Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155


 Sara Barber
 1 years ago
 Views:
Transcription
1 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 j i. (a) j (b) k k i In problems 3 through 8, let u 3i j k, v i 3j k and w i k. 3. Calculate u v. (a) 7 (b) 3 8,,. Calculate u v. (a), 0, (b) 8,,, 0, 7 5. Calculate u v w. (a) 30 (b) 8i j k. Calculate proj v u. 7, 7, 7 (a) (b) 7 7,, 7. Calculate cos where is the angle between u and v (a) (b) Which of the following vectors is orthogonal to both u and v? (a) k (b) 8i j k v u w 7
2 5 Chapter Vectors and the Geometry of Space 9. Which of the following is an orthogonal pair of vectors? (a) i j, i k (b) i j k, i j k 5i j k, i j 3k 3i k, j k 0. Which of the following is a set of parametric equations for the line through the points, 0, 3 and, 3, 3? (a) t (b) t t y t y 3t y t 3 t 3 3 t y 3t 3 3t. Write an equation of the plane that contains the line given by and is perpendicular to the line given by 7 y 5 3 y 3 7. (a) 7 y 7 0 (b) 3y 8 0 3y 0 7 y 7 0. Which of the following is a sketch of the plane given by y 3? (a) 3 (b) 3 3 y y 3. Calculate the distance from the point,, 3 to the line given by t, y, and t. 5 (a) (b) 3 3 y y 5
3 Chapter Test Bank 57. Find the center and the radius of the sphere given by y y 7 0. (a) Center:,, 3 (b) Center:,, 3 Center:,, 3 Radius: Radius: 37 Radius: Center:,, 3 Radius: Identify the quadric surface given by y. (a) Elliptic cone (b) Elliptic paraboloid Hyperbolic paraboloid Hyperboloid of one sheet (e) Hyperboloid of two sheets. Identify the quadric surface sketched at the right. (a) Hyperboloid of two sheets (b) Elliptic paraboloid Hyperboloid of one sheet Elliptic cone y 7. Find the equation of the surface of revolution if the generating curve, y, is revolved about the yais. (a) y y 0 (b) 3 0 y 0 3 y 0 8. Epress the cylindrical point, in spherical coordinates., 3 (a) 5, (b) 5,, 3, 0 5 5,, Find an equation in spherical coordinates for the surface given by the rectangular equation y. (a) (b) r sec 5,, 0.93 cot csc tan csc sec 0. Find the work done in moving a particle from P3,, 0 to Q, 3, if the magnitude and direction of the force are given by v 5,,. (a) 5 (b) cot csc sec
4 58 Chapter Vectors and the Geometry of Space Test Form B 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) i j 7k (b) i j k 5 i 5 j 7 5 k 5 i 5 j 5 k. Calculate j k. (a) i (b) k i j In problems 3 through 8, let u i j k, v 3i j k and w j k. 3. Calculate u v. (a) 7, 7, 7 (b) 3,,. Calculate u v. (a) 7, 7, 7 (b) 3,, 5. Calculate u v w. (a) (b) 0 i 8j. Calculate the component of u in the direction of v. (a) (b) 3 7. Calculate cos where is the angle between u and v. 9 9 (a) (b) 39 9 i 3 j 3 k
5 Chapter Test Bank Which of the following vectors is orthogonal to both u and w? (a) j (b) w u v j k 9. Which of the following statements is true about the vectors u v i 3 j k? i 3 j k and (a) u and v are orthogonal. (b) u and v are parallel. u is a unit vector of v. The angle between u and v is. 0. Which of the following is a set of parametric equations for the line through the points 3,, 0 and, 3, 3? (a) 7t (b) 3t 3 t y 3 t y 3 t y 5t 3 3t 3 3t 3 t y 3t 3t. Find an equation of the plane that passes through the points,,, 3,, 3 and,, 0. (a) y (b) 7 y y y 5 0. If the equation of a cylinder is given by 0, then (a) The rulings are parallel to the ais. (b) The graph is a parabola. The rulings are parallel to the ais. The generating curve is a parabola. 3. Calculate the distance from the point,, 3 to the plane given by 3 y 0. (a) (b) 5. Find the center and radius of the sphere given by the equation y y 0. (a) Center:,, (b) Center:,, Center:,, Radius: Radius: Radius: 0 Center:,, Radius: 0
6 0 Chapter Vectors and the Geometry of Space 5. Identify the quadric surface given by y. (a) Elliptic cone (b) Elliptic paraboloid Hyperbolic paraboloid Hyperboloid of one sheet (e) Hyperboloid of two sheets. Identify the quadric surface sketched at the right. (a) Hyperboloid of two sheets Hyperboloid of one sheet (b) Elliptic paraboloid Elliptic cone y 7. Find the equation of revolution if the generating curve, y, is revolved about the ais. (a) y y 0 (b) 3 0 y 0 3 y Epress the spherical point 3, in rectangular coordinates., (a) 3 (b) 33, 33, 3, 3, 33 3, 33, 3 3 3, 33, 3 9. Find an equation in cylindrical coordinates for the surface given by the rectangular equation y. (a) r (b) r sec cot csc cot csc sec 0. Find the work done in moving a particle from P0,, to Q3,, if the magnitude and direction of the force are given by v,, 3. (a) 5 (b) 8 3
7 Chapter Test Bank Test Form C Chapter Name Class Date Section. A vector v has initial point, 3 and terminal point,. Find the unit vector in the direction of v. 3 (a) (b) 5 i 5 i 5 j 5 j 5 i 3 5 j i j 5. Find a vector with magnitude 0 in the direction of v,. (a) (b) 0, 0,, 5, 5 3. Forces with magnitudes of 00 and 300 pounds act on a machine part at angles of 0 and 5, respectively, with the positive ais. Find the direction of the resultant forces. (a) 7. (b) Find the standard equation for the sphere that has points, 3, 5 and,, as endpoints of a diameter. (a) y 38 (b) y 3 y 3 5 y 3 5. Determine which vector is parallel to the vector v, 3,. (a),, (b) 3,, 3, 9, 3. Find the unit vector in the direction of the vector v i j k. (a) (b) i j k 9 i 9 j 5 i 5 j 3 5 k 9 k 3 i 3 j 3 k, 3,
8 Chapter Vectors and the Geometry of Space 7. Find u v if u 0, v 5, and the angle between vectors u and v is (a) 3003 (b) Find the vector component of u orthogonal to v for u i j and v i j. (a) 3 (b) i 3 i j j i j i 5 j 9. Calculate the angle that vector v 3i 5j k makes with the positive yais. (a) 59.5 (b) v v v (a) v (b) v proj u v v. Calculate u v for u 3i j k and v i j 3k. (a) i j k (b) 3i 7j 5k i j k i j k. Find the area of the parallelogram having vectors v i j k and v 3i j k as adjacent sides. (a) 3 (b) Find an equation of the plane determined by the points,, 3,, 3,, and 0,,. (a) 8i j 3k 0 (b) y 0 y 5 y 3 7. Find the point of intersection of the line given by the parametric equations 3 t, y 7 8t, and t with the yplane. (a) 0, 5, 7 (b) 0, 37 3, 3 3, 7, 7, 33, 7
9 Chapter Test Bank 3 5. Find a generating curve for the surface of revolution given by y 3. (a) y 3 (b) 3 y Both (a) and (b). Write the equation in standard form and identify the quadric surface given by 9y 3 0. (a) 3 9y, Cone (b) 9 y, Hyperbolic paraboloid 9 y, Hyperboloid of one sheet 9 y 9, Hyperbolic paraboloid 7. Epress the spherical coordinate point, in cylindrical coordinates., (a) 3 (b) 33, 3, 33, 3, 3 33,, , 3, 3 8. Find a rectangular equation for the graph represented by the spherical equation cos. (a) y (b) y y 0
10 Chapter Vectors and the Geometry of Space Test Form D 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,.. Calculate i k. In problems 3 through 8, let u i j k, v i j k and w 3i k. 3. Calculate u v.. Calculate u v. 5. Calculate u v w.. Calculate proj v u. 7. Calculate where is the angle between u and v. 8. Calculate a vector perpendicular to both u and w. 9. Let L : t, y 5t, 7t and L : t, y 5 t, t. Show that the vectors v and parallel to lines L and L, respectively, are orthogonal. v 0. Find parametric equations for the line through the point,, 3 and parallel to the line given by y 5.. Write an equation of the plane that is determined by the lines given by and y 3.. Sketch the plane given by 3 y. y 3. Find the numbers and y such that the point, y, lies on the line passing through the points, 5, 7 and 0, 3,.. Write the equation of the sphere, y y 8 9 0, in standard form and find the center and radius Identify and sketch the quadric surface given by y 0.
11 Chapter Test Bank 5. Sketch the surface represented by the equation y. 7. Find the equation of the surface of revolution if the generating curve, 3, is revolved about the ais. 8. Convert the point 3, 3, 7 from rectangular to cylindrical coordinates. 9. Find an equation in rectangular coordinates for the spherical coordinate equation and identify the surface. 9 csc csc 0. Find the area of the parallelogram having vectors u i j k and v 5i k as adjacent sides.
12 Chapter Vectors and the Geometry of Space Test Form E Chapter Name Class Date Section. A vector v has initial point, and terminal point, 3. Find the unit vector in the direction of v.. Find a vector with magnitude 3 in the direction of the vector v,. 3. Three forces with magnitude of 50, 0, and 0 pounds act on an object at angles of 0, 30, and 90, respectively, with the positive ais. Find the direction and magnitude of the resultant forces.. Find the standard equation for the sphere that has points, 5, and 3, 7, 3 as endpoints of a diameter. 5. Determine whether each vector is parallel to the vector v i j 5k. If it is, find c such that v cu. a. u 8i j 0k b. u 8i j 0k c. u 3i 3 d. u i j 5 j 5 k k. A vector v has initial point,, 3 and terminal point,,. a. Write v in the component form. b. Write v as a linear combination of the standard unit vectors. c. Find the magnitude of v. d. Find the unit vector in the direction of v. e. Find the unit vector in the direction opposite that of v. 7. Find u v if u 7, v, and the angle between u and v is. 8. Let u i j k and v 3i j k. (a) Find the projection of u onto v. (b) Find the vector component of u orthogonal to v. 9. Find direction cosines for the vector with initial point,, 3 and terminal point, 3, Determine a scalar, k, so that the vectors u 3i j and v i kj are orthogonal.
13 Chapter Test Bank 7. Let u i j k and v 3i j k. a. Find u v. b. Show that the vector u v is orthogonal to v.. Calculate the area of the triangle with vertices 3,,,,, 5, and,,. 3. Find parametric equations for the line perpendicular to the lines given by 3 and y passing through the point,, y Find the point of intersection of the line y 3 and the plane 3 3y Sketch the surface represented by y.. Identify each of the quadric surfaces: a. 3 3y 3 3y 0 b. 5 y c. y 7. Consider the surface whose rectangular equation is y. a. Find an equation for the surface in cylindrical coordinates. b. Find an equation for the surface in spherical coordinates. 8. Sketch the surface represented by the spherical equation csc.
Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationL 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 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 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 informationSolution: 2. Sketch the graph of 2 given the vectors and shown below.
7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit
More information2 Topics in 3D Geometry
2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and
More informationHomework 3 Model Solution Section
Homework 3 Model Solution Section 12.6 13.1. 12.6.3 Describe and sketch the surface + z 2 = 1. If we cut the surface by a plane y = k which is parallel to xzplane, the intersection is + z 2 = 1 on a plane,
More information42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections
2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
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 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 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
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 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 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 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 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 informationSurface Normals and Tangent Planes
Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to
More informationSphere centered at the origin.
A Quadratic surfaces In this appendix we will study several families of socalled quadratic surfaces, namely surfaces z = f(x, y) which are defined by equations of the type A + By 2 + Cz 2 + Dxy + Exz
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 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 informationLines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line
Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space
More informationHyperboloid of Two Sheets
Math 59 Winter 009 Recitation Handout 5: Sketching D Surfaces If you are simply given an equation of the form z = f(x, y) and asked to draw the graph, producing the graph can be a very complicated and
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 information3.6 Cylinders and Quadric Surfaces
3.6 Cylinders and Quadric Surfaces Objectives I know the definition of a cylinder. I can name the 6 quadric surfaces, write their equation, and sketch their graph. Let s take stock in the types of equations
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(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 + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationSection 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates
Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in
More informationChange of Coordinates in Two Dimensions
CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse
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 informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationSection 2.1 Rectangular Coordinate Systems
P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is
More informationHigher Mathematics Homework A
Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(3,1) and B(5,3) 2. Find the equation of the tangent to the circle
More informationThere are good hints and notes in the answers to the following, but struggle first before peeking at those!
Integration Worksheet  Using the Definite Integral Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the various types of problems. Important:
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationSECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xyplane. It is sketched in Figure 11.
SECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is
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 informationSurface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry
Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel
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 informationExtra Problems for Midterm 2
Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
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 informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationQuadratic curves, quadric surfaces
Chapter 3 Quadratic curves, quadric surfaces In this chapter we begin our study of curved surfaces. We focus on the quadric surfaces. To do this, we also need to look at quadratic curves, such as ellipses.
More informationMATHEMATICS FOR ENGINEERS & SCIENTISTS 7
MATHEMATICS FOR ENGINEERS & SCIENTISTS 7 We stress that f(x, y, z) is a scalarvalued function and f is a vectorvalued function. All of the above works in any number of dimensions. For instance, consider
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 information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationChapter 5 Applications of Integration
MA111 Application of Integration Asst.Prof.Dr.Supranee Lisawadi 1 Chapter 5 Applications of Integration Section 5.1 Area Between Two Curves In this section we use integrals to find areas of regions that
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
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 informationContents. 12 Applications of the Definite Integral Area Solids of Revolution Surface Area...
Contents 12 Applications of the Definite Integral 179 12.1 Area.................................................. 179 12.2 Solids of Revolution......................................... 181 12.3 Surface
More informationGeometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
More informationAngle  a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees
Angle  a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in
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 informationVectorValued Functions The Calculus of VectorValued Functions Motion in Space Curvature Tangent and Normal Vect. Calculus. VectorValued Functions
Calculus VectorValued Functions Outline 1 VectorValued Functions 2 The Calculus of VectorValued Functions 3 Motion in Space 4 Curvature 5 Tangent and Normal Vectors 6 Parametric Surfaces Vector Valued
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 information1. A plane passes through the apex (top point) of a cone and then through its base. What geometric figure will be formed from this intersection?
Student Name: Teacher: Date: District: Description: MiamiDade County Public Schools Geometry Topic 7: 3Dimensional Shapes 1. A plane passes through the apex (top point) of a cone and then through its
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 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
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 informationMVE041 Flervariabelanalys
MVE041 Flervariabelanalys 201516 This document contains the learning goals for this course. The goals are organized by subject, with reference to the course textbook Calculus: A Complete Course 8th ed.
More informationLines and Planes in R 3
.3 Lines and Planes in R 3 P. Daniger Lines in R 3 We wish to represent lines in R 3. Note that a line may be described in two different ways: By specifying two points on the line. By specifying one point
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More informationWhen Is the Sum of the Measures of the Angles of a Triangle Equal to 180º?
Sum of the Measures of Angles Unit 9 NonEuclidean Geometries When Is the Sum of the Measures of the Angles of a Triangle Equal to 180º? Overview: Objective: This activity illustrates the need for Euclid
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 115 Spring 2014 Written Homework 10SOLUTIONS Due Friday, April 25
Math 115 Spring 014 Written Homework 10SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain
More informationMATH 1231 S2 2010: Calculus. Section 1: Functions of severable variables.
MATH 1231 S2 2010: Calculus For use in Dr Chris Tisdell s lectures Section 1: Functions of severable variables. Created and compiled by Chris Tisdell S1: Motivation S2: Function of two variables S3: Visualising
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 informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationPractice Problems for Midterm 2
Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,
More informationApplications of the Integral
Chapter 6 Applications of the Integral Evaluating integrals can be tedious and difficult. Mathematica makes this work relatively easy. For example, when computing the area of a region the corresponding
More informationThe Dot Product. If v = a 1 i + b 1 j and w = a 2 i + b 2 j are vectors then their dot product is given by: v w = a 1 a 2 + b 1 b 2
The Dot Product In this section, e ill no concentrate on the vector operation called the dot product. The dot product of to vectors ill produce a scalar instead of a vector as in the other operations that
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 information*1. Understand the concept of a constant number like pi. Know the formula for the circumference and area of a circle.
Students: 1. Students deepen their understanding of measurement of plane and solid shapes and use this understanding to solve problems. *1. Understand the concept of a constant number like pi. Know the
More information7.4 Applications of Eigenvalues and Eigenvectors
37 Chapter 7 Eigenvalues and Eigenvectors 7 Applications of Eigenvalues and Eigenvectors Model population growth using an age transition matri and an age distribution vector, and find a stable age distribution
More informationx 2 would be a solution to d y
CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y
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 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 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 informationEngineering Geometry
Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the righthand rule.
More information1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers.
1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form a + by + c = 0, where a, b and c are integers. (b) Find the length of AB, leaving your answer in surd form.
More information1) Convert 13 32' 47" to decimal degrees. Round your answer to four decimal places.
PRECLCULUS FINL EXM, PRCTICE UNIT ONE TRIGONOMETRIC FUNCTIONS ) Convert ' 47" to decimal degrees. Round your answer to four decimal places. ) Convert 5.6875 to degrees, minutes, and seconds. Round to the
More informationGeometry Concepts. Figures that lie in a plane are called plane figures. These are all plane figures. Triangle 3
Geometry Concepts Figures that lie in a plane are called plane figures. These are all plane figures. Polygon No. of Sides Drawing Triangle 3 A polygon is a plane closed figure determined by three or more
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationa.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a
Bellwork a.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationOverview Mathematical Practices Congruence
Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason
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 informationGeneral tests Algebra
General tests Algebra Question () : Choose the correct answer :  If = then = a)0 b) 6 c)5 d)4  The shape which represents Y is a function of is :   V A B C D o   y   V o    V o    V o  if
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 information6. Vectors. 1 20092016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More information... ... . (2,4,5).. ...
12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,
More informationModule 8 Lesson 4: Applications of Vectors
Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems
More informationFigure 1: Volume between z = f(x, y) and the region R.
3. Double Integrals 3.. Volume of an enclosed region Consider the diagram in Figure. It shows a curve in two variables z f(x, y) that lies above some region on the xyplane. How can we calculate the volume
More informationGeometry Vocabulary. Created by Dani Krejci referencing:
Geometry Vocabulary Created by Dani Krejci referencing: http://mrsdell.org/geometry/vocabulary.html point An exact location in space, usually represented by a dot. A This is point A. line A straight path
More information