Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane
|
|
- Jonah Potter
- 6 years ago
- Views:
Transcription
1 Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential. Review: Parametric curves on the plane A curve on the plane is given in parametric form iff it is given b the set of points ( (t), (t) ), where the parameter t I R. Eample Describe the curve (t) = cosh(t), (t) = sinh(t), for t, ). Solution: (t)] 2 (t)] 2 = (t) cosh 2 (t) sinh 2 (t) = 1. (t) This is a portion of a hperbola with asmptotes = ±, starting at (1, ). 1
2 Review: Parametric curves on the plane A ccloid with parameter a > is the curve given b (t) = a(t sin(t)), (t) = a(1 cos(t)), t R. Remar: From the equation of the ccloid we see that (t) at = a sin(t), (t) a = a cos(t). Therefore, (t) at] 2 + (t) a] 2 = a 2. Remars: This is not the equation of a circle. The point ((t), (t)) belongs to a moving circle. The ccloid plaed an important role in designing precise pendulum clocs, needed for navigation in the 17th centur. Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential.
3 The slope of tangent lines to curves A curve defined b the parametric function values ( (t), (t) ), for t I R, is differentiable iff each function and is differentiable on the interval I. Theorem Assume that the curve defined b the graph of the function = f (), for (a, b), can be described b the parametric function values ( (t), (t) ), for t I R. If this parametric curve is differentiable and (t) for t I, then holds df d = (d/dt) (d/dt). Proof: Epress (t) = f ((t)), then d dt = df d d dt df d = (d/dt) (d/dt). The slope of tangent lines to curves Remar: The formula df d = (d/dt) provides an alternative wa (d/dt) to find the slope of the line tangent to the graph of the function f. = f ( ) f( ) = f()
4 The slope of tangent lines to curves Eample Find the slope of the tangent lines to a circle radius r at (, ). Solution: The equation of the circle is = r 2. One possible set of parametric equations are: (t) = r cos(nt), (t) = r sin(nt), n 1. The derivatives of the parametric functions are (t) = nr sin(nt), (t) = nr cos(nt). The slope of the tangent lines to the circle at = cos(nt ) is ( ) = (t ) (t ) = nr cos(nt ) nr sin(nt ) 1 ( ) = tan(nt ). Remar: In the first quadrant holds ( ) = 1 ( ) 2. Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential.
5 The arc-length of a curve The length or arc length of a curve in the plane or in space is the limit of the polgonal line length, as the polgonal line approimates the original curve. = f() L Theorem The arc-length of a continuousl differentiable curve ( (t), () ), for t a, b] is the number b L = (t) ] 2 + (t) ] 2 dt. a The arc-length of a curve Idea of the Proof: The curve length is the limit of the polgonal line length, as the polgonal line approimates the original curve. = f() L L N = N 1 n= ( ) 2 + ( ) 2 {a = t, t 1,, t N 1, t N = b}, L N N 1 n= L N N L = (t )] 2 + (t )] 2 t, b a (t) ] 2 + (t) ] 2 dt.
6 The arc-length of a curve Eample Find the length of the curve ( r cos(t), r sin(t) ), for r > and t π/4, 3π/4]. (Quarter of a circle.) Solution: Compute the derivatives ( r sin(t), r cos(t) ). The length of the curve is given b the formula L = 3π/4 π/4 r sin(t) ] 2 + r cos(t) ] 2 dt L = 3π/4 π/4 r 2( sin(t) ] 2 ] 2 ) 3π/4 + cos(t) dt = r dt. π/4 Hence, L = π 2 r. (The length of quarter circle of radius r.) The arc-length of a curve Eample Find the length of the spiral ( t cos(t), t sin(t) ), for t, t ]. Solution: The derivative of the parametric curve is ( (t), (t) ) = ( t sin(t) + cos(t) ], t cos(t) + sin(t) ]), ( ) 2 + ( ) 2 = t 2 sin 2 (t) + cos 2 (t) 2t sin(t) cos(t) ] + t 2 cos 2 (t) + sin 2 (t) + 2t sin(t) cos(t) ] We obtain ( ) 2 + ( ) 2 = t The curve length is given b L(t ) = t t 1 + t 2 dt = 1 + t ln( t t )] 2 t We conclude that L(t ) = t t ln( ) t t 2.
7 Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential. The arc-length function and differential Remar: The previous eample suggests to introduce the length function of a curve. The arc-length function of a continuousl differentiable curve given b ( (t), (t) ) for t t, t 1 ] is given b L(t) = t t (τ) ] 2 + (τ) ] 2 dτ. Remars: (a) The value L(t) of the length function is the length along the curve ( (t), (t) ) from t to t. (b) If the curve is the position of a moving particle as function of time, then the value L(t) is the distance traveled b the particle from the time t to t.
8 The arc-length function and differential Remar: The arc-length differential is the differential of the arc-length function, that is, This is a useful notation. dl = (t) ] 2 + (t) ] 2 dt. Eample Find the length of (t) = (2t + 1) 3/2 /3, (t) = t + t 2 for t, 1]. Solution: We first compute the length differential, 1 dl = 3 3 ] 2 2 (2t + ] 2 1)1/ t = (2t + 1) t + 4t 2 L = 1 (4t 2 + 6t + 2) dt = ( 4t 3 ) t2 + 2t = 19 3.
Vector-Valued Functions
Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationIntegrals along a curve in space. (Sect. 16.1)
Integrals along a curve in space. (Sect. 6.) Line integrals in space. The addition of line integrals. ass and center of mass of wires. Line integrals in space Definition The line integral of a function
More informationFind the rectangular coordinates for each of the following polar coordinates:
WORKSHEET 13.1 1. Plot the following: 7 3 A. 6, B. 3, 6 4 5 8 D. 6, 3 C., 11 2 E. 5, F. 4, 6 3 Find the rectangular coordinates for each of the following polar coordinates: 5 2 2. 4, 3. 8, 6 3 Given the
More information11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS
PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS. Parametric Equations Preliminar Questions. Describe the shape of the curve = cos t, = sin t. For all t, + = cos t + sin t = 9cos t + sin t =
More informationDirectional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables.
Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of
More informationMath 113 Final Exam Practice
Math Final Exam Practice The Final Exam is comprehensive. You should refer to prior reviews when studying material in chapters 6, 7, 8, and.-9. This review will cover.0- and chapter 0. This sheet has three
More information17.3. 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 informationLimits and the derivative function. Limits and the derivative function
The Velocity Problem A particle is moving in a straight line. t is the time that has passed from the start of motion (which corresponds to t = 0) s(t) is the distance from the particle to the initial position
More informationGeneralized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks:
Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n, d n, t < δ n (t) = n, t 3 d3 d n, t > n. d t The Dirac delta generalized function
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing
More informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationExercise. Exercise 1.1. MA112 Section : Prepared by Dr.Archara Pacheenburawana 1
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 1 Exercise Exercise 1.1 1 8 Find the vertex, focus, and directrix of the parabola and sketch its graph. 1. x = 2y 2 2. 4y +x 2 = 0 3. 4x 2 =
More informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE (SUPPLEMENTARY) EXAMINATION, FEBRUARY 2017 (2015 ADMISSION)
B116S (015 dmission) Pages: RegNo Name PJ BDUL KLM TECHNOLOGICL UNIVERSITY FIRST SEMESTER BTECH DEGREE (SUPPLEMENTRY) EXMINTION, FEBRURY 017 (015 DMISSION) MaMarks : 100 Course Code: M 101 Course Name:
More informationHW - Chapter 10 - Parametric Equations and Polar Coordinates
Berkeley City College Due: HW - Chapter 0 - Parametric Equations and Polar Coordinates Name Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify
More informationConservative fields and potential functions. (Sect. 16.3) The line integral of a vector field along a curve.
onservative fields and potential functions. (Sect. 16.3) eview: Line integral of a vector field. onservative fields. The line integral of conservative fields. Finding the potential of a conservative field.
More informationPARAMETRIC EQUATIONS AND POLAR COORDINATES
PARAMETRIC EQUATINS AND PLAR CRDINATES Parametric equations and polar coordinates enable us to describe a great variet of new curves some practical, some beautiful, some fanciful, some strange. So far
More informationSCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing
More informationAnswers to Some Sample Problems
Answers to Some Sample Problems. Use rules of differentiation to evaluate the derivatives of the following functions of : cos( 3 ) ln(5 7 sin(3)) 3 5 +9 8 3 e 3 h 3 e i sin( 3 )3 +[ ln ] cos( 3 ) [ln(5)
More information90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.
90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y
More information10.3 Parametric Equations. 1 Math 1432 Dr. Almus
Math 1432 DAY 39 Dr. Melahat Almus almus@math.uh.edu OFFICE HOURS (212 PGH) MW12-1:30pm, F:12-1pm. If you email me, please mention the course (1432) in the subject line. Check your CASA account for Quiz
More informationOpen the TI-Nspire file: Astroid. Navigate to page 1.2 of the file. Drag point A on the rim of the bicycle wheel and observe point P on the rim.
Astroid Student Activity 7 9 TI-Nspire Investigation Student min Introduction How is the motion of a ladder sliding down a wall related to the motion of the valve on a bicycle wheel or to a popular amusement
More informationChapter 12 and 13 Math 125 Practice set Note: the actual test differs. Given f(x) and g(x), find the indicated composition and
Chapter 1 and 13 Math 1 Practice set Note: the actual test differs. Given f() and g(), find the indicated composition. 1) f() = - ; g() = 3 + Find (f g)(). Determine whether the function is one-to-one.
More informationContents. About the Author. Preface to the Instructor. xxi. Acknowledgments. xxiii. Preface to the Student
About the Author v Preface to the Instructor v Acknowledgments i Preface to the Student iii Chapter 0 The Real Numbers 1 0.1 The Real Line 2 Construction of the Real Line 2 Is Ever Real Number Rational?
More informationThe region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.
Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.
More informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 2017 MA101: CALCULUS PART A
A B1A003 Pages:3 (016 ADMISSIONS) Reg. No:... Name:... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 017 MA101: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART
More information17.3. 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(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.
Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44
More informationMATH 162. Midterm 2 ANSWERS November 18, 2005
MATH 62 Midterm 2 ANSWERS November 8, 2005. (0 points) Does the following integral converge or diverge? To get full credit, you must justify your answer. 3x 2 x 3 + 4x 2 + 2x + 4 dx You may not be able
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More informationENGI Parametric Vector Functions Page 5-01
ENGI 3425 5. Parametric Vector Functions Page 5-01 5. Parametric Vector Functions Contents: 5.1 Arc Length (Cartesian parametric and plane polar) 5.2 Surfaces of Revolution 5.3 Area under a Parametric
More informationThings to Know and Be Able to Do Understand the meaning of equations given in parametric and polar forms, and develop a sketch of the appropriate
AP Calculus BC Review Chapter (Parametric Equations and Polar Coordinates) Things to Know and Be Able to Do Understand the meaning of equations given in parametric and polar forms, and develop a sketch
More informationDirectional derivatives and gradient vectors (Sect. 14.5) Directional derivative of functions of two variables.
Directional derivatives and gradient vectors (Sect. 14.5) Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of three
More informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationC H A P T E R 9 Topics in Analytic Geometry
C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas.................... 77 Section. Ellipses........................... 7 Section. Hperbolas......................... 7 Section. Rotation
More informationAP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions
AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions Question No. 1 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as
More information698 Chapter 11 Parametric Equations and Polar Coordinates
698 Chapter Parametric Equations and Polar Coordinates 67. 68. 69. 70. 7. 7. 7. 7. Chapter Practice Eercises 699 75. (a Perihelion a ae a( e, Aphelion ea a a( e ( Planet Perihelion Aphelion Mercur 0.075
More informationScalar functions of several variables (Sect. 14.1)
Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 2, 5 2 C) - 5 2
Test Review (chap 0) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) Find the point on the curve x = sin t, y = cos t, -
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, )
Midterm Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and graph the function. ) G(t) = t - 3 ) 3 - -3 - - 3 - - -3
More information9.4 CALCULUS AND PARAMETRIC EQUATIONS
9.4 Calculus with Parametric Equations Contemporary Calculus 1 9.4 CALCULUS AND PARAMETRIC EQUATIONS The previous section discussed parametric equations, their graphs, and some of their uses for visualizing
More informationAP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:
AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented
More informationReview Sheet for Exam 1 SOLUTIONS
Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma
More informationBC Exam 1 - Part I 28 questions No Calculator Allowed - Solutions C = 2. Which of the following must be true?
BC Exam 1 - Part I 8 questions No Calculator Allowed - Solutions 6x 5 8x 3 1. Find lim x 0 9x 3 6x 5 A. 3 B. 8 9 C. 4 3 D. 8 3 E. nonexistent ( ) f ( 4) f x. Let f be a function such that lim x 4 x 4 I.
More informationThis set of questions goes with the pages of applets and activities for Lab 09. Use the applets and activities there to answer the questions.
Page 1 of 5 Lab 09 Name: Start time: Number of questions: 11 This set of questions goes with the pages of applets and activities for Lab 09. Use the applets and activities there to answer the questions.
More informationIt s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4]
It s Your Turn Problems I. Functions, Graphs, and Limits. Here s the graph of the function f on the interval [ 4,4] f ( ) =.. It has a vertical asymptote at =, a) What are the critical numbers of f? b)
More informationF dr y 2. F r t r t dt. a sin t a sin t a cos t a cos t a 2 cos 2 t a 2 sin 2 t. P dx Q dy yy. x C. follows that F is a conservative vector field.
6 CHAPTER 6 VECTOR CALCULU We now easil compute this last integral using the parametriation given b rt a cos t i a sin t j, t. Thus C F dr C F dr Frt rt dt a sin ta sin t a cos ta cos t a cos t a sin t
More informationChapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus
Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 1. Parabola A parabola is the set of all points x, y ( ) that are equidistant from a fixed line and a fixed point
More informationis the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input x at x = b.
Uses of differentials to estimate errors. Recall the derivative notation df d is the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input at = b. Eamples.
More informationAP Calculus (BC) Summer Assignment (104 points)
AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationCalculus BC: Section I
Calculus BC: Section I Section I consists of 45 multiple-choice questions. Part A contains 28 questions and does not allow the use of a calculator. Part B contains 17 questions and requires a graphing
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function rt vector T t r r t t is the unit tangent vector to the curve C. Now define N
More information( ) = 1 t + t. ( ) = 1 cos x + x ( sin x). Evaluate y. MTH 111 Test 1 Spring Name Calculus I
MTH Test Spring 209 Name Calculus I Justify all answers by showing your work or by proviing a coherent eplanation. Please circle your answers.. 4 z z + 6 z 3 ez 2 = 4 z + 2 2 z2 2ez Rewrite as 4 z + 6
More informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012
The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,
More informationCALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt
CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental
More informationINTRINSIC COORDINATES
INTRINSIC CRDINATES Question 1 (**) s = 4asin m A bead of mass m is made to slide along a smooth wire, which is fied in a vertical plane, and is bent into the shape of an inverted ccloid with intrinsic
More information4.3 Mean-Value Theorem and Monotonicity
.3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such
More informationChapter 9. Derivatives. Josef Leydold Mathematical Methods WS 2018/19 9 Derivatives 1 / 51. f x. (x 0, f (x 0 ))
Chapter 9 Derivatives Josef Leydold Mathematical Methods WS 208/9 9 Derivatives / 5 Difference Quotient Let f : R R be some function. The the ratio f = f ( 0 + ) f ( 0 ) = f ( 0) 0 is called difference
More informationSample Final Exam Problems Solutions Math 107
Sample Final Eam Problems Solutions Math 107 1 (a) We first factor the numerator and the denominator of the function to obtain f() = (3 + 1)( 4) 4( 1) i To locate vertical asymptotes, we eamine all locations
More informationMTH3101 Spring 2017 HW Assignment 4: Sec. 26: #6,7; Sec. 33: #5,7; Sec. 38: #8; Sec. 40: #2 The due date for this assignment is 2/23/17.
MTH0 Spring 07 HW Assignment : Sec. 6: #6,7; Sec. : #5,7; Sec. 8: #8; Sec. 0: # The due date for this assignment is //7. Sec. 6: #6. Use results in Sec. to verify that the function g z = ln r + iθ r >
More informationMath 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math Activity (Due by end of class Jan. 6) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationExamples of the Accumulation Function (ANSWERS) dy dx. This new function now passes through (0,2). Make a sketch of your new shifted graph.
Eamples of the Accumulation Function (ANSWERS) Eample. Find a function y=f() whose derivative is that f()=. dy d tan that satisfies the condition We can use the Fundamental Theorem to write a function
More informationMath 1160 Final Review (Sponsored by The Learning Center) cos xcsc tan. 2 x. . Make the trigonometric substitution into
Math 60 Final Review (Sponsored by The Learning Center). Simplify cot csc csc. Prove the following identities: cos csc csc sin. Let 7sin simplify.. Prove: tan y csc y cos y sec y cos y cos sin y cos csc
More information+ 4 Ex: y = v = (1, 4) x = 1 Focus: (h, k + ) = (1, 6) L.R. = 8 units We can have parabolas that open sideways too (inverses) x = a (y k) 2 + h
Unit 7 Notes Parabolas: E: reflectors, microphones, (football game), (Davinci) satellites. Light placed where ras will reflect parallel. This point is the focus. Parabola set of all points in a plane that
More informationArc Length. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Arc Length Today 1 / 12
Philippe B. Laval KSU Today Philippe B. Laval (KSU) Arc Length Today 1 / 12 Introduction In this section, we discuss the notion of curve in greater detail and introduce the very important notion of arc
More informationParameterization and Vector Fields
Parameterization and Vector Fields 17.1 Parameterized Curves Curves in 2 and 3-space can be represented by parametric equations. Parametric equations have the form x x(t), y y(t) in the plane and x x(t),
More information1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: and. So the slopes of the tangent lines to the curve
MAT 11 Solutions TH Eam 3 1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: Therefore, d 5 5 d d 5 5 d 1 5 1 3 51 5 5 and 5 5 5 ( ) 3 d 1 3 5 ( ) So the
More informationFundamental Theorem of Calculus
Fundamental Theorem of Calculus MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Summer 208 Remarks The Fundamental Theorem of Calculus (FTC) will make the evaluation of definite integrals
More informationChapter 10: Conic Sections; Polar Coordinates; Parametric Equations
Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations Section 10.1 Geometry of Parabola, Ellipse, Hyperbola a. Geometric Definition b. Parabola c. Ellipse d. Hyperbola e. Translations f.
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 3 Solutions [Multiple Integration; Lines of Force]
ENGI 44 Advanced Calculus for Engineering Facult of Engineering and Applied Science Problem Set Solutions [Multiple Integration; Lines of Force]. Evaluate D da over the triangular region D that is bounded
More informationEdexcel 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 informationSOLUTIONS 1 (27) 2 (18) 3 (18) 4 (15) 5 (22) TOTAL (100) PROBLEM NUMBER SCORE MIDTERM 2. Form A. Recitation Instructor : Recitation Time :
Math 5 March 8, 206 Form A Page of 8 Name : OSU Name.# : Lecturer:: Recitation Instructor : SOLUTIONS Recitation Time : SHOW ALL WORK in problems, 2, and 3. Incorrect answers with work shown may receive
More information1 The Derivative and Differrentiability
1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped
More informationCalculus AB 2014 Scoring Guidelines
P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official
More informationMath120R: Precalculus Final Exam Review, Fall 2017
Math0R: Precalculus Final Eam Review, Fall 07 This study aid is intended to help you review for the final eam. Do not epect this review to be identical to the actual final eam. Refer to your course notes,
More informationMath 1A Test Fall 10 Show your work for credit. Write all responses on separate paper. Don t use a calculator.
Math A - 3.7 Test Fall 0 Name Show your work for credit. Write all responses on separate paper. Don t use a calculator.. Open cylindrical tubes resonate at the approimate frequencies, f, where nv f = L+
More informationMath 2414 Activity 1 (Due by end of class July 23) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math 44 Activity (Due by end of class July 3) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationSolutions to Practice Test 3
Solutions to Practice Test 3. (a) Find the equation for the plane containing the points (,, ), (, 2, ), and (,, 3). (b) Find the area of the triangle with vertices (,, ), (, 2, ), and (,, 3). Answer: (a)
More informationMath 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n.
. Find the following its (if they eist: sin 7 a. 0 9 5 b. 0 tan( 8 c. 4 d. e. f. sin h0 h h cos h0 h h Math 4 Final Eam Review g. h. i. j. k. cos 0 n nn e 0 n arctan( 0 4 l. 0 sin(4 m. cot 0 = n. = o.
More informationDepartment of Mathematical Sciences. Math 226 Calculus Spring 2016 Exam 2V2 DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO
Department of Mathematical Sciences Math 6 Calculus Spring 6 Eam V DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO SO NAME (Printed): INSTRUCTOR: SECTION NO.: When instructed, turn over this cover page
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Semester 1Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. 1) Which one of the equations below matches the graph? 1)
More informationDIFFERENTIAL EQUATION
MDE DIFFERENTIAL EQUATION NCERT Solved eamples upto the section 9. (Introduction) and 9. (Basic Concepts) : Eample : Find the order and degree, if defined, of each of the following differential equations
More information18.01 Final Answers. 1. (1a) By the product rule, (x 3 e x ) = 3x 2 e x + x 3 e x = e x (3x 2 + x 3 ). (1b) If f(x) = sin(2x), then
8. Final Answers. (a) By the product rule, ( e ) = e + e = e ( + ). (b) If f() = sin(), then f (7) () = 8 cos() since: f () () = cos() f () () = 4 sin() f () () = 8 cos() f (4) () = 6 sin() f (5) () =
More informationChapter 3 Differentiation Rules
Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,
More informationCHAPTER 11 Vector-Valued Functions
CHAPTER Vector-Valued Functions Section. Vector-Valued Functions...................... 9 Section. Differentiation and Integration of Vector-Valued Functions.... Section. Velocit and Acceleration.....................
More informationSee the figures below where we zoom in on f(x) = e x at x = 1:
The Differential Almost from the time when we first learned that the derivative of a function f() at a point = a is the slope of the tangent line at the point (a, f(a) we calculated the equation of the
More informationParametric Curves You Should Know
Parametric Curves You Should Know Straight Lines Let a and c be constants which are not both zero. Then the parametric equations determining the straight line passing through (b d) with slope c/a (i.e.
More informationMATH 1220 Midterm 1 Thurs., Sept. 20, 2007
MATH 220 Midterm Thurs., Sept. 20, 2007 Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the eam and nothing else. Calculators
More informationFUNCTIONS OF ONE VARIABLE FUNCTION DEFINITIONS
Page of 6 FUNCTIONS OF ONE VARIABLE FUNCTION DEFINITIONS 6. HYPERBOLIC FUNCTIONS These functions which are defined in terms of e will be seen later to be related to the trigonometic functions via comple
More informationI. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.
0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.
More information1998 AP Calculus AB: Section I, Part A
998 AP Calculus AB: 55 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number.. What is the -coordinate
More informationOverview. Properties of the exponential. The natural exponential e x. Lesson 1 MA Nick Egbert
Overview This lesson should alread be familiar to ou from precalculus. But for the sake of completeness and because of their crucial importance, we review some basic properties of the eponential and logarithm
More informationName: Date: Period: Calculus Honors: 4-2 The Product Rule
Name: Date: Period: Calculus Honors: 4- The Product Rule Warm Up: 1. Factor and simplify. 9 10 0 5 5 10 5 5. Find ' f if f How did you go about finding the derivative? Let s Eplore how to differentiate
More information(a) We split the square up into four pieces, parametrizing and integrating one a time. Right side: C 1 is parametrized by r 1 (t) = (1, t), 0 t 1.
Thursda, November 5 Green s Theorem Green s Theorem is a 2-dimensional version of the Fundamental Theorem of alculus: it relates the (integral of) a vector field F on the boundar of a region to the integral
More informationTOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12
NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus
More informationFaculty of Engineering, Mathematics and Science School of Mathematics
Faculty of Engineering, Mathematics and Science School of Mathematics SF Engineers SF MSISS SF MEMS MAE: Engineering Mathematics IV Trinity Term 18 May,??? Sports Centre??? 9.3 11.3??? Prof. Sergey Frolov
More information