Math 259 Winte 2009 Handout 6: In-class Review fo the Cumulative Final Exam The topics coveed by the cumulative final exam include the following: Paametic cuves. Finding fomulas fo paametic cuves. Dawing gaphs of cuves defined by paametic equations. Finding tangent lines to cuves defined by paametic equations. Finding the aea beneath (between the cuve and the x-axis) a paametic cuve. Finding the ac length of a paametic cuve. Pola coodinates fo the xy-plane. Identifying egions of the xy-plane descibed by pola coodinates. Conveting Catesian equations to pola equations. Conveting pola equations to Catesian equations. Sketching cuves in the xy-plane defined by pola equations. Finding fomulas fo tangent lines to cuves defined by pola equations. Finding aeas enclosed by pola cuves. Finding ac lengths of cuves defined by pola equations. Conic sections in Catesian and pola coodinates. Sketching conic sections defined by pola equations. Identifying eccenticity, diectix, etc. fom a pola equation. Classifying conic sections using eccenticity. Equations of lines, planes and sphees in 3D. Combining vectos. Magnitude of a vecto. Unit vectos. Applications of vectos in physics. Dot poduct of vectos. Angle between vectos. Othogonality. Vecto pojections. Coss poduct of vectos. Geomety of the coss poduct. Coss poduct and aeas. Calculating volumes with the scala tiple poduct. Finding equations fo lines and planes in 3D using the coss poduct. Distances fom points to lines and planes, and fom lines to planes. Symmetic equations. Sketching sufaces in 3D using contou plots. Intepeting contou plots. Classifying quadic sufaces. Recognizing planes, quadic sufaces and cylindes fom thei equations. Ceating and using vecto functions in 3D. Velocities and tangent vectos fo vecto functions (including unit tangent vectos). Showing that limits in 2D do not exist using a vaiety of stategies (e.g. y = mx, contou plots o tables). Evaluating and using functions with seveal input vaiables. Poving that limits do exist using the ε-δ definition. Calculating values fo limits/showing limits exist using the Squeezing Theoem. Calculating and intepeting patial deivatives. Finding equations fo tangent planes. Using the tangent plane to calculate a linea appoximation. Calculating total diffeentials fo functions. Using total diffeentials to estimate changes and eos. Using the Chain Rule fo functions of seveal vaiables. Calculating diectional deivatives fo functions. Calculating gadient vectos. Finding the diection of maximum ate of change (and magnitude of the maximum ate of change). Intepeting the pactical meaning of a diectional deivative. Finding and classifying (local maximum, local minimum, saddle point) the citical points of a function of seveal vaiables using patial deivatives and the Jacobian deteminant. Finding the global maximum and global minimum of a continuous function ove a egion in the xy-plane. Lagange Multiplies. Lagange Multiplies with two constaints. Wod poblems that can be solved using Lagange Multiplies.
Double Riemann sums. The Midpoint Rule fo double Riemann sums (no eo estimates). Setting up double integals. Changing the ode of integation in a double integal. Evaluating double integals. Applications of double integals to chemisty and physics (e.g. volume, mass, cente of mass). Pola coodinates. Conveting double integals fom Catesian to pola coodinates. Setting up double integals in pola coodinates. Evaluating double integals in pola coodinates. Setting up tiple integals. Changing the ode of integation fo a tiple integal. Evaluating tiple integals. Applications of tiple integals (e.g. calculating volume and mass). Setting up and evaluating tiple integals in cylindical/pola coodinates. Setting up and evaluating tiple integals in spheical coodinates. Dawing two-dimensional vecto fields. Associating fomulas fo two-dimensional vecto fields with visual epesentations. Detemining whethe a vecto field is consevative o not. Finding potential functions fo consevative vecto fields. Setting up paametic equations fo cuves in two dimensions. Setting up and evaluating (diectly) line integals of functions in two dimensions. Setting up and evaluating (diectly) line integals of vecto fields in two and thee dimensions. Calculating line integals of consevative vecto fields using the Fundamental Theoem. Calculating line integals in two dimensions using Geen s Theoem. Divegence and cul opeatos. Setting up and evaluating suface integals. Evaluating suface integals using Stoke s Theoem and the Divegence Theoem. 1. Find symmetic equations fo the line of intesection of the planes: x - y + z = 5 2x + y - 3z = 4.
2. Find the aeas of the plana (flat) shapes that have vetices located at: (a) (0, 0), (4, 1), (2, 3), and (6, 4). (b) (2, -3), (1, 1), (5, -6), and (4, -2). (c) (0, 0, 0), (0, 1, 0), and (1, 1, 0). (d) (1, 3, 0), (0, 2, 5), and (-1, 0, 2).
3. (a) Detemine values of λ and µ so that the points (-1, 3, 2), (-4, 2, -2) and (5, λ, µ) lie on a staight line. (b) Find a value of λ that will make the thee vectos: a = I + j + k b = 2I - 4k c = I + λj +3k coplana. (c) Suppose that the pais of vectos A and B, and C and D each detemine a plane. Show that if these planes ae paallel, then: (A B) (C D) = 0.
4. Conside the cuve in the plane defined by the paametic equations: x(t) = sin(t) y(t) = sin(2t) 0 t π. (a) Sketch the gaph of this cuve in the xy plane. (b) Find the total length of the cuve. (c) Find the aea enclosed by this cuve.
5. In this poblem we will conside the two lines given by: x = 1 - t, y = -2 + t, z = 2t and x = 4+ s, y = 1 + 3s, z = -3 + s. (a) Detemine whethe o not these lines ae paallel. (b) If the lines ae not paallel, do they intesect? If so, find the coodinates of the point of intesection. (c) Find the minimum distance between the two lines?
6. In each of the following cases, find the volume of the paallelpiped that has vetices at the points listed. (a) (1, 1, 1), (-4, 2, 7), (3, 5, 7) and (0, 1, 6). (b) (1, 6, 1), (-2, 4, 2), (3, 0, 0), and (2, 2, -4). (c) (1, 1, 1), (2, 2, 2), (6, 1, 3), and (-2, 4, 6).
7. An astoid (not to be confused with an asteoid) is a mathematical cuve with the equation, x 2 / 3 + y 2 / 3 =1. This cuve can be descibed by the paametic equations: x = cos 3 (t) and y = sin 3 (t) with 0 t 2π. (a) Sketch a gaph of the astoid. (b) Find a fomula fo dy dx the astoid. (c) What is the total length of the astoid? (d) Set up an integal that gives the aea enclosed by the astoid.
8. The position vecto of a paticle moving in the xy plane is given below. (t) = e t sin(t)i + e t cos(t)j (a) Sketch the path of the paticle beginning with t = 0. (b) Find the tangential component of the paticle s acceleation. (That is, the vecto pojection of the acceleation vecto onto the velocity vecto.) (c) Find the nomal component of the paticle s acceleation. (That is, the component of the acceleation vecto that is nomal to the velocity vecto.) (d) How could you deduce fom the sketch in pat (a) that the nomal component of the acceleation was going to be non-zeo? (e) Descibe the motion of a paticle that would have a zeo nomal component of acceleation.
9. (a) Find an equation fo the plane that includes both the point (1, 3, 0) and the line x, y,z = 2,7,1 + t " #1,1,1. Expess you final answe in the fom: ax + by + cz = d. (b) The diagam given below shows a weight of 10 pounds suspended fom a cod. Calculate the tension vecto, T 2, in the cod to the ight of the suspended weight. Calculate the vecto, not just the magnitude of the vecto. 45 o 30 o T 1 T 2 10 pounds
10. In this poblem u, v and w will always efe to the following vectos: (a) u = <1, 2, 3> Calculate v " w. v = < 7, 14, 21> w = <0, 1, 1>. (b) Calculate the volume of the paallelpiped whose sides ae fomed by the vectos u, v and w. (c) Calculate u " v. (d) What can you conclude about the vectos u and v? Cicle any of the statements that you believe to be tue. i. ii. iii. iv. v. u and v ae paallel (o anti-paallel). u and v ae both pependicula to the vecto v " w. u and v ae othogonal. u and v ae unit vectos. u is a unit vecto. v
11. Two lines ae defined below using thei symmetic equations. and x - 1 = 2(y + 1) = 3(z - 2) x - 3 = 2(y - 1) = 3(z + 1). (a) Show that these lines ae paallel. (b) Find an equation fo the plane that contains these lines.
12. Conside the suface defined by the equation, z = cos(xy). (a) Calculate the equations of the level cuves of this suface. (b) Sketch some of the level cuves in the xy plane. (c) Use the level cuves to sketch a pictue of the suface in thee dimensions.
13. Find and classify all of the citical points (local minimums, local maximums, saddle points, etc.) of the suface defined by the equation: f(x, y) = x 2 + xy + y 2 6x + 2.
14. In this poblem, g(x, y) will efe to the function defined by the fomula: g( x, y) = xy x 2 + y 2. (a) The gaph z = g(x, y) is shown below. Based on the appeaance of this gaph, do you think that g(x, y) has a limit as (x, y) (0, 0)? (b) Eithe pove that g(x, y) has a limit as (x, y) (0, 0) o show that the limit does not exist. ( Pove hee means use the ε δ definition.)
15. Descibe the level sufaces of the following functions. In each case, pove that the function has a limit as (x, y) (0, 0) o show that the limit does not exist. ( Pove hee means use the ε δ definition.) (a) f(x, y) = xy (b) f ( x, y) = x + y x " y (c) f ( x, y) = e 1/ ( x 2 +y 2 )
16. Find the equation of the tangent plane to the given suface at the given point. (a) z = y 2 - x 2. (-4, 5, 9) (b) z = ln(2x + y). (-1, 3, 0) (c) f ( x, y) = x 2 + 3y 2 " 2xy, (2, 2, 8).
17. The tempeatue, T, in a metal ball is invesely popotional to the distance fom the cente of the ball (which we will assume is located at the oigin). The tempeatue at the point (1, 2, 2) is 120 degees centigade. (a) Find the ate of change in T at (1, 2, 2) in the diection towads the point (2, 1, 3). (b) Show that at any point in the ball, the diection of geatest incease in tempeatue is a vecto pointing diectly towads the oigin. (c) Find a fomula fo the geatest possible ate of change of tempeatue at the point (x, y, z) (whee (x, y, z) (0, 0, 0)).
18. In this poblem the function f(x, y, z) will always efe to the function defined by the fomula: f ( x, y,z) =10e "0.01 ( 3x 2 "y 2 "2z 2 ). This function gives the tempeatue of a snake (on a plane) in degees Celsius ( o C). The coodinates of the snake s physical location (x, y, z) ae all measued in metes. (a) The snake is located at the point (1, 1, 1) and plans to slithe in the diection given by the vecto v = 1,2,3. What is the ate of change of tempeatue that the snake will expeience? Give appopiate units with you answe. (b) Snakes ae cold blooded and enjoy wamth. Conside the snake sitting at the point (1, 1, 1). In what diection should the snake slithe to maximize the ate of change of tempeatue? (c) What ate of change of tempeatue will the snake expeience if it stats at the point (1, 1, 1) and slithes in the diection you calculated in (b)? Give appopiate units with you answe.
19. In this poblem, z = f(x, y) is the function of x and y defined by the following fomula: z = f (x, y) = e "x +y 2. (a) Suppose that x and y ae both functions of t. All that you can assume about the functions x and y is listed below. Calculate z "(2). x(2) =1 y(2) = "1 x "(2) = #3 y "(2) = 1 2 (b) Suppose instead that x and y ae both functions of t and s, i.e. x = x(t, s) and y = y(t, s). All that you can assume about the functions x and y is listed below. x(2,0) =1 y(2,0) = "1 x t (2,0) = "3 y t (2,0) = 2 x s (2,0) = " 1 y 2 s (2,0) =1 Calculate z s ( 2,0). Show you wok!
20. The mantis shimp (Squilla empusa) is a small shimp-like custacean with vey poweful font claws. These shimp ae sometimes called thumb splittes because they can hit so had with thei claws that they sometimes split peoples thumbs open. A public aquaium is planning to exhibit mantis shimp. They will need to build a special tank with a slate bottom and glass sides. The eason fo this is that sometimes the shimp pound on the bottom of the tank with thei claws and can beak a glass bottom. The shimp don t jump, so the tank doesn t need a lid. The tank must have a volume of 1,000,000 cm 3 of wate. Glass costs 10 cents pe squae centimete, and slate costs 50 cents pe squae centimete. Find the dimensions of the least expensive tank that will hold 1,000,000 cm 3 of wate.
21. Find the coodinates of the point (o points) on the suface, that ae closest to the oigin (0, 0, 0). xy 2 z 3 = 2
22. When an electical cuent I entes two esistos with esistances R 1 and R 2, that ae connected in paallel (see below), it splits into two cuents I 1 and I 2 (with I = I 1 + I 2 ) so that the total electical powe, P = R 1 I 1 2 + R 2 I 2 2 is minimized. (a) Find fomulas fo I 1 and I 2 in tems of I and the two esistances, R 1 and R 2. (b) Show that the two esistances ae equivalent to a single esistance, R, whee: 1 R = 1 R 1 + 1 R 2.
23. A snake (on a plane) has found a wam metal plate to slithe aound on. The tempeatue of the plate (given in o C) at a point (x and y ae both measued in metes) is given by the function: T( x, y) = 4x 2 " 4xy + y 2. The snake slithes aound in a path that looks exactly like a cicle of adius 5 metes centeed on the oigin. What ae the highest and lowest tempeatues encounteed by the snake as it slithes aound this cicula path?
24. In this poblem the function f(x, y) will always efe to the function defined by the fomula: f ( x, y) = xy " x " y + 3. (a) Find the x and y coodinates of any citical points of f(x, y). (b) Classify the citical points that you found in Pat (a) as local maximums, local minimums o saddle points. (c) Find the global maximum and global minimum of f(x, y) on the tiangula egion of the fist quadant with vetices located at the points (0, 0), (2, 0) and (0, 4).
25. Each of the following integals is vey difficult to wok out. Revese the ode of integation to make the integal easie, and then evaluate. 1 3 (a) " " e x 2 dxdy 0 3y ( ) 3 9 (b) " " y cos x 2 dxdy 0 y 2 (c) # 1 0 " / 2 # cos( x) 1+ cos 2 ( x) dxdy acsin( y)
26. Find the volumes of the egions descibed below. (a) Unde the paaboloid z = x 2 + y 2 and above the egion bounded by y = x 2 and x = y 2. (b) Unde the suface z = xy and above the tiangle with vetices (1, 1), (4, 1) and (1, 2). (c) Bounded by the cylinde x 2 + y 2 = 9 and the planes x = 0, y = 0, x + 2y = 2 in the fist octant.
27. Use pola coodinates to evaluate the integals given below. (a) "" xyda whee R is the egion in the fist quadant between the cicles x 2 + y 2 = 4 and x 2 + y 2 = R 25. (b) e "x 2 "y ## 2 da whee D is the egion bounded by the semi-cicle x = 4 " y 2 and the y-axis. D (c) 1 x 2 x 2 4"x # # xydydx + # # xydydx + # # 2 xydydx. 1/ 2 1"x 2 1 0 2 0
28. Find the coodinates of the cente of mass of a unifom sheet of mateial in the shape of an isosceles ight tiangle. The two equal sides both have length a, and the density of the mateial is popotional to the squae of the distance fom the vetex opposite the hypoteneuse.
29. The aveage value of a function f(x, y, z) ove a solid egion E is defined to be: aveage = 1 V E ( ) """ f ( x, y,z)dv, E whee V(E) is the volume of the egion E. Find the aveage value of f(x, y, z) = xyz ove the cube with side length L with one vetex at the oigin and sides paallel to the coodinate axes.
30. (a) Calculate the value of the tiple integal: e " ( x 2 +y 2 +z 2 ) 3 2 ### dv, whee B is the solid sphee of adius a > 0 centeed at the oigin. You final answe should contain the lette a. B (b) Conside the vecto field F x, y,z ( ) defined by the fomula: F ( x, y,z) = "3x x 2 + y 2 + z 2 # e " ( x 2 +y 2 +z 2 ) 3 2,"3y x 2 + y 2 + z 2 # e " ( x 2 +y 2 +z 2 ) 3 2,"3z x 2 + y 2 + z 2 # e " ( x 2 +y 2 +z 2 ) 3 2 Calculate value of the line integal: 0, 0) and (0, 1, 1). " F d, whee C is the line segment joining the points (2, C
31. (a) The suface S is the pat of the paaboloid z = x 2 + y 2 that lies inside the cylinde x 2 + y 2 = 4. Use the axes povided below to make an accuate sketch of the suface S. z x y (b) Set up a tiple integal in x, y, z coodinates that will give the volume enclosed by the suface S, the cylinde x 2 + y 2 = 4 and the plane z = 0. (c) Convet you integal fom Pat (b) to cylindical coodinates and use this to calculate the volume enclosed by the suface S, the cylinde x 2 + y 2 = 4 and the plane z = 0.
32. The diagams given below show fou vecto fields. Match each pictue of a vecto field with one of the fomulas given below. You should have one unmatched fomula at the end of the poblem. DIAGRAM A DIAGRAM B DIAGRAM C DIAGRAM D (a) F x, y ( ) = x, 1 y MATCHING PICTURE: (b) F ( x, y) = 1 y,y MATCHING PICTURE: (c) (d) F x, y ( ) = sin x ( ), y MATCHING PICTURE: F ( x, y) = 1 y,x MATCHING PICTURE: (e) F x, y ( ) = y, x MATCHING PICTURE:
# diectly (i.e. not using Geen s C Theoem) whee C is the cuve shown in the diagam given below. 33. (a) Evaluate the line integal: y 2 dx + 3xy " dy 2 y (!1, 1.5) (1, 1.5) 1 x -1 1 (b) Evaluate the line integal: y 2 dx + 3xy " dy # using Geen s Theoem. C
"", whee S is the suface of the ectangula S pism shown in the diagam given below with positive oientation. (The vetex of the pism obscued in the diagam is ( 4, 2, 1).) 34. Evaluate the suface integal 3x,4 y,5z ds z (-4, -2, 1) (-4, 2, 1) (4, -2, 1) (4, 2, 1) (-4, 2, -1) y (4, -2, -1) (4, 2, -1) x
"", whee f x, y 35. (a) Evaluate the suface integal f ( x, y,z)ds of the paaboloid z = x 2 + y 2 that lies inside the cylinde x 2 + y 2 = 4. S ( ) =1+ xy and S is the pat NOTE: The following integal fomula may be helpful: " 2 + 4 4 d = ( 1+ 42) 2 + 4 4 12 + C. (b) Evaluate the suface integal: ## xz,"yz,"5 ds, whee S is the hemispheical suface z = 4 " x 2 " y 2. S NOTE: The following integal fomula may be helpful: " sin 2 ( x)dx = 1 x # 1 sin 2x 2 4 ( ) + C and cos2 x " ( )dx = 1 x + 1 2 4 sin ( 2x ) + C.