos(t); plot3(,,); label(''),label('' ),label('') The above graphic shows the vector function that we found above, that is the intersection of two surfaces. But it doesn t show the surfaces themselves. The graphic below shows the original two surfaces themselves:
5 CHAPTR 3. VCTOR FUNCTIONS 5 t = linspace(,*pi); = *cos(t); = *sin(t); = 4*sin(t).*cos(t); plot3(,,, 'k', ' LineWidth', 3); hold on [c,c,c]=clinder(,5) surf(c,c,3*c) surf(c,c,-3*c) alpha(.3) [u,v]=meshgrid( linspace(-,)); =u.*v; mesh(u,v,) label(''),label(''),label('' ) hold off
Section Vector Functions and Space Curves
Section 13.1 Section 13.1 Goals: Graph certain plane curves. Compute limits and verify the continuity of vector functions. Multivariable Calculus 1 / 32 Section 13.1 Equation of a Line The equation of
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More information(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.
Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44
More informationTo produce exactly what I ve graphed above you need to change various appearences:
Appendi A Matlab eamples 2. z The simplest wa to graph this is: = linspace(,); = linspace(-5,5); z=.^2; plot3(,,z); To produce eactl what I ve graphed above ou need to change various appearences: = linspace(,);
More informationEXAM 2 ANSWERS AND SOLUTIONS, MATH 233 WEDNESDAY, OCTOBER 18, 2000
EXAM 2 ANSWERS AND SOLUTIONS, MATH 233 WEDNESDAY, OCTOBER 18, 2000 This examination has 30 multiple choice questions. Problems are worth one point apiece, for a total of 30 points for the whole examination.
More informationMATH107 Vectors and Matrices
School of Mathematics, KSU 20/11/16 Vector valued functions Let D be a set of real numbers D R. A vector-valued functions r with domain D is a correspondence that assigns to each number t in D exactly
More informationCHAPTER 11 Vector-Valued Functions
CHAPTER Vector-Valued Functions Section. Vector-Valued Functions...................... 9 Section. Differentiation and Integration of Vector-Valued Functions.... Section. Velocit and Acceleration.....................
More informationMath 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following:
Math 323 Eam 2 - Practice Problem Solutions 1. Given the vectors a = 2,, 1, b = 3, 2,4, and c = 1, 4,, compute the following: (a) A unit vector in the direction of c. u = c c = 1, 4, 1 4 =,, 1+16+ 17 17
More informationCHAPTER 4 DIFFERENTIAL VECTOR CALCULUS
CHAPTER 4 DIFFERENTIAL VECTOR CALCULUS 4.1 Vector Functions 4.2 Calculus of Vector Functions 4.3 Tangents REVIEW: Vectors Scalar a quantity only with its magnitude Example: temperature, speed, mass, volume
More informationDifferentiation of Parametric Space Curves. Goals: Velocity in parametric curves Acceleration in parametric curves
Block #2: Differentiation of Parametric Space Curves Goals: Velocity in parametric curves Acceleration in parametric curves 1 Displacement in Parametric Curves - 1 Displacement in Parametric Curves Using
More informationMA 351 Fall 2007 Exam #1 Review Solutions 1
MA 35 Fall 27 Exam # Review Solutions THERE MAY BE TYPOS in these solutions. Please let me know if you find any.. Consider the two surfaces ρ 3 csc θ in spherical coordinates and r 3 in cylindrical coordinates.
More informationD = 2(2) 3 2 = 4 9 = 5 < 0
1. (7 points) Let f(, ) = +3 + +. Find and classif each critical point of f as a local minimum, a local maimum, or a saddle point. Solution: f = + 3 f = 3 + + 1 f = f = 3 f = Both f = and f = onl at (
More information13.1: Vector-Valued Functions and Motion in Space, 14.1: Functions of Several Variables, and 14.2: Limits and Continuity in Higher Dimensions
13.1: Vector-Value Functions an Motion in Space, 14.1: Functions of Several Variables, an 14.2: Limits an Continuity in Higher Dimensions TA: Sam Fleischer November 3 Section 13.1: Vector-Value Functions
More informationCoordinate goemetry in the (x, y) plane
Coordinate goemetr in the (x, ) plane In this chapter ou will learn how to solve problems involving parametric equations.. You can define the coordinates of a point on a curve using parametric equations.
More informationKevin James. MTHSC 206 Section 13.2 Derivatives and Integrals of Vector
MTHSC 206 Section 13.2 Derivtives nd Integrls of Vector Functions Definition Suppose tht r(t) is vector function. We define its derivtive by [ ] dr r(t + h) r(t) dt = r (t) = lim h 0 h Definition Suppose
More information13.3 Arc Length and Curvature
13 Vector Functions 13.3 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. We have defined the length of a plane curve with parametric equations x = f(t),
More informationStudy guide for Exam 1. by William H. Meeks III October 26, 2012
Study guide for Exam 1. by William H. Meeks III October 2, 2012 1 Basics. First we cover the basic definitions and then we go over related problems. Note that the material for the actual midterm may include
More informationVector-Valued Functions
Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1
More informationSection 14.1 Vector Functions and Space Curves
Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a
More informationCalculus Vector Principia Mathematica. Lynne Ryan Associate Professor Mathematics Blue Ridge Community College
Calculus Vector Principia Mathematica Lynne Ryan Associate Professor Mathematics Blue Ridge Community College Defining a vector Vectors in the plane A scalar is a quantity that can be represented by a
More informationMATH 12 CLASS 5 NOTES, SEP
MATH 12 CLASS 5 NOTES, SEP 30 2011 Contents 1. Vector-valued functions 1 2. Differentiating and integrating vector-valued functions 3 3. Velocity and Acceleration 4 Over the past two weeks we have developed
More informationVectors and the Geometry of Space
Chapter 12 Vectors and the Geometr of Space Comments. What does multivariable mean in the name Multivariable Calculus? It means we stud functions that involve more than one variable in either the input
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More informationMATH141: Calculus II Exam #1 review 6/8/2017 Page 1
MATH: Calculus II Eam # review /8/7 Page No review sheet can cover everything that is potentially fair game for an eam, but I tried to hit on all of the topics with these questions, as well as show you
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 214 (R1) Winter 2008 Intermediate Calculus I Solutions to Problem Set #8 Completion Date: Friday March 14, 2008 Department of Mathematical and Statistical Sciences University of Alberta Question 1.
More informationPreface.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationTopic 3 Notes Jeremy Orloff
Topic 3 Notes Jerem Orloff 3 Line integrals and auch s theorem 3.1 Introduction The basic theme here is that comple line integrals will mirror much of what we ve seen for multivariable calculus line integrals.
More informationLater in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.
10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing
More information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, )
Midterm Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and graph the function. ) G(t) = t - 3 ) 3 - -3 - - 3 - - -3
More informationMATH Max-min Theory Fall 2016
MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions
More informationLet F be a field defined on an open region D in space, and suppose that the (work) integral A
16.3 1 16.3 Path Independence and Conservative Fields Definition. Path Independence Let F be a field defined on an open region D in space, and suppose B that the work) integral A F dr is the same for all
More informationMath 261 Solutions to Sample Final Exam Problems
Math 61 Solutions to Sample Final Eam Problems 1 Math 61 Solutions to Sample Final Eam Problems 1. Let F i + ( + ) j, and let G ( + ) i + ( ) j, where C 1 is the curve consisting of the circle of radius,
More informationLecture for Week 6 (Secs ) Derivative Miscellany I
Lecture for Week 6 (Secs. 3.6 9) Derivative Miscellany I 1 Implicit differentiation We want to answer questions like this: 1. What is the derivative of tan 1 x? 2. What is dy dx if x 3 + y 3 + xy 2 + x
More informationLecture Notes for MATH2230. Neil Ramsamooj
Lecture Notes for MATH3 Neil amsamooj Table of contents Vector Calculus................................................ 5. Parametric curves and arc length...................................... 5. eview
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More informationThis set of questions goes with the pages of applets and activities for Lab 09. Use the applets and activities there to answer the questions.
Page 1 of 5 Lab 09 Name: Start time: Number of questions: 11 This set of questions goes with the pages of applets and activities for Lab 09. Use the applets and activities there to answer the questions.
More informationPower Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.
.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x
More informationName: Class: Math 7B Date:
1. Match the given differential equations to their families of solutions. 2. Match the given differential equations and the graphs of their solutions. PAGE 1 3. Match the differential equation with its
More informationPractice Problems: Exam 2 MATH 230, Spring 2011 Instructor: Dr. Zachary Kilpatrick Show all your work. Simplify as much as possible.
Practice Problems: Exam MATH, Spring Instructor: Dr. Zachary Kilpatrick Show all your work. Simplify as much as possible.. Write down a table of x and y values associated with a few t values. Then, graph
More informationMTH 234 Exam 1 February 20th, Without fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More information1. Vectors and Matrices
E. 8.02 Exercises. Vectors and Matrices A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by dir A = A, (A 0); A it is the unit vector lying along A and pointed like
More informationVector Functions. EXAMPLE Describethecurves cost,sint,0, cost,sint,t,and cost,sint,2t.
13 Vector Functions ½ º½ ËÔ ÙÖÚ We have already seen that a convenient way to describe a line in three dimensions is to provide a vector that points to every point on the line as a parameter t varies,
More informationExam 1 Review SOLUTIONS
1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make
More informationMATH 151, FALL SEMESTER 2014 COMMON EXAMINATION I - VERSION B GUIDELINES
MATH151-14c MATH 151, FALL SEMESTER 2014 COMMON EXAMINATION I - VERSION B Time Allowed: 2 hours Last name, First name (print): Signature: Instructor s name: Section No: GUIDELINES In Part 1 (Problems 1
More informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 1 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH SOME SOLUTIONS TO EXAM 1 Fall 018 Version A refers to the regular exam and Version B to the make-up 1. Version A. Find the center
More informationSection 8.4 Plane Curves and Parametric Equations
Section 8.4 Plane Curves and Parametric Equations Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called parametric equations).
More informationThe slope, m, compares the change in y-values to the change in x-values. Use the points (2, 4) and (6, 6) to determine the slope.
LESSON Relating Slope and -intercept to Linear Equations UNDERSTAND The slope of a line is the ratio of the line s vertical change, called the rise, to its horizontal change, called the run. You can find
More informationMATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane
MATH 100 WORKSHEET 1.1 & 1. Vectors in the Plane Find the vector v where u =, 1 and w = 1, given the equation v = u w. Solution. v = u w =, 1 1, =, 1 +, 4 =, 1 4 = 0, 5 Find the magnitude of v = 4, 3 Solution.
More informationMath Exam IV - Fall 2011
Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and
More informationUnit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount.
Name Unit #17: Spring Trig Unit Notes #1: Basic Trig Review I. Unit Circle A circle with center point and radius. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that
More informationMa 227 Final Exam Solutions 5/8/03
Ma 7 Final Eam Solutions 5/8/3 Name: Lecture Section: I pledge m honor that I have abided b the Stevens Honor Sstem. ID: Directions: Answer all questions. The point value of each problem is indicated.
More informationExam 2 Review Solutions
Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon
More informationMidterm 1 Review. Distance = (x 1 x 0 ) 2 + (y 1 y 0 ) 2.
Midterm 1 Review Comments about the midterm The midterm will consist of five questions and will test on material from the first seven lectures the material given below. No calculus either single variable
More information698 Chapter 11 Parametric Equations and Polar Coordinates
698 Chapter Parametric Equations and Polar Coordinates 67. 68. 69. 70. 7. 7. 7. 7. Chapter Practice Eercises 699 75. (a Perihelion a ae a( e, Aphelion ea a a( e ( Planet Perihelion Aphelion Mercur 0.075
More informationMcKinney High School AP Calculus Summer Packet
McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work
More informationMath 261 Solutions To Sample Exam 2 Problems
Solutions to Sample Eam Problems Math 6 Math 6 Solutions To Sample Eam Problems. Given to the right is the graph of a portion of four curves:,, and + 4. Note that these curves divide the plane into separate
More informationCalculus III: Practice Final
Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains
More informationVector fields, line integrals, and Green s Theorem
Vector fields, line integrals, and Green s Theorem Line integrals The problem: Suppose ou have a surface = f(, ) defined over a region D. Restrict the domain of the function to the values of and which
More informationTriple integrals in Cartesian coordinates (Sect. 15.5)
Triple integrals in Cartesian coordinates (Sect. 5.5) Triple integrals in rectangular boes. Triple integrals in arbitrar domains. Volume on a region in space. Triple integrals in rectangular boes Definition
More informationMATH 151, FALL 2017 COMMON EXAM I - VERSION A
MATH 11, FALL 017 COMMON EXAM I - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN OFF cell phones
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More information32 +( 2) ( 4) ( 2)
Math 241 Exam 1 Sample 2 Solutions 1. (a) If ā = 3î 2ĵ+1ˆk and b = 4î+0ĵ 2ˆk, find the sine and cosine of the angle θ between [10 pts] ā and b. We know that ā b = ā b cosθ and so cosθ = ā b ā b = (3)(
More informationHonors Calculus Homework 1, due 9/8/5
Honors Calculus Homework 1, due 9/8/5 Question 1 Calculate the derivatives of the following functions: p(x) = x 4 3x 3 + 5 x 4x 1 3 + 23 q(x) = (1 + x)(1 + x 2 )(1 + x 3 )(1 + x 4 ). r(t) = (1 + t)(1 +
More information3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line.
PreCalculus Review Review Questions 1 The following transformations are applied in the given order) to the graph of y = x I Vertical Stretch by a factor of II Horizontal shift to the right by units III
More informationMath 11 Fall 2016 Section 1 Monday, September 19, Definition: A vector parametric equation for the line parallel to vector v = x v, y v, z v
Math Fall 06 Section Monay, September 9, 06 First, some important points from the last class: Definition: A vector parametric equation for the line parallel to vector v = x v, y v, z v passing through
More informationThe Calculus of Vec- tors
Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 3 1 The Calculus of Vec- Summary: tors 1. Calculus of Vectors: Limits and Derivatives 2. Parametric representation of Curves r(t) = [x(t), y(t),
More informationMath 323 Exam 1 Practice Problem Solutions
Math Exam Practice Problem Solutions. For each of the following curves, first find an equation in x and y whose graph contains the points on the curve. Then sketch the graph of C, indicating its orientation.
More informationMath 343 Lab 7: Line and Curve Fitting
Objective Math 343 Lab 7: Line and Curve Fitting In this lab, we explore another use of linear algebra in statistics. Specifically, we discuss the notion of least squares as a way to fit lines and curves
More informationLecture for Week 2 (Secs. 1.3 and ) Functions and Limits
Lecture for Week 2 (Secs. 1.3 and 2.2 2.3) Functions and Limits 1 First let s review what a function is. (See Sec. 1 of Review and Preview.) The best way to think of a function is as an imaginary machine,
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationRectangular box of sizes (dimensions) w,l,h wlh Right cylinder of radius r and height h r 2 h
Volumes: Slicing Method, Method of Disks and Washers -.,.. Volumes of Some Regular Solids: Solid Volume Rectangular bo of sizes (dimensions) w,l,h wlh Right clinder of radius r and height h r h Right cone
More informationMath 1A Chapter 3 Test Typical Problems Set Solutions
Math 1A Chapter 3 Test Typical Problems Set Solutions 1. Use the definition of the derivative to compute each of the following limits. a. b. c.. Find equations of the lines tangent to the curve which are
More informationMA 110 Algebra and Trigonometry for Calculus Fall 2016 Exam 4 12 December Multiple Choice Answers EXAMPLE A B C D E.
MA 110 Algebra and Trigonometry for Calculus Fall 2016 Exam 4 12 December 2016 Multiple Choice Answers EXAMPLE A B C D E Question Name: Section: Last 4 digits of student ID #: This exam has twelve multiple
More information( ) ( ) Math 17 Exam II Solutions
Math 7 Exam II Solutions. Sketch the vector field F(x,y) -yi + xj by drawing a few vectors. Draw the vectors associated with at least one point in each quadrant and draw the vectors associated with at
More informationMTHE 227 Problem Set 2 Solutions
MTHE 7 Problem Set Solutions 1 (Great Circles). The intersection of a sphere with a plane passing through its center is called a great circle. Let Γ be the great circle that is the intersection of the
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function rt vector T t r r t t is the unit tangent vector to the curve C. Now define N
More informationAnswers and Solutions to Section 13.7 Homework Problems 1 19 (odd) S. F. Ellermeyer April 23, 2004
Answers and olutions to ection 1.7 Homework Problems 1 19 (odd). F. Ellermeyer April 2, 24 1. The hemisphere and the paraboloid both have the same boundary curve, the circle x 2 y 2 4. Therefore, by tokes
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 11. Show all your work on the standard
More informationMultiple Choice. 1.(6 pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
Multiple Choice.(6 pts) Find smmetric equations of the line L passing through the point (, 5, ) and perpendicular to the plane x + 3 z = 9. (a) x = + 5 3 = z (c) (x ) + 3( 3) (z ) = 9 (d) (e) x = 3 5 =
More informationWorksheet 1.7: Introduction to Vector Functions - Position
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position From the Toolbox (what you need from previous classes): Cartesian Coordinates: Coordinates of points in general,
More informationMa 227 Final Exam Solutions 12/13/11
Ma 7 Final Exam Solutions /3/ Name: Lecture Section: (A and B: Prof. Levine, C: Prof. Brady) Problem a) ( points) Find the eigenvalues and eigenvectors of the matrix A. A 3 5 Solution. First we find the
More informationLecture D2 - Curvilinear Motion. Cartesian Coordinates
J. Peraire 6.07 Dynamics Fall 2004 Version. Lecture D2 - Curvilinear Motion. Cartesian Coordinates We will start by studying the motion of a particle. We think of a particle as a body which has mass, but
More informationChapter 14: Vector Calculus
Chapter 14: Vector Calculus Introduction to Vector Functions Section 14.1 Limits, Continuity, Vector Derivatives a. Limit of a Vector Function b. Limit Rules c. Component By Component Limits d. Continuity
More information13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:
8 CHAPTER VECTOR FUNCTIONS N Some computer algebra sstems provide us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes in
More informationLecture 4: Partial and Directional derivatives, Differentiability
Lecture 4: Partial and Directional derivatives, Differentiability Rafikul Alam Department of Mathematics IIT Guwahati Differential Calculus Task: Extend differential calculus to the functions: Case I:
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationMA227 Surface Integrals
MA7 urface Integrals Parametrically Defined urfaces We discussed earlier the concept of fx,y,zds where is given by z x,y.wehad fds fx,y,x,y1 x y 1 da R where R is the projection of onto the x,y - plane.
More informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 2017 MA101: CALCULUS PART A
A B1A003 Pages:3 (016 ADMISSIONS) Reg. No:... Name:... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 017 MA101: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationIn general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute
alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In
More information3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2
More informationStaple or bind all pages together. DO NOT dog ear pages as a method to bind.
Math 3337 Homework Instructions: Staple or bind all pages together. DO NOT dog ear pages as a method to bind. Hand-drawn sketches should be neat, clear, of reasonable size, with axis and tick marks appropriately
More information2. Evaluate C. F d r if F = xyî + (x + y)ĵ and C is the curve y = x 2 from ( 1, 1) to (2, 4).
Exam 3 Study Guide Math 223 Section 12 Fall 2015 Instructor: Dr. Gilbert 1. Which of the following vector fields are conservative? If you determine that a vector field is conservative, find a valid potential
More informationScienceWord and PagePlayer Graphical representation. Dr Emile C. B. COMLAN Novoasoft Representative in Africa
ScienceWord and PagePlayer Graphical representation Dr Emile C. B. COMLAN Novoasoft Representative in Africa Emails: ecomlan@scienceoffice.com ecomlan@yahoo.com Web site: www.scienceoffice.com Graphical
More informationMath 8 Winter 2010 Midterm 2 Review Problems Solutions - 1. xcos 6xdx = 4. = x2 4
Math 8 Winter 21 Midterm 2 Review Problems Solutions - 1 1 Evaluate xcos 2 3x Solution: First rewrite cos 2 3x using the half-angle formula: ( ) 1 + cos 6x xcos 2 3x = x = 1 x + 1 xcos 6x. 2 2 2 Now use
More informationParametric Curves You Should Know
Parametric Curves You Should Know Straight Lines Let a and c be constants which are not both zero. Then the parametric equations determining the straight line passing through (b d) with slope c/a (i.e.
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2018, WEEK 1 JoungDong Kim Week 1 Vectors, The Dot Product, Vector Functions and Parametric Curves. Section 1.1 Vectors Definition. A Vector is a quantity that
More informationExtra Problems Chapter 7
MA11: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 1 Etra Problems hapter 7 1. onsider the vector field F = i+z j +z 3 k. a) ompute div F. b) ompute curl F. Solution a) div F = +z +3z b) curl F = i
More information