1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.


 Catherine Black
 2 years ago
 Views:
Transcription
1 .(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3 5 = z + = y 3 5 = z + The symmetric equations of line are given by (x x 0 /a = (y y 0 /b = (z z 0 /c, where (x 0, y 0, z 0 is a point on the line and a, b, c is a direction vector. Since L is perpendicular to the plane x + 3y z = 9, then we can take the normal to the plane as the direction vector, this is,, 3, is a direction vector of L. Therefore, the symmetric equations are x = y + 5 = z 3. = 9.(6pts The two curves below intersect at the point (, 4, = r (0 = r (. Find the cosine of the angle of intersection ( r (t = e 3t i + 4 sin t + π j + (t k r (t = ti + 4j + (t k (a 5 (b 0 (c 5 (d e e + 4 (e 3 Note r (t = ( 3e 3t, 4 cos t + π, t r (t =, 0, t To compute the angle of intersection we find r (0 = 3, 0, 0 r ( =, 0, so that cos θ = r (0 r ( r (0 r ( = =. 5 3.(6pts Find the projection of the vector,, 5 onto the vector,, 4 (a,, 4 (b,, 5 (c 6, 3, (d 3, 3, 5 (e,, 5 5 proj,,4 (,, 5 =,, 4,, 5,, 4,, 4,, 4 =,, 4 =,, 4
2 4.(6pts Find Recall: r(xdx where sec x dx = tan x + C. r(x = (sec xi + e x k (a (tan x + C i + C j + (e x + C 3 k (b (tan x + C i + (e x + C k (c (tan xi + e x k (d tan x + e x + C (e (tan x + Ci + Cj + (e x + Ck ((sec r(xdx = xi + e x k dx ( ( = sec xdx i + ( 0dx j + e x dx k = (tan x + C i + C j + (e x + C 3 k 5.(6ptsThe curvature of the function y = sin x at x = π is (a (b 0 (c (d (e Does not exist. ( π Let r(t = t, sin t, 0. Then r (t =, cos t, 0, r (t = 0, sin t, 0, r = ( π, 0, 0, and r = 0,, 0. One of the formulas for curvature says that κ = r r r 3. But r r at t = π is 0 0 = 0, 0, and 0, 0, = (6pts What is the (approximate normal component of the acceleration for a car traveling 9 m/s around a curve with curvature κ = /3 m. (a 54.3 m/s (b.9 m/s (c m/s (d 400. m/s (e 40.3 m/s Using the formula that a N = κv. Plugging in the values from the problem we find that a N = (6pts Find the area of the triangle formed by the three points (, 0,, (, 0, and (3, 3, 3. (a 3 (b 0 (c 4 (d 3 (e.
3 Two vectors which form two sides of the triangle are, 0, =, 0,, 0, and, 3, = 3, 3, 3, 0,. Hence 0 = 3 0, (, 3 0 = 3, 0, 3 3 The area of the parallelogram is 3, 0, 3 = = 3 and the area of the triangle is half this. 8.(6pts Find the volume of the parallelepiped spanned by the three vectors,,, 0,, and 3,,. (a (b 3 (c 9 (d 3 (e 0 Answer is the absolute value of the triple product 0 3 = ( = = 9.(0pts Find an equation for the plane which goes through the point (,, 5 and contains the line,, t + 3, 4,. One vector in the plane is,,. A second is 3, 4,,, 5 =,, 4. Hence a normal vector for the plane is N = 4 = + 4, ( + 8, 4 =, 6, Hence an equation is, 6, x, y, z =, 6,,, 5 = 0 = 0 or x + 6y + z = 0 or x + 3y + z = 0. 0.(0pts Let Π be the plane passing through the point Q = (5, 0, 3 with normal vector n = 3,,. Let P be the point on Π which is closest to the origin. Let l be the line passing through the origin and P. Hint: parts (a and (b can be answered independently of each other. (a Find the coordinates of P. (b Find a vector equation for l. We find the position vector of P as OP. = Proj n OQ = OQ n n n n = 3,, = 6,, and so P = (6,,. The line l has direction n, and passes through the origin, so has vector equation 3,, x, y, z = 0
4 .(0pts Suppose the curve C has parametric equations: x(t = t 3 t, y(t = t, z(t = t + t (a Let P = (0,,. Find t 0 so that P = (x(t 0, y(t 0, z(t 0. (b Let r(t = t 3 t, t, t + t. Find r (t 0, the tangent vector to the above curve C at the point P = (0,,. (c Find the parametric equation for the tangent line to the above curve C at the point P = (0,,. (a 0 = t 3 0 t 0 = t 0 (t 0 (t 0 + implies t = 0,, or. = t 0 implies t 0 = i.e. t 0 =. (b r (t = 3t, t, t + r ( =,, 3 (c Let v(t =,, 3. Then the vector equation for the tangent line to C at P is given by tv( + 0,, = t, t, 3t + 0,, = t, t, 3t +. Then the parametric equation for the tangent line to C at P is given by x(t = t, y(t = t, z(t = 3t +.
5 .(0pts Suppose a particle has acceleration function a(t = j (sin tk for t 0. (t + Suppose that the initial position is r(0 = k and the initial velocity is v(0 = i + j + k. (a Find the velocity function. (b Find the position function. First find the velocity function as the integral of the acceleration function ( v(t = a(tdt = C i + t + + C j + (cos t + C 3 k Next we plug in t = 0 and compare with the initial velocity ( v(0 = C i C j + (cos 0 + C 3 k = C i + ( + C j + ( + C 3 k and by comparison to the given initial velocity, v(0 = i + j + k, we have C =, C = C 3 = 0 So, the velocity function is v(t = i + j + (cos tk t + The position function is the integral of the velocity function r(t = v(tdt = (t + D i + (ln(t + + D j + (sin t + D 3 k Next, we plug in t = 0 and compare with the initial position r(0 = (0 + D i + (ln(0 + + D j + (sin 0 + D 3 k = D i + D j + D 3 k and by comparison to the given initial position, r(0 = k, we have Thus, the position function is D = D = 0 D 3 = r(t = ti + ln(t + j + + sin tk OR using vector arithmetic v(t = 0, (t +, sin t dt = 0, Then v(0 = 0,, + C =,, so C =, 0, 0 and v(t =, t +, cos t t +, cos t + C Then r(t =, t +, cos t dt = t, ln t +, sin t + D Then r(0 = 0, 0, 0 + D = 0, 0, 0 so D = 0, 0, and r(t = t, ln t +, + sin t 3.(0pts Find an equation for the osculating plane of r(t = t, cos t, e t when t = 0. r (t =, sin t, e t, r (t = 0, cos t, e t, r(0 = 0,,, r (0 =, 0,, and r (0 = 0,,. The normal vector is, 0, 0,, =,,. So an equation is (x 0 (y (z = 0 or x y z =.
L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationAssignment 3. Solutions. Problems. February 22.
Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationMidterm Exam I, Calculus III, Sample A
Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More informationSAMPLE TEST 1a: SOLUTIONS
LAST (family) NAME: Test # 1 FIRST (given) NAME: Math 2Q04 ID # : TUTORIAL #: Instructions: You must use permanent ink. Tests submitted in pencil will not be considered later for remarking. This exam consists
More informationMotion in Space: Velocity and Acceleration
Study Sheet (13.4) Motion in Space: Velocity and Acceleration Here, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study: The motion of an object, including
More informationPractice Problems for Midterm 1
Practice Problems for Midterm 1 Here are some problems for you to try. A few I made up, others I found from a variety of sources, including Stewart s Multivariable Calculus book. (1) A boy throws a football
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 2537, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 537, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More information12.1. VectorValued Functions. VectorValued Functions. Objectives. Space Curves and VectorValued Functions. Space Curves and VectorValued Functions
12 VectorValued Functions 12.1 VectorValued Functions Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Analyze and sketch a space curve given
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More informationAB2.2: Curves. Gradient of a Scalar Field
AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationPROBLEM SET. Practice Problems for Exam #1. Math 2350, Fall 2004. Sept. 30, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam #1 Math 350, Fall 004 Sept. 30, 004 ANSWERS i Problem 1. The position vector of a particle is given by Rt) = t, t, t 3 ). Find the velocity and acceleration vectors
More informationMath 21a Old Exam One Fall 2003 Solutions Spring, 2009
1 (a) Find the curvature κ(t) of the curve r(t) = cos t, sin t, t at the point corresponding to t = Hint: You ma use the two formulas for the curvature κ(t) = T (t) r (t) = r (t) r (t) r (t) 3 Solution:
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More information10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.
SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationIntegration and projectile motion (Sect. 13.2)
Interation and projectile motion (Sect. 13.) Interation of vector functions. Application: Projectile motion. Equations of a projectile motion. Rane, Heiht, Fliht Time. Interation of vector functions Definition
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More informationComponents of Acceleration
Components of Acceleration Part 1: Curvature and the Unit Normal In the last section, we explored those ideas related to velocity namely, distance, speed, and the unit tangent vector. In this section,
More informationAssignment 5 Math 101 Spring 2009
Assignment 5 Math 11 Spring 9 1. Find an equation of the tangent line(s) to the given curve at the given point. (a) x 6 sin t, y t + t, (, ). (b) x cos t + cos t, y sin t + sin t, ( 1, 1). Solution. (a)
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationMath 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).
Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationChapter 1 Vectors, lines, and planes
Simplify the following vector expressions: 1. a (a + b). (a + b) (a b) 3. (a b) (a + b) Chapter 1 Vectors, lines, planes 1. Recall that cross product distributes over addition, so a (a + b) = a a + a b.
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More information= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L.
Math 21a Lines and lanes Spring, 2009 Lines in Space How can we express the equation(s) of a line through a point (x 0 ; y 0 ; z 0 ) and parallel to the vector u ha; b; ci? Many ways: as parametric (scalar)
More informationPROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS
PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationChapter 2. Parameterized Curves in R 3
Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,
More informationUNIVERSITY OF ALABAMA Department of Physics and Astronomy. PH 125 / LeClair Spring Problem Set 2
UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 125 / LeClair Spring 2009 Instructions: Problem Set 2 1. Answer all questions below. Follow the problemsolving template provided. 2. Some problems
More informationProblem set on Cross Product
1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j  3 k ) 2 Calculate the vector product of i  j and i + j (Ans ) 3 Find the unit vectors that are perpendicular
More informationParametric Curves, Vectors and Calculus. Jeff Morgan Department of Mathematics University of Houston
Parametric Curves, Vectors and Calculus Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu Online Masters of Arts in Mathematics at the University of Houston http://www.math.uh.edu/matweb/grad_mam.htm
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationParametric Equations and the Parabola (Extension 1)
Parametric Equations and the Parabola (Extension 1) Parametric Equations Parametric equations are a set of equations in terms of a parameter that represent a relation. Each value of the parameter, when
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More information2012 HSC Mathematics Sample Answers
2012 HSC Mathematics Sample Answers When examination committees develop questions for the examination, they may write sample answers or, in the case of some questions, answers could include. The committees
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationMath 113 HW #7 Solutions
Math 3 HW #7 Solutions 35 0 Given find /dx by implicit differentiation y 5 + x 2 y 3 = + ye x2 Answer: Differentiating both sides with respect to x yields 5y 4 dx + 2xy3 + x 2 3y 2 ) dx = dx ex2 + y2x)e
More informationMath 211, Multivariable Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions
Math 211, Multivariable Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions For each question, I give a model lution ( you can see the level of detail that I expect from you in the exam) me comments.
More informationVectorsAlgebra and Geometry
Chapter Two VectorsAlgebra and Geometry 21 Vectors A directed line segment in space is a line segment together with a direction Thus the directed line segment from the point P to the point Q is different
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More informationApplications of Integration to Geometry
Applications of Integration to Geometry Volumes of Revolution We can create a solid having circular crosssections by revolving regions in the plane along a line, giving a solid of revolution. Note that
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationMA261A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM
MA6A Calculus III 6 Fall Homework Solutions Due 9//6 :AM 9. # Find the parametric euation and smmetric euation for the line of intersection of the planes + + z = and + z =. To write down a line euation,
More informationPeople s Physics book 3e Ch 251
The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate
More informationSection 3.1 Calculus of VectorFunctions
Section 3.1 Calculus of VectorFunctions De nition. A vectorvalued function is a rule that assigns a vector to each member in a subset of R 1. In other words, a vectorvalued function is an ordered triple
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationVector Fields and Line Integrals
Vector Fields and Line Integrals 1. Match the following vector fields on R 2 with their plots. (a) F (, ), 1. Solution. An vector, 1 points up, and the onl plot that matches this is (III). (b) F (, ) 1,.
More information3.1 VectorValued Functions
3.1 VectorValued Functions Vectors differ from regular numbers because they have both a magnitude (length) and a direction. In our three dimensional world, vectors have one component for each direction,
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationThis function is symmetric with respect to the yaxis, so I will let  /2 /2 and multiply the area by 2.
INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,
More informationGiven three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);
1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationWinter 2012 Math 255. Review Sheet for First Midterm Exam Solutions
Winter 1 Math 55 Review Sheet for First Midterm Exam Solutions 1. Describe the motion of a particle with position x,y): a) x = sin π t, y = cosπt, 1 t, and b) x = cost, y = tant, t π/4. a) Using the general
More information1 3 4 = 8i + 20j 13k. x + w. y + w
) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationCHAPTER FIVE. 5. Equations of Lines in R 3
118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
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 information2 Using the definitions of acceleration and velocity
Physics I [P161] Spring 2008 Review for Quiz # 3 1 Main Ideas Two main ideas were introduced since the last quiz. 1. Using the definitions of acceleration and velocity to obtain equations of motion (chapter
More informationGreen s theorem is true, in large part, because. lim
Recall the statement of Green s Theorem: If is the smooth positively oriented boundary of a simply connected region D in the xyplane, and if F is a vector field in the xyplane with continuous first partial
More informationMATH 2030: ASSIGNMENT 1 SOLUTIONS
MATH 030: ASSIGNMENT SOLUTIONS Geometry and Algebra of Vectors Q.: pg 6, q,. For each vector draw the vector in standard position and with its tail at the point (, 3): [ [ [ ] [ 3 3 u =, v =, w =, z =.
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationNotice that v v w (4)( 15) ( 3)( 20) (0)(2) ( 2)( 15) (2)( 20) (5)(2)
The Cross Product When discussing the dot product, we showed how two vectors can be combined to get a number. Here, we shall see another way of combining vectors, this time resulting in a vector. This
More informationC relative to O being abc,, respectively, then b a c.
2 EPProgram  Strisuksa School  Roiet Math : Vectors Dr.Wattana Toutip  Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 2. Vectors A
More informationCatholic Schools Trial Examination 2004 Mathematics
0 Catholic Trial HSC Examination Mathematics Page Catholic Schools Trial Examination 0 Mathematics a If x 5 = 5000, find x correct to significant figures. b Express 0. + 0.. in the form b a, where a and
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More informationThis makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5
1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More information