Calc & Analyt Geom IV (HONORS)
Calc & Analyt Geom IV (HONORS) MATH 2443
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Math 2443 Calculus 55 Analytic Geometry 4 Murphy Spring 2005 1 review Find an equation for the sphere centered at the point 1 23 and containing the point 021 2 review What shape in 3E3 does the equation have a 23 0 175 c 9572 d 2y25 3 review Find the traces of the surface 9x2 7 y2 7 22 9 in the planes z k y k and z k Then identify the surface and sketch it 4 review a Describe the surface given by the equation 2 2 yz b Describe the surface given by the equation 2 y2 3 c Find the curve of intersection of the two surfaces given above Describe the curve be sure to give information about important characteristics of the curve Cf review a Describe the surface given by the equation 2 v5 7 x2 7 yz b Identify the surface given by the equation x2 y2 1 c Find the curve of intersection of the two surfaces given above Describe the curve be sure to give information about important characteristics of the curve 6 151 Fill in the table of values on the blue sheets 7 151 Construct a contour diagram for the function fy 2 7 y 8 152 Show that lirnmya00 y does not exist w2y2 9 153 from McCallum7 W G7 Hughes Hallett7 D7 Gleason7 A M7 et al 19967 Multi variable Calculus Preliminary Edition7 p 4 Suppose you are in a stadium where the audience is doing the wave This is a ritual in which members of the audience stand up and down in such a way as to create a wave that moves around the stadium Normally a single wave travels all the way around the stadium7 but we will assume there is a continuous sequence of waves What sort of function will describe the motion of the audience To keep things simple7 we will consider just one row of spectators We consider the function which describes the motion of each individual in the row This is a function of two variables 71 the seat number and t the time in seconds For each value of n and t we write hnt for the height in feet above the ground of the head of the spectator in seat number n at time t seconds For purposes of analyzing the situation7 suppose that 71 takes on continuous rather than discrete values is this reasonable and that hn7 t 5 cos05n 7 t a Explain the signi cance of h25 in terms of the wave b Explain the signi cance of h2t in terms of the wave Find the period of h2t What does this period represent Explain the signi cance of hn7 5 in terms of the wave Find the period of hx7 5 What does this period represent c Find hnnt and htnt d Explain the signi cance of each of the following in terms of the wave hn27 t hnn7 5 hn27 5 ht27 t htn7 5 ht25 O 153 Given fy zy nd the rst partial derivatives H 153 Given fy zy nd the second partial derivatives E0 153 The paraboloid z 6 7 m 7 x2 7 2y2 intersects the plane z 1 in a parabola Find parametric equations for the tangent line to this parabola at the point 17 27 74 H 03 H I H 00 20 21 2 2 2 4 2 Cf F 9 154 Find the equation of the tangent plane to fy 2717 5l 7x2 7 y2 at the point 154 The dimensions of a closed rectangular box are measured as 80 cm 60 cm and 50 cm respectively with a possible error of 02 cm in each dimension Use differentials to estimate the maximum error in calculating the surface area of the box 155 Use the Chain Rule Case 1 to nd for z 6 where z cost and y 62 155 Use the Chain Rule Case 2 to nd and for z sarctanzy where z t2 and y set 155 The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out The resistance R is slowly increasing as the resistor heats up Use Ohm7s Law V IR to nd how the current I is changing at the moment when R 400 Q I 008A 7001 Vs and 003 Qs 156 Find the directional derivative of fy sinx 2y at the point 4 72 in the direction indicated by the angle 6 7quot 156 see also the supplemental graphics provided on the Calc 4 website Find the rate of change of fy em sing at the point 16 in the direction of the vector Si 7 1 A 2 A 156 see also the supplemental graphics provided on the Calc 4 website Find the gradient of fy emsz39ny and evaluate it at the point 1 156 see also the supplemental graphics provided on the Calc 4 website Find the maximum rate of increase of fy emsz39ny and evaluate it at the point 1 E and the direction in which it occurs 157 Find and classify the critical points of fy 3x 7 3 7 Bzyz 157 Find and classify the critical points of fy sinx sing in the square 0 lt z lt 2 7T 0ltylt27r 158 Use Lagrange multipliers to nd the maximum and minimum values of fy zzy subject to the constraint x2 y2 1 158 Optimize fy 2x2 z y2 7 2 subject to the constraint 2 y2 4 26 27 161 Use a double Riemann sum to approximate the volume under the surface fx y x2 4y over the rectangle R l 0 S x 3 20 3 y S 3 with partition lines z 1 y 1 y 2 and taking the center of the subrectangleRlj 161 0 5 10 15 20 25 30 A 20 ft by 30 ft swimming pool is lled with water The depth is measured at 5 ft inter vals starting at one corner of the pool and the values are recorded in the table Estimate the volume of water in the pool 162 Use an iterated integral to calculate the volume under the surface fy x24y over the rectangle R l 0 S x S 2 0 S y S 3 Use Eubini7s Theorem to reverse the order of integration and then calculate the new integral to verify that you get the same result and to practice integrating 162 Calculate the double integral ffzemy dA where R 01 gtlt 01 R 163 Consider the double integral ye dA D where D is the triangular region with vertices 00 24 and 6 0 a Sketch the region D b Set up two iterated intergrals set up one then reverse the order of integration to set up the other that are equivalent to the double integral in the problem c Evaluate one of your iterated integrals 163 Consider the solid bounded by the paraboloid z x2 y2 4 how do you know that7s a paraboloid and the planes z 0 y 0 z 0 and z y 1 how do you know these are planes7 a Set up two iterated intergrals that could be used to calculate the volume of this solid set up one then reverse the order of integration to set up the other You really should sketch the domain of integration to help you gure out the limits for your integrals b Evaluate one of your iterated integrals 32 163 Consider the integral a flt7 3 d1quot dy 1 m 0 my dydz b f6 f6 M721 drr dy Which of the integrals is equivalent C y dx dy d i i M y doc dy 33 164 Use polar coordinates to evaluate the integral ffydA where R is the region in the rst quadrant bounded by the circle x2 y2 9 and the lines y z and y 0 34 164 Evaluate the integral by converting to polar coordinates a W L 0 2 y232 Ch dy 35 165 A lamina occupies the part of the disk 2 y2 S 1 in the rst quadrant Find its center of mass if the density at a point is proportional to the square of its distance from the origin 36 1657 adapted from McCallum A forest has shape given by 7139 37139 Dr6l1 r 4 6 a Sketch the region D that represents the shape of the forest b Find the area of the forest c Suppose the population density of bunnies in the forest in thousands of bunnies per square mile is given by pr7 t9 2 Estimate the number of bunnies in the forest d Suppose the population density of bunnies in the forest in thousands of bunnies per square mile is given by pr7 t9 4 7 r2 cos 9 Estimate the number of bunnies in the forest 37 166 Set up a double iterated integral to nd the surface area of z 20 7 2 7 y2 over the rectangle 713 gtlt 733 you need NOT evaluate the integral 38 166 Find the surface area of the part of the paraboloid z 4 7 x2 7 y2 that lies above the xy plane 39 gm 7 A cube 1 A eemmemen an ands end 1 meae nimelmel that hes meme dmmy Kane me enhe cube at the mgn end the edpcml men ere an we Pmuve e L m the mam yang me cube s dm y m yncma at 271172 1 ya 7 7 my 1 271 nd me ms 0 me cube 0 gm 7 Came the xuh hounded by the Plane a gg 1 m Lha rstcmsm whosedsnr my 1 yvem w my 2 21 Set up meted mmye in en ex me archer Pm sxhle adms a new ere 172147 me ere equelmlhe meted mcegex me me blue shees memeemeeam jw mamsz n gm 7 Why dos me new n j m zwm he make sew 2 gm 7 5m me regal ameng m we anlA new L 271de u gm 7 se up en elem anle new e nd me vnlume emee sdld balm a by me Perehnlm cylm ae daz1fendlheplsneszoendzzl YmnesiNOTeveluele em I m mlwel See 1 m dssnneles ee up he eeme mmyel Discuss your 4 km thssounds m mam anyway u gm 7 Set up M19191 hhuegexs in en d the mhex humble den 0 Magellan how many humble mdms a 1nle ere thehev het exe equeh m the imam mega n A 3 mtsz A5 gm setuhethhle mtegeltehhathemmmeexthe xuhd mdnssibylhepexehnlmdsz urehaz a 2 7t Ymheeahmevehhetethemegeh e gm Came the mind emme by the hem 17th haw u 2 o thh amty New W set up mte gels to mkulele the msss eha the eehteh nimsssdthlssnhd Wheedhhtevehhete the hem 4 I 4 00 49 168 Set up the integral ff ay z dV where E is inside the sphere x2 y2 22 16 E and outside the cylinder x2 y2 4 168 Set up the integralfffzyz dV where E is above the cone 2 xx2 y2 and E below the sphere 2 y2 22 1 169 Consider the parametric equations 169 Consider the parametric equations 9 Hint Use the transformation z au7 y by First verify that this transformation yields 9 z the desired ellipse 31 7 cos 6 an 7 sin 6 7 2 an 7 cos 6 sin b 373 r sin6 sin b 3y 2 cos b E g Bu 169 Use a transformation to nd the area of the ellipse 52 169 Use a transformation to nd the area of the ellipsoid 22 CZ 2 yz gdLbiZJr 1 Calculate the determinant 31 34 34 343 31 372 31 Calculate the determinant 31 3w 3w g 3w 5 5 4 5 CT 5 57 5 00 59 9 03 review from Calc 3 Pararnetrize the curve z Represent the curve as a vector function And draw the curve using the vector function representation 172 Evaluate the line integral 0 y em d5 where C is the line segment joining the point 1 2 to the point 4 7 172 Evaluate the line integral 0 sin z dxcosy dy where C is the top half of the circle x2 y2 1 from 10 to 710 and the line segment from 710 to 723 171 Sketch the vector eld z y 2 j 171 Find the gradient of the function fy Sketch the vector eld Fwy WW4 x2 ys 171 Consider the vector eld 226 3ij from the example above Find the function fy for which this vector eld is the gradient This means that you need a function f so that 2x and 3y2 How many potential functions are there 171 Try to nd a potential function for the vector eld 4y i 2x j 171 Try to nd a potential function for 3x2 7 4y i 4y2 7 2x j 172 From the de nition of line integral calculate the work done by the force eld V Fy xi y 2j in moving an object along one arch of the cycloid Ft t 7 sint 17 cost j for 0 S t S 27139 173 Use the FTLl to calculate the work done by the force eld x y 2 j in moving an object along one arch of the cycloid Ft t 7 sin t 1 7 cost j for 0 S t S 27139 How would the work done by the vector eld in moving an object from 00 to 27T0 change if the path was a curve other than the cycloid Why 63 173 Consider the vector eld 6xy 7 y3 4y 3x2 7 Bzyz j a Without nding a potential function show that the eld is conservative b Now that we believe a potential function exists call it fy so that we are all consistent with our notation for the moment 7 to minimize confusion and nd it c Use your work to help you evaluate the line integral of P along y 2 71 g z lt 2 d Would your answer to c be different if the path were the line segment from 711 to 24 e What would be the value of a line integral around a closed smooth oriented path 64 173 related to the next two problems 6 Cf a Give an example of a region which is neither open nor closed Look up these terms on a previous handout if you need to b Give an example of a connected region which is not simply connected 173 What is the circulation of any conservative vector eld around any closed path Look up circulation77 on a previous handout if you need to 66 173 Consider the vector eld V F MI 96 iA 7 72yzZ2ya7 a Where is F not de ned b Show that i everywhere that F is de ned Remember to tell me how you computed the derivatives and why you chose the tool you used Does this equality also hold where F is not de ned Why or why not c Compute C P d directly from the dot product where C is the unit circle centered at the origin and traversed counterclockwise d ls F conservative How do you know What went wrong ie I thought that if Q 7 Q a am 7 3y then F is conservative 67 173 INSERT Calculus Problems for a New Century Xii19 p 199 68 173 INSERT Calculus Problems for a New Century Xii110 p 200 69 173 INSERT Calculus Problems for a New Century Xii111 p 201 70 review 172 and 173 lead into 174 The vector eld F 7y zj is sketched below a Add to the graph the curve C1givenby 15 a A 2 A I I k f memwxossl l rrLLL rernernbertoincludethe ll orientation ofthe curvel 1 A I x x X X l 1 4 lt V V x X X A 4 V V V X K b Use the dot product form to O 5 A 4 g g k calculate the line integral of k R k R Fover C1 39 V x 39 4 V x A K l l l y y 7 l p k A i i i T l c Add to the graph the curve 0 V g 1 A A A 4 l l l l 39 Cg given by v 4 4 A A 4 4 f l f a 7 A 3A x v v 4 4 l 4 I f f f f rt7tztg0t1 05 L b V I 4 l f f f f I 0 Ln 0 0 Ln H H Ln d Use the dot product form to calculate the line integral of P over Cg e ls F a conservative vector eld Why or why not 71 174 Let C Cg 7 C1 where C1 and Cg are given as in the problem above Note that C is piecewise srnooth7 sirnple7 closed7 and positively oriented why7 Note that F has continuous partial derivatives at every point in the region enclosed by C why7 Use Green7s Theorem to evaluate the integral 0 F dF 72 174 What does Green7s Theorem tell us about conservative vector elds 73 Use Green7s Theorem to compute f0 FdF 7 7 9 CT where 2siy4i5y3xi6j and C is the triangle with vertices 00 30 and 32 traversed counterclockwise Could you have used the Fundamental Theorem for Line Integrals instead Why Use the Greens Theorem to compute C F d where x y 6 x 7 y j and C is the triangle with vertices 00 30 and 32 traversed counterclockwise Could you have used the Fundamental Theorem for Line lntegrals instead Why Use the Fundamental Theorem for Line lntegrals to compute C F d where 9w x y x 7 y j and C is the parabola y x2 1 traversed from 01 to 12 Could you have used Green7s Theorem instead Why Compute f0 FdF where 2 7y 6 yz x j and C is the parabola y 21 traversed from 01 to 12 Could you have used the Fundamental Theorem for Line lntegrals Why Could you have used Green7s Theorem Why 77 175Cmsxdgx the veto ten new 22y1yijzzz e Find mm b Find an 78 175Cmsxdgx the veto hem Fwy 7212 27 e mm mm Dos your answer mehe smse 7 Why or Why mm b mm M Does your emweh mehe sme 7 Why or Why mm 79 175Cmsldex the veto heme shown In the gephe helm e h the van heh sheath an the man hem which museum does the curl vettm ea oeomhtt h In the team eld shown m the RIGHTheloNJs the dwelymoe mtwem hegtwe et the pmht at m At the punt them M the pmht 2 2y unveu Nu so hummus equme mungmeemcm gem manned y esthva m m m mm mmngm Mg and no mm can be mm as mum 1 m M 0 mu 7 am where a 15 the speed mug Use these equsums R wave that wwxs a gne M the tenyml plsne m the surface my u 1w27mwk at the 17mm msm me 17an 5 mm m Cenemsn Mmees 32 gm m the stead me pan 0 the surface my u21w27mw7c mmmmmmsw 5 us 1 72 5w 5 2 83 176machesurmeexeedchemmsgvenby my 3aswcmu13cmwsmujsnwk m o 5 K m o 9 g 27 Thsmeymree vmnwmkmle lhe ey u 176Cmsldex ammum z mg show um m ths spams me AS 1amp2 2 A 85 177 Evaluate the surface integral ffde where S is the surface 2 g fl2 1132 for S 0 1and0 y 1 86 177 Evaluate the surface integral ffzdS where S is the surface z y 222 for S 0 y 1and0 z 1 87 177 Evaluate the surface integral ffsyz dS where S is the part of the sphere x2 y2 S 22 1 that lies above the cone 2 xx2 yz 88 177 Use the ux formula to calculate the ux of the given vector eld through a rectangular net that is 30m long by 10rn high 177 Evaluate where Eyz x2y 73y2j4yglg and S is z z2y2 7 9 S below the rectangle 0 S x S 27 0 S y S 1 with downward orientation 8 90 179 Find the ux of the vector eld Fthrough out of the sphere with radius R centered at the origin 91 179 Find the ux of E through S where Ezyz 516 2x22j 31228 and S is bounded by z 4 7 x2 7 y2 and the zy plane7 and is oriented outward 92 178 Calculate ffcurlE dS where Ezyz y2 26 2j x2 yzlg and S is the part of z 2 y2 that lies inside the cylinder x2 y2 17 oriented upward