e) D œ < f) D œ < b) A parametric equation for the line passing through Ð %, &,) ) and (#,(, %Ñ.

Transcription

1 Page 1 Calculus III : Bonus Problems: Set 1 Grade /42 Name Due at Exam 1 6/29/ (2 points) Give the equations for the following geometrical objects : a) A sphere of radius & centered at the point ( ", (, ) ). b) A parametric equation for the line passing through Ð %, &,) ) and (#,(, %Ñ. d) The plane containing ( #, "), #Ñ, Ð ", ", &Ñ, and Ð &,', %Ñ. e) A parametric equation for the line of intersection of the two planes : B # C D œ # and B C D œ ". f) The angle between these same two planes. 2. (2 points) For each equation below in either cylindrical ( <ß) ßDÑ or spherical Ð3 ß) ß9Ñ coordinates identify and/or describe the surface. a) < œ +! b) 3 œ +! 1 1 c) 9 œ % d) ) œ ' # # # e) D œ < f) D œ < # # # # # # # # g) D < œ + h) 3 Ðsin 9 sin ) cos 9Ñ œ + 3. (1 point) Sketch the surface generated by each equation. # # # # # a) C D œ % b) B C D œ %

2 Page 2 Calculus III : Bonus Problems Due at Exam 1 6/29/ Þ (2 points) Match each figure with an equation from the list below which could generate the surface shown. Write the answer in the blank beneath the figure. # # # # # # # # # # # a) C D œ " b) B C D œ! c) B C D œ " d) B C D œ " # # # # # # # # # # e) B C D œ! f) B C D œ! g) B C D œ " h) B C D œ " # # # # # # # # # i) B C D œ! j) B C D œ! k) B C œ " l) B D œ " Figure A Figure B Figure A Figure C Figure B Figure D Figure C Figure D

3 Page 3 Calculus III : Bonus Problems Due at Exam 1 6/29/2018 Figure E Figure F Figure E Figure F Figure G Figure H Figure G Figure H

4 Page 4 Calculus III : Bonus Problems Due at Exam 1 6/29/ (3 points) In cylindrical coordinates <ß ) ß and D one can form the following vectors : Y< œ cos) s3 sin ) s4 ; Y) œ sin) s3 cos ) s4 ; a) Evaluate the following : Y Y œ Y Y œ Y 5 s œ Y Y œ < < < ) < ) ) Y 5 s œ Y Y œ Y 5 s œ s5 Y œ ) < ) ) < p p b) The position vector V œ Bs3 C s4 D s5 can also be written as V œ < Y s < D5. What are the transformation equations for the Cartesian coordinates Bß Cß and D in terms of <ß ) ß and D? In spherical coordinates 3ß9 ß and ) one can form the following vectors : Y s3 s4 s5 Y s3 s4 s 3 œ sin 9 cos) sin 9 sin) cos 9 ; 9 œ cos 9 cos) cos 9 sin) sin 95 ; Y) œ sin) s3 cos ) s4 ; c) Evaluate the following : Y 3 Y3 œ Y 3 Y9 œ Y 3 Y) œ Y 9 Y9 œ Y 9 Y) œ Y ) Y) œ Y 3 Y9 œ Y ) Y3 œ Y 9 Y) œ p p d) The position vector V œ Bs3 C s4 D5s can also be written as V œ 3Y 3. What are the transformation equations for the Cartesian coordinates Bß Cß and D in terms of 3ß9, and )?

5 Page 5 Calculus III : Bonus Problems Due at Exam 1 6/29/ (2 points) For each equation below i) Sketch the surface generated. ii) Give a valid parametric representation of the surface with a single set of three equations of the form: B œ 0Ð?ß>Ñ C œ 1Ð?ß>Ñ D œ 2Ð?ß>ÑÞ Include in your representation the domain of the parameters? and >. iii) Generate and attach with your project a Winplot 3D graph of your parameterization. # # # # # a) The cylinder: B D œ % with $ Ÿ C Ÿ $ b) The elliptic hyperboloid B C D œ % with lbl Ÿ ' 7. (2 points) Consider the curve C œ Ecos Ð,BÑ, where E and, are positive constants. a) Determine the curvature,,, as a function of B, E and,. b) Determine the radius of curvature, 3, as a function of B, E and,., œ 3 œ c) For what values of B is the curvature a maximum? Explain the relationship between this answer and the graph of the curve. d) For what values of B is the curvature a minimum? Explain the relationship between this answer and the graph of the curve.

6 Page 6 Calculus III : Bonus Problems Due at Exam 1 6/29/ (2 points) Consider the parabola C œ,b, where, is a positive constant. a) Determine the curvature,,, as a function of B and,., œ b) What is the maximum value of the curvature and where does this occur on the parabola? c) Determine the unit normal vector to the parabola at the point Ð>ß,> # Ñ. # d) Determine the radius of curvature, 3, as a function of > and, at the point Ð>ß,> ÑÞ 3 œ # e) Determine the coordinates of the center of curvature to the parabola at the point Ð>ß,> ÑÞ f) Using Winplot generate and attach the following two graphs. i) For, œ " the plot of the parabola and a parametric plot showing all the centers of curvature (called the "evolute") to the parabola. ii) For, œ # the plot of the parabola and a parametric plot showing all the centers of curvature to the parabola. It is possible to view and generate computer plots of three dimensional trajectories. In WinPlot use the 3-dim Window and the Curve space curve option of the Equa menu to enter equations for the Bß C, and D components of the curve in terms of the parameter >. 9. (1 point) For the curve below find the parametric equation for the tangent line to the curve at the given point and at this point calculate the curvature,. V Ð > Ñ œ #/ > #>/ > 4 Ð> # 3s s > $Ñ s 5 at Ð#ß!ß $Ñ Tangent line equation, œ

7 Page 7 Calculus III : Bonus Problems Due at Exam 1 6/29/2018 In the remaining problems the following notations will be used: V Ð>Ñ œ BÐ>Ñs3 CÐ>Ñs4 D Ð>Ñ 5s, where V is the position vector of a given object at time >. The Cartesian components of V are also considered to be functions of time. ± V l œ É V V œ ÈB2 C2 D 2 is the length of the position vector at time œ d V V B C 4 D d> Š œ œ Ð>Ñs3 Ð>Ñs Ð>Ñs 5 is the instantaneous velocity vector of the object. ± œ È B. 2 C. 2 D. 2 is the speed of the object. + œ d d> is the instantaneous acceleration vector of the object. 10. (2 points) A positively charged particle moving in a uniform magnetic field directed along the positive D axis will travel in a helical trajectory of the form : V Ð> Ñ œ <! cos Ð = > Ñ 3s <! sin Ð = > Ñ > s! 5, where and = are positive constants. a) What is the particle's velocity in Cartesian coordinates? b) What is the particle's acceleration in Cartesian coordinates? c) What is the particle's speed? d) What is the unit tangent vector of the particle's motion at time >? 21 e) How far does the particle travel along its trajectory in one turn of the helix (i.e., over any = time interval )? f) What is the curvature,,, of the trajectory? g) For fixed = and, what is the maximum value of, as v varies? r!! h) What is the radius of curvature, 3, of the trajectory? i) What is the principal unit normal vector of the trajectory? j) What is the binormal vector of the trajectory?

8 Page 8 Calculus III : Bonus Problems Due at Exam 1 6/29/2018 k) What is the torsion, 7, of the trajectory? l) For fixed = and r!, what is the maximum value of l7l as v! varies? m) What is the tangential component of acceleration? n) What is the normal component of acceleration? 11. (2 points) The following table gives astronomical data for the four inner planets of our solar system in units where the earth's distance to the sun is taken as "Þ!! and time is measured in earth years. As a simplification, all the planets are assumed to orbit the sun in the ecliptic, the plane of the earth's orbit (i.e., the angle of inclination is taken as zero). The angle?= is the sum of the longitude of ascending node and the longitude of perihelion of each planet minus the longitude of perihelion of the earth. In the approximation that the angle of inclination of a planet is zero,?= is the angular distance between the perihelion of the earth and the perihelion of the planet. The radial distance from the sun at perihelion is designated by r!. Planet Semimajor Axis r! Eccentricity Orbital Period?= Mercury!Þ$)(!Þ#!&' ##Þ#( Venus!Þ(#$!Þ!!') "!%Þ*) Earth "Þ!!!!Þ!"'( "Þ!!!!Þ!! Mars "Þ&#%!Þ!*$% #)#Þ") a) Determine the relationship between the semimajor axis, <!, and the eccentricity. From this equation calculate and fill in the values of missing from the above table. r! b) From Kepler's third law calculate and fill in the values of the orbital period missing from the above table. c) Using a computer plotting program generate and attach with your project a plot of the orbits of the four inner planets using the data in the above table. For convenience, take the perihelion of earth to occurr at a polar angle of ) œ!. Remember to convert all angles to radian measure before inputing to a trig function.

9 Page 9 Calculus III : Bonus Problems Due at Exam 1 6/29/ (2 points) Simplify the following expressions : d a) d> œ b) d d> ŠV V œ c) d l œ d) d d> l œ 13. (2 points) Given V œ < s< D s 5, with s< œ cos) s 3 sin ) s 4 ; s) œ sin) s 3 cos ) s 4 ; ) œ = >, = a constant ; ) D œ, Ò" cosð # ÑÓ ; < œ + Ò" cosð) Ñ Óà + and, are positive constants. a) Using a computer plotting program generate a three dimensional graph of this parametric space curve when + œ, œ ". b) What is the period of this trajectory? c) in terms of s< ß s ) and s 5. d) Express + in terms of s< ß s ) and 5 s. e) Is this trajectory the result of a central force? Explain.

10 Page 10 Calculus III : Bonus Problems Due at Exam 1 6/29/ (2 points) A carnival ride called the "Screeches" consists of a central arm of length %Þ& m which rotates at an angular velocity of & rpm (revolutions per minute). Attached to the outer end of the central arm is a second arm of radius $Þ! m which rotates in the same direction with an angular velocity of #) rpm. A passenger carrying gondola is located at the outer end of this second arm. a) What is the maximum magnitude of the velocity in meters per second experienced by a person riding in the gondola? b) What is the minimum magnitude of the velocity in meters per second experienced by a person riding in the gondola? c) What is the maximum magnitude of the acceleration experienced by a person riding in the gondola? Express the answer in units of 1 œ *Þ)! m per sec 2 Þ d) What is the minimum magnitude of the acceleration experienced by a person riding in the gondola? Express the answer in units of 1 œ *Þ)! m per sec 2 Þ e) What is the time interval between the experience of maximum magnitude of acceleration and minimum magnitude of acceleration?

11 Page 11 Calculus III : Bonus Problems Due at Exam 1 6/29/ (3 points) The Twisted Cubic. Consider the space curve defined parametrically as # #> V Ð>Ñ œ ) > ß $ ß> a) Determine an expression for the unit tangent vector, X s Ð:Ñ, at > œ : Þ $ b) Determine an expression the curvature,,ð:ñ, at > œ :. c) Determine an expression for the unit normal normal vector, R s Ð:Ñ, at > œ : Þ d) Determine an expression for the unit normal binormal vector, F s Ð:Ñ, at > œ : Þ e) Determine an expression for coordinates of the center of curvature at > œ :Þ f) In Winplot generate a graph of of this space curve for # Ÿ > Ÿ #. The Winplot graph needs to include the TNB vectors displayed as attached to the trajectory at the point V Ð:Ñ for # Ÿ : Ÿ #Þ This Winplot file should be attached to an sent to me at alehnen@matcmadison.edu. I need to receive this file the day the Group Project is due. Extra Bonus (3 points): g) Include in your Winplot graph the osculating circle attached to the trajectory at the point V Ð:Ñ for # Ÿ : Ÿ #Þ

12 Page 12 Calculus III : Bonus Problems Due at Exam 1 6/29/ (3 points) In this problem the following notations are used: The relative position vector from mass Q to mass 7 at time > is V Ð>Ñ œ B Ð>Ñ s 3 C Ð>Ñ s4 D Ð>Ñ5 s, where the Cartesian components of V are functions of time. The instantaneous separation velocity vector œ d V V B C 4 D d> Š œ œ Ð>Ñs3 Ð>Ñs Ð>Ñ5s. The instantaneous separation acceleration vector is + œ d d> After removing the motion of the center of mass from the problem, Newton's Second Law of Motion states F œ.+, where F is Q7 the net force acting on mass 7 due to mass Q and. is the "reduced" mass given by. œ Q7. If the force on 7 due to Q is gravitational, it is given by F œ KQ7 V, where K is the universal gravitational constant. l V l$ An object of constant mass, 7, is acted upon bya constant mass Q by some force F. The angular momentum of the system is defined as P œ dp a) Show that d> œ V F b) Suppose that F is a central force. This means that F 0 Ð ± V œ V ± Ñ, where 0 is a function only of the distance from lvl Q to 7 and not the direction. Show that P must now be a constant vector. c) Explain why this proves that in a central force the vector V is confined to a plane. d) Show that d d> Š ± V V ± V±. e) The kinetic energy, K, is given by the expression : O do. Show that œ F.

13 Page 13 Calculus III : Bonus Problems Due at Exam 1 6/29/2018 Page 13 2 f) Consider an object of mass, 7, attracted by gravity to an object of mass, Q. The potential energy is given by GMm dy V Y œ ; Show that d œ. ± V± > œ ± Vl $ g) Show that the total energy, I œ O Y, is a constant. In establishing Kepler's first law, the eccentricity of the orbit took the form, & œ 2! GÐM7Ñ ". h) What is the total energy, I, of the orbit at perihelion? i) What is the total energy at any other point along the orbit? j) Show that & " œ 2r! I KQ7. k) If I )!, what do you conclude about the orbit? l) If I œ!, what do you conclude about the orbit? m) If I!, what do you conclude about the orbit? o) The probability of a comet having a parabolic orbit is zero. Explain why.

14 Page 14 Calculus III : Bonus Problems Due at Exam 1 6/29/ (2 points) Below are eight contour diagrams of surfaces each made with a constant difference between contours. Match each contour with the correct 3-D surface graph on pages 3 and 4 as well as an equation from the list below which could generate the surface shown. Write the two answers for each contour in the blank provided. # # B C # # B C # # # a) D œ % * b) D œ * % c) B C D œ " d) D œ cosðbcñ # # # # e) D œ cos ÐBCÑ f) D œ cosðbñcos ÐCÑ g) D œ expð B ÐC "Ñ Ñ exp Ð B ÐC "Ñ Ñ # # # # # # h) D œ B C i) D œ B C j) D œ C B k) D œ BC Contour A Contour B Contour C Contour D

15 Page 15 Calculus III : Bonus Problems Due at Exam 1 6/29/2018 Page 15 2 Contour E Contour F Contour G Contour H

16 Page 16 Calculus III : Bonus Problems Due at Exam 1 6/29/2018 Surface Graph I Surface Graph II Surface Graph III Surface Graph IV

17 Page 17 Calculus III : Bonus Problems Due at Exam 1 6/29/2018 Surface Graph V Surface Graph VI Surface Graph VII Surface Graph VIII

18 Page 18 Calculus III : Bonus Problems Due at Exam 1 6/29/ (2 points) For the given functions (1) State the functions's domain (2) State the functions's range. (3) Sketch the level curves of the function. (4) Sketch the surface D œ 0ÐBß CÑ a) 0ÐBßCÑ œ È B# C# * 10 y x b) 0ÐBßCÑ œ È " B# C# y x

19 Page 19 Calculus III : Bonus Problems Due at Exam 1 6/29/ (4 points) For the given functions evaluate the following : # a) 0ÐBßCÑ œ B cosðbcñ lim ÐBßCÑ Ð"ß1Ñ 0ÐBßCÑ `0 `B `0 `C # ` 0 `C# # ` 0 `C`B ` `B Ð Ñ `0 `C # # b) 0ÐBßCÑ œ / 5ÐB C Ñ, where 5 is a positive constant. lim ÐBßCÑ Ð!ß!Ñ "0ÐBßCÑ B# C# `0 `B # ` 0 `C# ` `C Ð Ñ `0 `B c) 0ÐBßCÑ œ $ # #C $BC B# C# lim 0ÐBßCÑ ÐBßCÑ Ð!ß! Ñ `0 `B `0 `C d) " 0ÐBßCßDÑ œ È B# C# D# à T œ ÐBßCßDÑ lim 0ÐBßCßDÑ T Ð!ß!ß! Ñ `0 `B # ` 0 `D# # # # ` 0 ` 0 ` 0 `B# `C# `D#