(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.

Size: px
Start display at page:

Download "(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter."

Transcription

1 Solutions Review for Eam #1 Math 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 = = 44 = (b) Find the midpoint between P and Q. Solution. midpoint = 1 2 (P + Q) = 1 2 (6, 4, 0) = (3, 2, 0). (c) Find a parametriation of the line through P and Q. Solution. The vector from P to Q is v = 2, 6, 2, so one parametriation of the line is Note that L(0) = P and L(1) = Q. L(t) = P + tv = 2 + 2t, 5 + 6t, 1 + 2t. (d) Find the equation of the sphere that has the line segment from P to Q as a diameter. Solution. If the segment from P to Q is a diameter, then its midpoint is the center of the sphere and the half the distance from P to Q is the radius. So, the sphere is centered at (3, 2, 0) with radius 11, which satisfies the equation ( 3) 2 + ( + 2) = 11. (e) Sketch the points P and Q, the midpoint, and the line. P M Q

2 2. Consider the plane given b the equation = 6. (a) Find the -, -, and -intercepts of the plane. Solution. The -intercept is = 1, the -intercept is = 2, and the -intercept is = 6. (b) Find a vector n perpendicular (normal) to the plane. Solution. A normal vector to the plane is n = 6, 3, 1. (c) Sketch the plane along with its normal vector. n 3. Consider the curve in R 3 given b r(t) = 3 sin(t 2 ), sin(t 2 ), 2 cos(t 2 ). (a) Find the arclength of r(t) from t = 0 to t = π. Solution. The arclength is (b) π 0 r dt = = π 0 π 0 12t2 cos 2 (t 2 ) + 4t 2 cos 2 (t 2 ) + 16t 2 sin(t 2 ) dt 4t = 2t 2 π = 2π. Compute the derivative r (t) and second derivative r (t) of the curve r(t). 0 Solution. The derivative is r (t) = 2t 3 cos(t 2 ), 2t cos(t 2 ), 4t sin(t 2 ).

3 The second derivative is r (t) = 4t 2 3 sin(t 2 ) cos(t 2 ), 4t 2 sin(t 2 ) + 2 cos(t 2 ), 8t 2 cos(t 2 ) 4 sin(t 2 ). (c) Find the point r = r( π), the tangent vector r = r ( π), and the acceleration vector r = r ( π) at time t = π. Solution. At t = π, we have r = 0, 0, 2, r = 2 3π, 2 π, 0, and r = 2 3, 2, 8π. (d) Find the unit tangent vector T at time t = π. Solution. At t = π, we have T = r r = π, 2 π, 0 = π 3 2, 1 2, 0. (e) Find the tangential component a T of the acceleration at time t = π. Solution. At t = π, the tangential component of the accleration is 3 a T = T r = 2, 1 2, 0 2 3, 2, 8π = 4. (f) Find the normal component a N of the acceleration at time t = π. Solution. At t = π, the normal component of the acceleration is a N = T r = 4π, 4π 3, 0 = 8π. (g) Find the unit normal vector N at time t = π. Solution. At t = π, the normal vector is N = 1 (r a T T) = 1 [ 2 3, 2, 8π a N 8π (h) Find the binormal vector B at time t = π. 2 ] 3, 2, 0 = 0, 0, 1. Solution. At t = π, the binormal vector is B = T N = 1 3 2, 2, 0. (i) Find the curvature κ at time t = π. Solution. At t = π, the curvature is κ = a N r 2 = 8π 16π = 1 2.

4 4. Consider the parabola in R 2 given b r(t) = t 2 2t + 1, 4t 4. (a) Carefull sketch the curve from t = 0 to t = 3. Label the point p = r(2). Solution. The point is p = 1, 4. v p a (b) Find the tangent vector v to the curve at t = 2. Sketch the tangent vector on the graph in part (a) starting from the point p. Solution. The derivative is r (t) = 2t 2, 4, so the tangent vector is v = 2, 4. (c) Find the parametric equation of the tangent line to the curve at t = 2. Sketch the line on the graph in part (a). Solution. The tangent line at t = 2 is given b L(t) = p + tv = 1 + 2t, 4 + 4t. (d) Find the acceleration vector a to the curve at t = 2. Sketch the accleration vector on the graph in part (a) starting from the point p. Solution. The acceleration is r (t) = 2, 0, so the at t = 2, we have a = 2, 0. (e) Compute the curvature at t = 2. Solution. The curvature at t = 2 is κ = v a v 3 = 0, 0, = /2 =

5 5. The following is the equation of a sphere. Find its center and radius. Solution. Completing the square, we get = 0. ( ) + ( ) + ( ) = ( + 4) 2 + ( 1) 2 + ( 5) 2 = 42. So, it is a sphere with center = ( 4, 1, 5) and radius = Consider the vectors u = 1, 2, 3 and v = 2, 1, 1. (a) Find a vector a which is perpendicular to both u and v. Solution. Take the cross product to get one such vector a = u v = 1, 7, 5. (b) Find the equation of the plane through the origin that contains both of the vectors u and v. Solution. Since the plane contains both u and v as points on the plane, then its normal vector must be perpendicular to both u and v, so we ma take a as its normal vector. The equation of the resulting plane is = 0. (c) Find the equation of the plane parallel to the one from part (b) that goes through the point (2, 1, 4). Solution. Since this plane is parallel to the other, it has the same normal vector; in other words, it is the same plane, but shifted to contain the point (2, 1, 4). Its equation is ( 2) + 7( + 1) 5( 4) = 0 or = Find the equation of the plane through the points (3, 2, 0), (0, 1, 2), and ( 2, 0, 5). Solution. The plane contains the directional vectors from each point to each other point. Consider, for instance the vectors from the second point to the first and third points, namel u = 3, 1, 2 and v = 2, 1, 7. If we take the cross product, we will get a vector n which is perpendicular to both u and v, and hence is perpendicular to the plane containing the three points: n = u v = 9, 25, 1. Taking the second point as a base point, we see that the equation of the plane is 9 25( 1) ( + 2) = 0 or 9 25 = 23. If ou plug in the first and third point, ou can easil check that this equation is correct.

6 8. Suppose that a surface has the following level curves from = 0 to = 4. Describe and sketch the surface. = 0 = 1 = 2 = 3 = 4 Solution. The surface is part of a cone, starting as a point at the origin, and with the height and radius growing at the same rate. 9. Find and sketch the level curves of the surface = 2 for -values = 2, 1, 0, 1, 2. Solution. Setting = k, we get the parabola = 2 + k. Here are the graphs of the level curves. = 2 = 1 = 0 = 1 = 2

7 10. Consider the surface given b = 36. (a) Find and sketch the equation of the curve obtained b intersecting the surface with the -plane. Solution. Setting = 0, we get = 1, which is an ellipse. 36 (b) Find and sketch the equation of the curve obtained b intersecting the surface with the -plane. Solution. Setting = 0, we get = 1, which is a hperbola. 9 (c) Find and sketch the equation of the curve obtained b intersecting the surface with the -plane. Solution. Setting = 0, we get = 1, which is a hperbola. 9 (d) For a -value fied to be constant, = k, find the equation of the level curve and identif what tpe of curve it is. Solution. Setting = k, we get = k 2. Dividing b the right-hand side (which is alwas positive), we get an ellipse for each value of k. Sketches of coordinate plane curves along with the level curve = 4. -plane -plane = plane 6 4 2

8 (e) Draw a rough sketch of the surface. 11. Convert the equations for the following surfaces from spherical coordinates to both clindrical and Cartesian coordinates. Use the Cartesian coordinate representation to identif the surface. (a) ρ sin φ = 2 cos θ Solution. Since ρ sin φ = r, the clindrical equation is r = 2 cos θ. If we multipl both sides of this b r, we get r 2 = 2r cos θ, which becomes the Cartesian coordinate equation = 2 or ( 1) = 1, so it is an upright clinder that intersects the -plane in the circle of radius 1 centered at (1, 0). (b) ρ = 6 sin φ sin θ Solution. If we multipl both sides b ρ, we get ρ 2 = 6ρ sin φ sin θ. Since r = ρ sin φ, and ρ 2 = r 2 + 2, we get the clindrical equation r = 6r sin θ. Now, since = r sin θ and r 2 = 2 + 2, we get the Cartesian equation = 6 or 2 + ( 3) = 9,, which is a sphere of radius 3 centered at (0, 3, 0).

9 12. Convert the equations for the following surfaces from Cartesian coordinates to both clindrical and spherical coordinates. (a) = 2 Solution. First we note that if = 0, then = 0 and can be anthing, so the -ais is contained in this surface. If we factor 2 out of the left-hand terms, we get 2 ( ) = 2 or = 2 2. Note that the second equation is onl valid awa from the -ais. We convert to clindrical coordinates b using = r 2 and / = tan θ to get the clindrical equation Similarl, the spherical equation is r = tan 2 θ. ρ 2 = tan 2 θ. This doesn t reall help us to sketch it, but it was good practice! (b) = 0 Solution. The clindrical equation is r 2 = 2. In other words, r = ±. If ou think about this long enough, ou ll realie that the surface is both the lower and upper pieces of the cone that ou sketched in problem 8, since each level curve had a circle of that same radius. To find the spherical representation, we can just add 2 to both sides to get r = 2 2, which becomes the spherical equation ρ 2 = 2ρ 2 sin 2 φ. If ρ = 0, it is the origin, otherwise we cancel ρ 2 and solve for φ to get 1 = 2 sin 2 φ φ = ± π 4. If ou think about this long enough, ou ll realie it also describes the cone from problem 8, since φ is fied at a 45 angle and ρ and θ are allowed to be anthing. Another wa to visualie this is to take the line = in the -plane (which is at a 45 angle) and rotate it around the -ais to get the cone.

10 13. Consider the vectors u = 3, 1 and v = 2, 1. (a) Find the projection of u onto v. Solution. (b) Find the projection of v onto u. pr v (u) = ( u v ) v = 5 v v 5 v = v. Solution. pr u (v) = ( v u ) u = 5 u u 10 u = 1 2 u. (c) Find the angle between u and v. Solution. The cosine of the angle between them is cos θ = u v u v = 5 = Therefore, the angle between them is θ = π 4. (d) Sketch the vectors u, v, and the projections. v = pr v (u) θ pr u (v) u

Triple Integrals. y x

Triple Integrals. y x Triple Integrals. (a) If is an solid (in space), what does the triple integral dv represent? Wh? (b) Suppose the shape of a solid object is described b the solid, and f(,, ) gives the densit of the object

More information

SOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003

SOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003 SOLUTIONS TO SECOND PRACTICE EXAM Math a, Spring 3 Problem ) ( points) Circle for each of the questions the correct letter. No justifications are needed. Your score will be C W where C is the number of

More information

Math 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 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 information

MA 351 Fall 2007 Exam #1 Review Solutions 1

MA 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 information

Vector Fields. Field (II) Field (V)

Vector Fields. Field (II) Field (V) Math 1a Vector Fields 1. Match the following vector fields to the pictures, below. Eplain our reasoning. (Notice that in some of the pictures all of the vectors have been uniforml scaled so that the picture

More information

SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253

SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253 SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the

More information

MATHEMATICS 200 December 2014 Final Exam Solutions

MATHEMATICS 200 December 2014 Final Exam Solutions MATHEMATICS 2 December 214 Final Eam Solutions 1. Suppose that f,, z) is a function of three variables and let u 1 6 1, 1, 2 and v 1 3 1, 1, 1 and w 1 3 1, 1, 1. Suppose that at a point a, b, c), Find

More information

MATHEMATICS 200 December 2011 Final Exam Solutions

MATHEMATICS 200 December 2011 Final Exam Solutions MATHEMATICS December 11 Final Eam Solutions 1. Consider the function f(, ) e +4. (a) Draw a contour map of f, showing all tpes of level curves that occur. (b) Find the equation of the tangent plane to

More information

3 = arccos. A a and b are parallel, B a and b are perpendicular, C a and b are normalized, or D this is always true.

3 = arccos. A a and b are parallel, B a and b are perpendicular, C a and b are normalized, or D this is always true. Math 210-101 Test #1 Sept. 16 th, 2016 Name: Answer Key Be sure to show your work! 1. (20 points) Vector Basics: Let v = 1, 2,, w = 1, 2, 2, and u = 2, 1, 1. (a) Find the area of a parallelogram spanned

More information

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere. MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

More information

D = 2(2) 3 2 = 4 9 = 5 < 0

D = 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 information

Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general

Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate

More information

MATHEMATICS 200 December 2013 Final Exam Solutions

MATHEMATICS 200 December 2013 Final Exam Solutions MATHEMATICS 2 December 21 Final Eam Solutions 1. Short Answer Problems. Show our work. Not all questions are of equal difficult. Simplif our answers as much as possible in this question. (a) The line L

More information

Tangent and Normal Vector - (11.5)

Tangent 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 information

Math 261 Solutions to Sample Final Exam Problems

Math 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 information

Multiple 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. 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 information

Math 261 Solutions To Sample Exam 2 Problems

Math 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 information

EXAM 2 ANSWERS AND SOLUTIONS, MATH 233 WEDNESDAY, OCTOBER 18, 2000

EXAM 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 information

Vectors and the Geometry of Space

Vectors 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 information

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is

1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order

More information

MATH Final Review

MATH Final Review MATH 1592 - Final Review 1 Chapter 7 1.1 Main Topics 1. Integration techniques: Fitting integrands to basic rules on page 485. Integration by parts, Theorem 7.1 on page 488. Guidelines for trigonometric

More information

4 The Cartesian Coordinate System- Pictures of Equations

4 The Cartesian Coordinate System- Pictures of Equations The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean

More information

ES.182A Topic 36 Notes Jeremy Orloff

ES.182A Topic 36 Notes Jeremy Orloff ES.82A Topic 36 Notes Jerem Orloff 36 Vector fields and line integrals in the plane 36. Vector analsis We now will begin our stud of the part of 8.2 called vector analsis. This is the stud of vector fields

More information

B. Incorrect! Presence of all three variables in the equation makes it not cylindrical.

B. Incorrect! Presence of all three variables in the equation makes it not cylindrical. Calculus III - Problem Drill 08: Clindrical and Quadric Surfaces Question No. of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the. Which

More information

Math 21a: Multivariable calculus. List of Worksheets. Harvard University, Spring 2009

Math 21a: Multivariable calculus. List of Worksheets. Harvard University, Spring 2009 Math 2a: Multivariable calculus Harvard Universit, Spring 2009 List of Worksheets Vectors and the Dot Product Cross Product and Triple Product Lines and Planes Functions and Graphs Quadric Surfaces Vector-Valued

More information

Vector Functions & Space Curves MATH 2110Q

Vector 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

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse

More information

ES.182A Problem Section 11, Fall 2018 Solutions

ES.182A Problem Section 11, Fall 2018 Solutions Problem 25.1. (a) z = 2 2 + 2 (b) z = 2 2 ES.182A Problem Section 11, Fall 2018 Solutions Sketch the following quadratic surfaces. answer: The figure for part (a) (on the left) shows the z trace with =

More information

Conic Section: Circles

Conic Section: Circles Conic Section: Circles Circle, Center, Radius A circle is defined as the set of all points that are the same distance awa from a specific point called the center of the circle. Note that the circle consists

More information

APPM 2350, Summer 2018: Exam 1 June 15, 2018

APPM 2350, Summer 2018: Exam 1 June 15, 2018 APPM 2350, Summer 2018: Exam 1 June 15, 2018 Instructions: Please show all of your work and make your methods and reasoning clear. Answers out of the blue with no supporting work will receive no credit

More information

a Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).

a Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8). Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates

More information

1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.

1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3. Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient

More information

MTHE 227 Problem Set 10 Solutions. (1 y2 +z 2., 0, 0), y 2 + z 2 < 4 0, Otherwise.

MTHE 227 Problem Set 10 Solutions. (1 y2 +z 2., 0, 0), y 2 + z 2 < 4 0, Otherwise. MTHE 7 Problem Set Solutions. (a) Sketch the cross-section of the (hollow) clinder + = in the -plane, as well as the vector field in this cross-section. ( +,, ), + < F(,, ) =, Otherwise. This is a simple

More information

MTHE 227 Problem Set 2 Solutions

MTHE 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 information

Vectors, dot product, and cross product

Vectors, dot product, and cross product MTH 201 Multivariable calculus and differential equations Practice problems Vectors, dot product, and cross product 1. Find the component form and length of vector P Q with the following initial point

More information

APPM 1360 Final Exam Spring 2016

APPM 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 information

Coordinate goemetry in the (x, y) plane

Coordinate 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 information

BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON

BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON Name: Problem 1 In this problem, we will use vectors to show that an angle formed by connecting a point on a circle to two

More information

Math 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas

Math 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas Math 180 Chapter 10 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 10.1 Section 10.1 Parabolas Definition of a Parabola A parabola is the set of all points in a plane

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=

More information

MATH H53 : Mid-Term-1

MATH H53 : Mid-Term-1 MATH H53 : Mid-Term-1 22nd September, 215 Name: You have 8 minutes to answer the questions. Use of calculators or study materials including textbooks, notes etc. is not permitted. Answer the questions

More information

VECTOR FUNCTIONS. which a space curve proceeds at any point.

VECTOR FUNCTIONS. which a space curve proceeds at any point. 3 VECTOR FUNCTIONS Tangent vectors show the direction in which a space curve proceeds at an point. The functions that we have been using so far have been real-valued functions. We now stud functions whose

More information

MATH 12 CLASS 5 NOTES, SEP

MATH 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 information

e x3 dx dy. 0 y x 2, 0 x 1.

e x3 dx dy. 0 y x 2, 0 x 1. Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=

More information

MATH 223 FINAL EXAM STUDY GUIDE ( )

MATH 223 FINAL EXAM STUDY GUIDE ( ) MATH 3 FINAL EXAM STUDY GUIDE (017-018) The following questions can be used as a review for Math 3 These questions are not actual samples of questions that will appear on the final eam, but the will provide

More information

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.

More information

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:

13.1. 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 information

Math 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Math 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 1 chapter si practice eam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Which equation matches the given calculator-generated graph and description?

More information

BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON. a b. Name: Problem 1

BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON. a b. Name: Problem 1 BROWN UNIVERSITY MATH 0350 MIDTERM 19 OCTOBER 2017 INSTRUCTOR: SAMUEL S. WATSON Name: Problem 1 In this problem, we will use vectors to show that an angle formed by connecting a point on a circle to two

More information

1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...

1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates... Contents 1 Geometry of R 1.1 Conic Sections............................................ 1. Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................

More information

DIFFERENTIATION. 3.1 Approximate Value and Error (page 151)

DIFFERENTIATION. 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 information

Department of Mathematical and Statistical Sciences University of Alberta

Department of Mathematical and Statistical Sciences University of Alberta MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #5 Completion Date: Frida Februar 5, 8 Department of Mathematical and Statistical Sciences Universit of Alberta Question. [Sec.., #

More information

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

More information

ARE YOU READY FOR CALCULUS?? Name: Date: Period:

ARE YOU READY FOR CALCULUS?? Name: Date: Period: ARE YOU READY FOR CALCULUS?? Name: Date: Period: Directions: Complete the following problems. **You MUST show all work to receive credit.**(use separate sheets of paper.) Problems with an asterisk (*)

More information

The Distance Formula. The Midpoint Formula

The Distance Formula. The Midpoint Formula Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x

More information

MATH107 Vectors and Matrices

MATH107 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 information

Triple Integrals in Cartesian Coordinates. Triple Integrals in Cylindrical Coordinates. Triple Integrals in Spherical Coordinates

Triple Integrals in Cartesian Coordinates. Triple Integrals in Cylindrical Coordinates. Triple Integrals in Spherical Coordinates Chapter 3 Multiple Integral 3. Double Integrals 3. Iterated Integrals 3.3 Double Integrals in Polar Coordinates 3.4 Triple Integrals Triple Integrals in Cartesian Coordinates Triple Integrals in Clindrical

More information

Section 8.5 Parametric Equations

Section 8.5 Parametric Equations 504 Chapter 8 Section 8.5 Parametric Equations Man shapes, even ones as simple as circles, cannot be represented as an equation where is a function of. Consider, for eample, the path a moon follows as

More information

17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes

17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes Parametric Curves 17.3 Introduction In this section we eamine et another wa of defining curves - the parametric description. We shall see that this is, in some was, far more useful than either the Cartesian

More information

Sample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs

Sample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs Sample Questions to the Final Eam in Math 1111 Chapter Section.1: Basics of Functions and Their Graphs 1. Find the range of the function: y 16. a.[-4,4] b.(, 4],[4, ) c.[0, ) d.(, ) e.. Find the domain

More information

Analytic Geometry in Three Dimensions

Analytic Geometry in Three Dimensions Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used

More information

MATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions

MATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions MATH 55 Applied Honors Calculus III Winter 11 Midterm 1 Review Solutions 11.1: #19 Particle starts at point ( 1,, traces out a semicircle in the counterclockwise direction, ending at the point (1,. 11.1:

More information

Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function 2 Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

More information

Practice problems. 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b.

Practice problems. 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b. Practice problems 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b. 1, 1 = c 1 3, 2 + c 2 2, 1. Solve c 1, c 2. 2. Suppose a is a vector in the plane. If the component of the a in

More information

Figure 10: Tangent vectors approximating a path.

Figure 10: Tangent vectors approximating a path. 3 Curvature 3.1 Curvature Now that we re parametrizing curves, it makes sense to wonder how we might measure the extent to which a curve actually curves. That is, how much does our path deviate from being

More information

Review Test 2. c ) is a local maximum. ) < 0, then the graph of f has a saddle point at ( c,, (, c ) = 0, no conclusion can be reached by this test.

Review Test 2. c ) is a local maximum. ) < 0, then the graph of f has a saddle point at ( c,, (, c ) = 0, no conclusion can be reached by this test. eview Test I. Finding local maima and minima for a function = f, : a) Find the critical points of f b solving simultaneousl the equations f, = and f, =. b) Use the Second Derivative Test for determining

More information

APPENDIX D Rotation and the General Second-Degree Equation

APPENDIX D Rotation and the General Second-Degree Equation APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the

More information

CURVATURE AND RADIUS OF CURVATURE

CURVATURE AND RADIUS OF CURVATURE CHAPTER 5 CURVATURE AND RADIUS OF CURVATURE 5.1 Introduction: Curvature is a numerical measure of bending of the curve. At a particular point on the curve, a tangent can be drawn. Let this line makes an

More information

Grade 12 Mathematics. unimaths.co.za. Revision Questions. (Including Solutions)

Grade 12 Mathematics. unimaths.co.za. Revision Questions. (Including Solutions) Grade 12 Mathematics Revision Questions (Including Solutions) unimaths.co.za Get read for universit mathematics b downloading free lessons taken from Unimaths Intro Workbook. Visit unimaths.co.za for more

More information

Exercises for Multivariable Differential Calculus XM521

Exercises 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 information

Tangent and Normal Vectors

Tangent and Normal Vectors Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in

More information

In order to take a closer look at what I m talking about, grab a sheet of graph paper and graph: y = x 2 We ll come back to that graph in a minute.

In order to take a closer look at what I m talking about, grab a sheet of graph paper and graph: y = x 2 We ll come back to that graph in a minute. Module 7: Conics Lesson Notes Part : Parabolas Parabola- The parabola is the net conic section we ll eamine. We talked about parabolas a little bit in our section on quadratics. Here, we eamine them more

More information

Parametric Equations and Polar Coordinates

Parametric Equations and Polar Coordinates Parametric Equations and Polar Coordinates Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. Another

More information

Math 317 M1A, October 8th, 2010 page 1 of 7 Name:

Math 317 M1A, October 8th, 2010 page 1 of 7 Name: Math 317 M1A, October 8th, 2010 page 1 of 7 Name: Problem 1 (5 parts, 30 points): Consider the curve r(t) = 3 sin(t 2 ), 4t 2 + 7, 3 cos(t 2 ), 0 t < a) (5 points) Find the arclength function s(t) giving

More information

Solution Midterm 2, Math 53, Summer (a) (10 points) Let f(x, y, z) be a differentiable function of three variables and define

Solution Midterm 2, Math 53, Summer (a) (10 points) Let f(x, y, z) be a differentiable function of three variables and define Solution Midterm, Math 5, Summer. (a) ( points) Let f(,, z) be a differentiable function of three variables and define F (s, t) = f(st, s + t, s t). Calculate the partial derivatives F s and F t in terms

More information

17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes

17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes Parametric Curves 7.3 Introduction In this Section we eamine et another wa of defining curves - the parametric description. We shall see that this is, in some was, far more useful than either the Cartesian

More information

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component

More information

1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...

1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates... Contents 1 Geometry of R 2 2 1.1 Conic Sections............................................ 2 1.2 Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................

More information

Department of Mathematical and Statistical Sciences University of Alberta

Department 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 information

Math 210, Exam 1, Practice Fall 2009 Problem 1 Solution

Math 210, Exam 1, Practice Fall 2009 Problem 1 Solution Math 20, Exam, Practice Fall 2009 Problem Solution. Let A = (,,2), B = (0,,), C = (2,,). (a) Find the vector equation of the plane through A, B, C. (b) Find the area of the triangle with these three vertices.

More information

Ma 227 Final Exam Solutions 5/8/03

Ma 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 information

APPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I

APPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I APPENDIXES A Numbers, Inequalities, and Absolute Values B Coordinate Geometr and Lines C Graphs of Second-Degree Equations D Trigonometr E F Sigma Notation Proofs of Theorems G The Logarithm Defined as

More information

Lesson 9.1 Using the Distance Formula

Lesson 9.1 Using the Distance Formula Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

More information

HOMEWORK 2 SOLUTIONS

HOMEWORK 2 SOLUTIONS HOMEWORK SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Find parametric equations for the tangent line of the graph of r(t) = (t, t + 1, /t) when t = 1. Solution: A point on this line is r(1) = (1,,

More information

Exam 1 Review SOLUTIONS

Exam 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 information

8.8 Conics With Equations in the Form

8.8 Conics With Equations in the Form 8.8 Conics With Equations in the Form ax + b + gx + f + c = 0 The CF-18 Hornet is a supersonic jet flown in Canada. It has a maximum speed of Mach 1.8. The speed of sound is Mach 1. When a plane like the

More information

Worksheet 1.7: Introduction to Vector Functions - Position

Worksheet 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 information

1 Vectors and 3-Dimensional Geometry

1 Vectors and 3-Dimensional Geometry Calculus III (part ): Vectors and 3-Dimensional Geometry (by Evan Dummit, 07, v..55) Contents Vectors and 3-Dimensional Geometry. Functions of Several Variables and 3-Space..................................

More information

CALCULUS 3 February 6, st TEST

CALCULUS 3 February 6, st TEST MATH 400 (CALCULUS 3) Spring 008 1st TEST 1 CALCULUS 3 February, 008 1st TEST YOUR NAME: 001 A. Spina...(9am) 00 E. Wittenbn... (10am) 003 T. Dent...(11am) 004 J. Wiscons... (1pm) 005 A. Spina...(1pm)

More information

CHAPTER 11 Vector-Valued Functions

CHAPTER 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 information

Solutions for the Practice Final - Math 23B, 2016

Solutions for the Practice Final - Math 23B, 2016 olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy

More information

MATH 423/ Note that the algebraic operations on the right hand side are vector subtraction and scalar multiplication.

MATH 423/ Note that the algebraic operations on the right hand side are vector subtraction and scalar multiplication. MATH 423/673 1 Curves Definition: The velocity vector of a curve α : I R 3 at time t is the tangent vector to R 3 at α(t), defined by α (t) T α(t) R 3 α α(t + h) α(t) (t) := lim h 0 h Note that the algebraic

More information

Tangent and Normal Vector - (11.5)

Tangent and Normal Vector - (11.5) Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out bythe vector-valued function r!!t # f!t, g!t, h!t $. The vector T!!t r! % r! %!t!t is the unit tangent vector

More information

IAS 3.1 Conic Sections

IAS 3.1 Conic Sections Year 13 Mathematics IAS 3.1 Conic Sections Robert Lakeland & Carl Nugent Contents Achievement Standard.................................................. The Straight Line.......................................................

More information

Math 3c Solutions: Exam 2 Fall 2017

Math 3c Solutions: Exam 2 Fall 2017 Math 3c Solutions: Exam Fall 07. 0 points) The graph of a smooth vector-valued function is shown below except that your irresponsible teacher forgot to include the orientation!) Several points are indicated

More information

Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations.

Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. 1. Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. x + y = 5, z = 4 Choose the correct description. A. The circle with center (0,0, 4)

More information

Mathematics Trigonometry: Unit Circle

Mathematics Trigonometry: Unit Circle a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and

More information

Tom Robbins WW Prob Lib1 Math , Fall 2001

Tom Robbins WW Prob Lib1 Math , Fall 2001 Tom Robbins WW Prob Lib Math 220-2, Fall 200 WeBWorK assignment due 9/7/0 at 6:00 AM..( pt) A child walks due east on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 7 miles

More information

Review Sheet for the Final

Review Sheet for the Final Review Sheet for the Final Math 6-4 4 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 the absence

More information

Math 114, Section 003 Fall 2011 Practice Exam 1 with Solutions

Math 114, Section 003 Fall 2011 Practice Exam 1 with Solutions Math 11, Section 003 Fall 2011 Practice Exam 1 with Solutions Contents 1 Problems 2 2 Solution key 8 3 Solutions 9 1 1 Problems Question 1: Let L be the line tangent to the curve r (t) t 2 + 3t + 2, e

More information