Parametric Equations and Polar Coordinates

Size: px
Start display at page:

Download "Parametric Equations and Polar Coordinates"

Transcription

1 Parametric Equations and Polar Coordinates

2 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 way to describe a curve is by expressing both coordinates as functions of a third variable t.

3 Parametric Equations

4 t I is a parameter for the curve. is the parameter interval. If I is a closed interval, a t b, ( f a ga) ( f b gb) ( ), ( ) is the initial point of the curve and ( ), ( ) is the terminal point of the curve.

5 Example 1: Sketch the curve defined by the parametric equations x= t 2, y = t+ 1, < t <.

6

7

8 Example 2: Identify geometrically the curve in Eg. 1 (Fig. 11.2) by eliminating the parameter t and obtaining an algebraic equation in x and y.

9

10 Example 3: Graph the parametric curves ( a) x= cos t, y = sin t, 0 t 2 π. ( b) x= acos t, y = asin t, 0 t 2π

11 Example 4: The position P( x, y) of a particle moving in the xy plane is given by the equations and parameter interval x= t, y = t, t 0. Identify the path traced by the particle and describe the motion.

12

13 Example 5: 2 A parametrization of the graph of the function f( x) is given by x = t y = f t = t < t < 2, (),. When t 0, this parametrization gives the same path in the xy-plane as we had in Eg. 4. However, since the parameter t here can now also be negative, we obtain the left-hand part of the parabola as well; that is, we have the entire parabolic curve. = x For this parametrization, there is no starting point and no terminal point. (Fig. 11.5).

14

15 Example 6: Find a parametrization for the line through the point (a, b) having the slope, m. Solution: A Cartesian equation of the line is y b= mx ( a). If we set the parameter t = x a, we find that x = a + t and y b = mt. That is, x = a+ t, y = b+ mt, < t < parametrizes the line. This parametrization differs from the one we would obtain by the technique used in Eg. 5 when t = x. However, both parametrizations give the same line.

16 Example 7: Sketch and identify the path traced by the point Pxy (, ) if 1 1 x = t +, y t, t 0. t = t >

17

18

19 Exercise 11.1:

20 Calculus with Parametric Curves In this section, we apply calculus to parametric curves.

21 Tangents A parametrized curve x= f( t) and y= gt ( ) is differentiable at tif fand gare differentiable at t. At a point in a differentiable parametrized curve where y is also a differentiable function of x, the derivatives dy dt, dx dt, and dy dx are related by the Chain Rule: dy dy dx = dt dx dt If dx dt = 0, we may divide both sides of this equation by dx dt to solve for dy dx.

22

23 Example 1: Find the tangent to the curve π π x = sec t, y = tan t, < t < 2 2 ( ) at the point 2, 1, where t = π 4 (Fig 11.12).

24

25 Example 2: 2 d y 2 3 Find as a function of t if x = t t, y = t t. 2 dx

26 Length of a Parametrically Defined Curve

27

28 Using Leibniz notation: L 2 2 b dx dy = + a dt dt dt (3)

29 Example 4: Using the definition, find the length of the circle of radius r defined parametrically by x = rcost and y = rsin t, 0 t 2 π.

30 Example 5: Find the length of the astroid (Fig ) 3 3 x = cos t, y = sin t, 0 t 2 π.

31 Length of a Curve y = f(x) The length formula in section 6.3 is a special case of Eq. (3). Given a continuously differentiable function y = f( x), a x b, we can assign x = t as a parameter. The graph of the function f is then the curve C defined parametrically by x = t and y = f( t), a t b, a special case of what we considered before. Then, dx dy = 1 and = f ( t). dt dt From equation (1), we have dy dx giving dy dt = = f ( t), dx dt 2 dy dx + dt dt 2 [ f t ] [ f x ] 2 2 = 1 + ( ) = 1 + ( ).

32 Exercises 11.2:

33 Polar Coordinates

34 Definition of Polar Coordinates

35

36

37

38 Example 1: Find all the polar coordinates of the point P(2, π 6).

39 Corresponding coordinate pairs of P are: 2, 1, 0,, , 2, 1, 0,, 2 6 2, ± ± = + ± ± = + n n n n π π π π

40 Exercise: Mark down the given polar coordinates on the diagram. A(3, π 3) B( 2, π 4) C( 4, π 2) D(1, π 6) E(2, 5π 6)

41 Polar Equations and Graphs

42 If ris fixed at a constant value r= a 0, the point Pr (, θ ) will lie a units from the origin O. As θ varies over any interval of length 2 π, P then traces a circle of radius a centered at O. If θ is fixed at a constant value θ= θ and let r vary between 0 and, the point Pr (, θ ) traces the line through O that makes an angle of measure θ0 with the initial ray.

43 Example 2: ( a) r = 1 and r = 1 are equations for the circle of radius1 centered at O. ( b) θ = π 6, θ = 7π 6, and θ = 5π 6 are equations for the line in Fig Equations of the form r = a and θ= θ can be combined to define regions, segments and rays. 0

44 Example 3: Graph the sets of points whose polar corrdinates satisfy the following conditions: ( a) 1 r 2 and π 0 θ 2 ( b) 3 r 2 and π θ = 4 2π 5π ( c) θ (no restrictions on r) 3 6

45

46 Relating Polar and Cartesian Coordinates

47 The first two of these equations uniquely determine the Cartesian coordinates x and y given the polar coordinates r and θ. If x and y are given, the third equation gives two possible choices for r (a positive and a negative). ( x y) ( ) θ [ π) For each, 0, 0, there is a unique 0,2 satisfying the first two equations, each then giving a polar coordinate representation of the Cartesian point ( x, y). The other polar coordinate representations for the point can be determined from these two as in Eg. 1.

48

49 2 2 Find a polar equationfor for the circle ( 3) 9. x y + = Example 5: θ θ θ 6sin 0 6sin or 0 0 sin Solution: = = = = = + = + + r r r r r y y x y y x

50 Example 6: Replace the following polar equations by equivalent Cartesian equations,and identify their graphs. (a) r cosθ = 4 (b) r 2 = 4r cosθ 4 (c) r = 2cosθ sinθ

51 Solution: (a) (b) r cosϑ = 4 Graph : Vertical line through x = 4 on the x - axis r 2 Graph :Circle, radius 2, center ( h,k) = (2, 0). (c) r(2cosϑ sinϑ) = 4 2r cosϑ r sinϑ = 4 2x = 4r cosϑ and x = 4 x x x ( x y = 4 y = 2x x + 4x 4 = 4 Graph : Line, slope m = 2, y - intercept b = ) y 2 2 = 4x y y 2 = 0 y 2 = 4

52 Exercises 11.3:

53 Graphing in Polar Coordinates This section describes techniques for graphing equations in polar coordinates.

54 Symmetry

55

56 Slope The slope of a polar curve r = f ( θ ) is given by dy dx, not r = df / Think of graph f as the graph of parametric equations: x = r cosθ = f ( θ )cosθ, y = r sinθ = f ( θ )sinθ If f is a differentiable function of θ, then so are x and y and when dx dθ 0, we can calculate dy dx from the parametric formula : dθ dy dx = dy dx dθ = dθ = d dθ d dθ df sinθ + dθ df cosθ dθ ( f ( θ ) sinθ ) ( f ( θ ) cosθ ) f ( θ )cosθ f ( θ )sinθ 56

57 If then dy dx the curve r = f ( θ ) ( 0, θ ) 0 0 = = 0, and the slope equation gives: f f ( θ0 )sinθ0 ( θ )cosθ 0 f ( θ ) passes through the origin at θ = θ, 0 = tanθ 0 0

58 Example 1: A cardioid Graph the curve r = 1 cos θ. Solution: The curve is symmetric about the x-axis because ( r, θ) on the graph r = 1 cosθ r = 1 cos( θ ) ( r, θ ) on the graph.

59

60 Example 2: 2 Graph the curve 4cos. r Solution: The equation requires cosθ 0, so we get the entire graph by running θ from π 2 to π 2. 2 (, ) on the graph 4cos r ϑ = r = 2 = r θ 4cos( ϑ) ϑ ( r, ϑ) on the graph. { symmetric about the x axis} 2 (, ) on the graph 4cos r ϑ r = r = 2 ( ) 4cos ϑ { } ( r, ϑ) on the graph. symmetric about the origin These 2 symmetries implies symmetry about the y axis. Make a short table of values, plot the corresponding points, and use information about symmetry and tangents to guide us in connecting the points with a smooth curve. ϑ

61

62 A technique for Graphing Method Make a table of ( r, θ ) -values. 2. Plot the corresponding points. 3. Connect them in order of increasing. Method Graph r = f (θ ) in the Cartesian rθ - plane. 2. Use the Cartesian graph as a table and guide to sketch the polar coordinate graph. Method 2 is better in the sense that its drawing shows where r is positive, negative, nonexistent, and also where r is increasing/decreasing. θ

63 Example 3: A lemniscate Graph the lemniscate curve r 2 = sin 2θ

64 Exercise 11.4:

10.1 Curves Defined by Parametric Equation

10.1 Curves Defined by Parametric Equation 10.1 Curves Defined by Parametric Equation 1. Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the form y = f (x) because C fails the Vertical

More information

PARAMETRIC EQUATIONS AND POLAR COORDINATES

PARAMETRIC EQUATIONS AND POLAR COORDINATES 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES We have seen how to represent curves by parametric equations. Now, we apply the methods of calculus to these parametric

More information

10.1 Review of Parametric Equations

10.1 Review of Parametric Equations 10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 2, 5 2 C) - 5 2

MULTIPLE 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 information

Example 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph:

Example 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph: Section 10.3: Polar Coordinates The polar coordinate system is another way to coordinatize the Cartesian plane. It is particularly useful when examining regions which are circular. 1. Cartesian Coordinates

More information

Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)

Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x) Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h)

More information

a k 0, then k + 1 = 2 lim 1 + 1

a 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 information

Edexcel past paper questions. Core Mathematics 4. Parametric Equations

Edexcel past paper questions. Core Mathematics 4. Parametric Equations Edexcel past paper questions Core Mathematics 4 Parametric Equations Edited by: K V Kumaran Email: kvkumaran@gmail.com C4 Maths Parametric equations Page 1 Co-ordinate Geometry A parametric equation of

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

4.1 Analysis of functions I: Increase, decrease and concavity

4.1 Analysis of functions I: Increase, decrease and concavity 4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval

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

Math 2300 Calculus II University of Colorado Final exam review problems

Math 2300 Calculus II University of Colorado Final exam review problems Math 300 Calculus II University of Colorado Final exam review problems. A slope field for the differential equation y = y e x is shown. Sketch the graphs of the solutions that satisfy the given initial

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

Power Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.

Power Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n. .8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x

More 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

C3 papers June 2007 to 2008

C3 papers June 2007 to 2008 physicsandmathstutor.com June 007 C3 papers June 007 to 008 1. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04* physicsandmathstutor.com June 007 x + 3 9+

More information

POLAR FORMS: [SST 6.3]

POLAR FORMS: [SST 6.3] POLAR FORMS: [SST 6.3] RECTANGULAR CARTESIAN COORDINATES: Form: x, y where x, y R Origin: x, y = 0, 0 Notice the origin has a unique rectangular coordinate Coordinate x, y is unique. POLAR COORDINATES:

More information

Calculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science

Calculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science Calculus III George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 251 George Voutsadakis (LSSU) Calculus III January 2016 1 / 76 Outline 1 Parametric Equations,

More information

Chapter 9 Overview: Parametric and Polar Coordinates

Chapter 9 Overview: Parametric and Polar Coordinates Chapter 9 Overview: Parametric and Polar Coordinates As we saw briefly last year, there are axis systems other than the Cartesian System for graphing (vector coordinates, polar coordinates, rectangular

More information

worked out from first principles by parameterizing the path, etc. If however C is a A path C is a simple closed path if and only if the starting point

worked out from first principles by parameterizing the path, etc. If however C is a A path C is a simple closed path if and only if the starting point III.c Green s Theorem As mentioned repeatedly, if F is not a gradient field then F dr must be worked out from first principles by parameterizing the path, etc. If however is a simple closed path in the

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

AP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:

AP 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 information

INTEGRAL CALCULUS DIFFERENTIATION UNDER THE INTEGRAL SIGN: Consider an integral involving one parameter and denote it as

INTEGRAL CALCULUS DIFFERENTIATION UNDER THE INTEGRAL SIGN: Consider an integral involving one parameter and denote it as INTEGRAL CALCULUS DIFFERENTIATION UNDER THE INTEGRAL SIGN: Consider an integral involving one parameter and denote it as, where a and b may be constants or functions of. To find the derivative of when

More information

Green s Theorem. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Green s Theorem

Green s Theorem. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Green s Theorem Green s Theorem MATH 311, alculus III J. obert Buchanan Department of Mathematics Fall 2011 Main Idea Main idea: the line integral around a positively oriented, simple closed curve is related to a double

More information

You can learn more about the services offered by the teaching center by visiting

You can learn more about the services offered by the teaching center by visiting MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources

More information

8.2 Graphs of Polar Equations

8.2 Graphs of Polar Equations 8. Graphs of Polar Equations Definition: A polar equation is an equation whose variables are polar coordinates. One method used to graph a polar equation is to convert the equation to rectangular form.

More information

Find the rectangular coordinates for each of the following polar coordinates:

Find 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 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

MATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval.

MATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval. MATH 8 Test -Version A-SOLUTIONS Fall 4. Consider the curve defined by y = ln( sec x), x. a. (8 pts) Find the exact length of the curve on the given interval. sec x tan x = = tan x sec x L = + tan x =

More information

11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes

11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes Parametric Differentiation 11.6 Introduction Sometimes the equation of a curve is not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this Section we see how to calculate the

More information

Math156 Review for Exam 4

Math156 Review for Exam 4 Math56 Review for Eam 4. What will be covered in this eam: Representing functions as power series, Taylor and Maclaurin series, calculus with parametric curves, calculus with polar coordinates.. Eam Rules:

More information

SECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.

SECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),

More information

Learning Objectives for Math 166

Learning Objectives for Math 166 Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the

More information

Edexcel Core Mathematics 4 Parametric equations.

Edexcel Core Mathematics 4 Parametric equations. Edexcel Core Mathematics 4 Parametric equations. Edited by: K V Kumaran kumarmaths.weebly.com 1 Co-ordinate Geometry A parametric equation of a curve is one which does not give the relationship between

More information

NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system.

NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system. NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system. Duplicating, selling, or otherwise distributing this product

More information

9.1 (10.1) Parametric Curves ( 參數曲線 )

9.1 (10.1) Parametric Curves ( 參數曲線 ) 9.1 (10.1) Parametric Curves ( 參數曲線 ) [Ex]Sketch and identify the curve defined by parametric equations x= 6 t, y = t, t 4 (a) Sketch the curve by using the parametric equations to plot points: (b) Eliminate

More information

Trigonometric Functions. Section 1.6

Trigonometric Functions. Section 1.6 Trigonometric Functions Section 1.6 Quick Review Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Radian

More information

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 10 (Second moments of an arc) A.J.Hobson

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 10 (Second moments of an arc) A.J.Hobson JUST THE MATHS UNIT NUMBER 13.1 INTEGRATION APPLICATIONS 1 (Second moments of an arc) by A.J.Hobson 13.1.1 Introduction 13.1. The second moment of an arc about the y-axis 13.1.3 The second moment of an

More information

Math 113 Final Exam Practice

Math 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 information

Math 323 Exam 1 Practice Problem Solutions

Math 323 Exam 1 Practice Problem Solutions Math Exam Practice Problem Solutions. For each of the following curves, first find an equation in x and y whose graph contains the points on the curve. Then sketch the graph of C, indicating its orientation.

More information

False. 1 is a number, the other expressions are invalid.

False. 1 is a number, the other expressions are invalid. Ma1023 Calculus III A Term, 2013 Pseudo-Final Exam Print Name: Pancho Bosphorus 1. Mark the following T and F for false, and if it cannot be determined from the given information. 1 = 0 0 = 1. False. 1

More information

(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim

(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)

More information

Calculus II Practice Test 1 Problems: , 6.5, Page 1 of 10

Calculus II Practice Test 1 Problems: , 6.5, Page 1 of 10 Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,

More information

Section 8.4 Plane Curves and Parametric Equations

Section 8.4 Plane Curves and Parametric Equations Section 8.4 Plane Curves and Parametric Equations Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called parametric equations).

More information

1 The Derivative and Differrentiability

1 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 information

MATH 100 REVIEW PACKAGE

MATH 100 REVIEW PACKAGE SCHOOL OF UNIVERSITY ARTS AND SCIENCES MATH 00 REVIEW PACKAGE Gearing up for calculus and preparing for the Assessment Test that everybody writes on at. You are strongly encouraged not to use a calculator

More information

MAC 2311 Calculus I Spring 2004

MAC 2311 Calculus I Spring 2004 MAC 2 Calculus I Spring 2004 Homework # Some Solutions.#. Since f (x) = d dx (ln x) =, the linearization at a = is x L(x) = f() + f ()(x ) = ln + (x ) = x. The answer is L(x) = x..#4. Since e 0 =, and

More information

Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document

Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.

More information

Mathematics Engineering Calculus III Fall 13 Test #1

Mathematics Engineering Calculus III Fall 13 Test #1 Mathematics 2153-02 Engineering Calculus III Fall 13 Test #1 Instructor: Dr. Alexandra Shlapentokh (1) Which of the following statements is always true? (a) If x = f(t), y = g(t) and f (1) = 0, then dy/dx(1)

More information

There are some trigonometric identities given on the last page.

There are some trigonometric identities given on the last page. MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during

More information

Math 20C Homework 2 Partial Solutions

Math 20C Homework 2 Partial Solutions Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we

More information

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1. MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:

More information

Precalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( )

Precalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( ) Syllabus Objectives: 5.1 The student will explore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.

More information

MATH 280 Multivariate Calculus Fall Integrating a vector field over a curve

MATH 280 Multivariate Calculus Fall Integrating a vector field over a curve MATH 280 Multivariate alculus Fall 2012 Definition Integrating a vector field over a curve We are given a vector field F and an oriented curve in the domain of F as shown in the figure on the left below.

More information

IYGB Mathematical Methods 1

IYGB Mathematical Methods 1 IYGB Mathematical Methods Practice Paper B Time: 3 hours Candidates may use any non programmable, non graphical calculator which does not have the capability of storing data or manipulating algebraic expressions

More information

Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus

Chapter 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 information

School of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester.

School of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester. School of Dtance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc Mathematics 0 Admsion Onwards IV Semester Core Course CALCULUS AND ANALYTIC GEOMETRY QUESTION BANK The natural logarithm

More information

HW - Chapter 10 - Parametric Equations and Polar Coordinates

HW - 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 information

CHAPTER 4 Stress Transformation

CHAPTER 4 Stress Transformation CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z

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

Math Section 4.3 Unit Circle Trigonometry

Math Section 4.3 Unit Circle Trigonometry Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise

More information

11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes

11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes Parametric Differentiation 11.6 Introduction Often, the equation of a curve may not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this section we see how to calculate the

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

+ 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 information

MAC Calculus II Spring Homework #6 Some Solutions.

MAC Calculus II Spring Homework #6 Some Solutions. MAC 2312-15931-Calculus II Spring 23 Homework #6 Some Solutions. 1. Find the centroid of the region bounded by the curves y = 2x 2 and y = 1 2x 2. Solution. It is obvious, by inspection, that the centroid

More information

University of Alberta. Math 214 Sample Exam Math 214 Solutions

University of Alberta. Math 214 Sample Exam Math 214 Solutions University of Alberta Math 14 Sample Exam Math 14 Solutions 1. Test the following series for convergence or divergence (a) ( n +n+1 3n +n+1 )n, (b) 3 n (n +1) (c) SOL: n!, arccos( n n +1 ), (a) ( n +n+1

More information

Messiah College Calculus I Placement Exam Topics and Review

Messiah College Calculus I Placement Exam Topics and Review Messiah College Calculus I Placement Exam Topics and Review The placement exam is designed to test a student s knowledge of material that is essential to the Calculus I course. Students who score less

More information

Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions function Characteristics of a function from set A to set B

Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions function Characteristics of a function from set A to set B Review of Topics in Algebra and Pre-Calculus I. Introduction to Functions A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in set B.

More information

(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3

(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III

More information

Parametric Curves. Calculus 2 Lia Vas

Parametric Curves. Calculus 2 Lia Vas Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider

More information

SOLUTIONS FOR PRACTICE FINAL EXAM

SOLUTIONS FOR PRACTICE FINAL EXAM SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable

More information

2.2 The derivative as a Function

2.2 The derivative as a Function 2.2 The derivative as a Function Recall: The derivative of a function f at a fixed number a: f a f a+h f(a) = lim h 0 h Definition (Derivative of f) For any number x, the derivative of f is f x f x+h f(x)

More information

2017 YEAR 5 PROMOTION EXAMINATION MATHEMATICS 9758

2017 YEAR 5 PROMOTION EXAMINATION MATHEMATICS 9758 RAFFLES INSTITUTION 07 YEAR 5 PROMOTION EXAMINATION MATHEMATICS 9758 September/October 07 Total Marks: 00 3 hours Additional materials: Answer Paper List of Formulae (MF6) READ THESE INSTRUCTIONS FIRST

More information

y = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx

y = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,

More information

Math 120: Precalculus Autumn 2017 A List of Topics for the Final

Math 120: Precalculus Autumn 2017 A List of Topics for the Final Math 120: Precalculus Autumn 2017 A List of Topics for the Final Here s a fairly comprehensive list of things you should be comfortable doing for the final. Really Old Stuff 1. Unit conversion and rates

More information

Arc Length and Surface Area in Parametric Equations

Arc Length and Surface Area in Parametric Equations Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length

More information

b g 6. P 2 4 π b g b g of the way from A to B. LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON ASSIGNMENT DUE

b g 6. P 2 4 π b g b g of the way from A to B. LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON ASSIGNMENT DUE A Trig/Math Anal Name No LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS (Brown Book) ASSIGNMENT DUE V 1 1 1/1 Practice Set A V 1 3 Practice Set B #1 1 V B 1

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9 MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

More information

Created by T. Madas LINE INTEGRALS. Created by T. Madas

Created by T. Madas LINE INTEGRALS. Created by T. Madas LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )

More information

Semester 2 Final Review

Semester 2 Final Review Name: Date: Per: Unit 6: Radical Functions [1-6] Simplify each real expression completely. 1. 27x 2 y 7 2. 80m n 5 3. 5x 2 8x 3 y 6 3. 2m 6 n 5 5. (6x 9 ) 1 3 6. 3x 1 2 8x 3 [7-10] Perform the operation

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

Math 111D Calculus 1 Exam 2 Practice Problems Fall 2001

Math 111D Calculus 1 Exam 2 Practice Problems Fall 2001 Math D Calculus Exam Practice Problems Fall This is not a comprehensive set of problems, but I ve added some more problems since Monday in class.. Find the derivatives of the following functions a) y =

More information

Calculus and Parametric Equations

Calculus and Parametric Equations Calculus and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Given a pair a parametric equations x = f (t) y = g(t) for a t b we know how

More information

Math 190 (Calculus II) Final Review

Math 190 (Calculus II) Final Review Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the

More information

2.8 Linear Approximation and Differentials

2.8 Linear Approximation and Differentials 2.8 Linear Approximation Contemporary Calculus 1 2.8 Linear Approximation and Differentials Newton's method used tangent lines to "point toward" a root of the function. In this section we examine and use

More information

MATH 152, Fall 2017 COMMON EXAM II - VERSION A

MATH 152, Fall 2017 COMMON EXAM II - VERSION A MATH 15, Fall 17 COMMON EXAM II - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN OFF cell phones

More information

4 The Cartesian Coordinate System- Pictures of Equations

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

More information

BHASVIC MαTHS. Skills 1

BHASVIC MαTHS. Skills 1 Skills 1 Normally we work with equations in the form y = f(x) or x + y + z = 10 etc. These types of equations are called Cartesian Equations all the variables are grouped together into one equation, and

More information

1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2

1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2 Math 51 Exam Nov. 4, 009 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown.. Any numerical answers should be left in exact form,

More information

The choice of origin, axes, and length is completely arbitrary.

The choice of origin, axes, and length is completely arbitrary. Polar Coordinates There are many ways to mark points in the plane or in 3-dim space for purposes of navigation. In the familiar rectangular coordinate system, a point is chosen as the origin and a perpendicular

More information

AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions

AP 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 information

9.1. Click here for answers. Click here for solutions. PARAMETRIC CURVES

9.1. Click here for answers. Click here for solutions. PARAMETRIC CURVES SECTION 9. PARAMETRIC CURVES 9. PARAMETRIC CURVES A Click here for answers. S Click here for solutions. 5 (a) Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the

More information

MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically

MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically 1 MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram

More information

C3 A Booster Course. Workbook. 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. (3) b) Hence, or otherwise, solve the equation

C3 A Booster Course. Workbook. 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. (3) b) Hence, or otherwise, solve the equation C3 A Booster Course Workbook 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. b) Hence, or otherwise, solve the equation x = 2x 3 (3) (4) BlueStar Mathematics Workshops (2011) 1

More information

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.

Candidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required. Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;

More information

Module 7 : Applications of Integration - I. Lecture 21 : Relative rate of growth of functions [Section 21.1] Objectives

Module 7 : Applications of Integration - I. Lecture 21 : Relative rate of growth of functions [Section 21.1] Objectives Module 7 : Applications of Integration - I Lecture 21 : Relative rate of growth of functions [Section 211] Objectives In this section you will learn the following : How to compare the rate of growth of

More information

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant

More information

Chapter 0 Preliminaries

Chapter 0 Preliminaries Chapter 0 Preliminaries MA1101 Mathematics 1A Semester I Year 2017/2018 FTMD & FTI International Class Odd NIM (K-46) Lecturer: Dr. Rinovia Simanjuntak 0.1 Real Numbers and Logic Real Numbers Repeating

More information

Math Test #3 Info and Review Exercises

Math Test #3 Info and Review Exercises Math 181 - Test #3 Info and Review Exercises Fall 2018, Prof. Beydler Test Info Date: Wednesday, November 28, 2018 Will cover sections 10.1-10.4, 11.1-11.7. You ll have the entire class to finish the test.

More information

DuVal High School Summer Review Packet AP Calculus

DuVal High School Summer Review Packet AP Calculus DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and

More information

Chapter 8. Complex Numbers, Polar Equations, and Parametric Equations. Section 8.1: Complex Numbers. 26. { ± 6i}

Chapter 8. Complex Numbers, Polar Equations, and Parametric Equations. Section 8.1: Complex Numbers. 26. { ± 6i} Chapter 8 Complex Numbers, Polar Equations, and Parametric Equations 6. { ± 6i} Section 8.1: Complex Numbers 1. true. true. true 4. true 5. false (Every real number is a complex number. 6. true 7. 4 is

More information