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# Analytic Geometry and Calculus 3 MATH 2400

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This 12 page Study Guide was uploaded by Cydney Conroy on Thursday October 29, 2015. The Study Guide belongs to MATH 2400 at University of Colorado at Boulder taught by Joshua Wiscons in Fall. Since its upload, it has received 162 views. For similar materials see /class/231821/math-2400-university-of-colorado-at-boulder in Mathematics (M) at University of Colorado at Boulder.

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Date Created: 10/29/15

MATH 2400 CALCULUS 3 Review for Final CH 12 3 D Space Vectors Vectors De nition and properties of dot product 7 Angle between vectors Orthogonal projections De nition and properties of cross product 7 De nition and properties of scalar triple product Parametric equations of lines Planes in 3 space Quadric Surfaces Cylindrical and spherical coordinates CH 13 Vector Valued Functions Differentiating and integrating VVf7s Limits and continuity of VVf7s Arclength change of parameter arclength parametrization CH 14 Partial Derivatives Limits and continuity of rnultivariable functions Partial differentiation and notation Implicit partial differentiation Differentials and local linearity Chain rule gtk lrnplicit differentiation gtk Related rates Directional derivatives gradients Tangent planes and normal vectors local linearity revisited Multivariable optirnization 2nd partials test Lagrange multipliers CH 15 Multiple lntegrals Double integrals rectangular and polar Pararnetric surfaces surface area Triple integrals rectangular7 cylindrical7 spherical 7 Change of variable Jacobians CH 16 Vector Calculus Vector elds gtk Conservative vector elds and potential functions gtk divF and curlF Line integrals work performed by a force eld Conservative vector eld test Fundamental theorem of line integrals Green7s Theorem 7 Surface integrals and ux Divergence theorem Stokes7 theorem D H 03 flaw Review Sheet for Midterm 1 Let v1 lt111 v2 lt7111 Find the angle between v1 and v2 in terms of arccosw A wagon is pulled horizontally by exerting a force of 10 lbs on the handle at an angle of 60 How much work is done by moving the wagon 50 feet Let v lt111gt b lt220gt a Find the orthogonal projection of V onto b b Find the component of V orthogonal to b Show that 12 gtlt 12 O for any vector 12 Let 131 lt1 23 132 lt321 Find a vector that is orthogonal to both 131 and 132 Find the volume ofthe tetrahedron de ned by P10 0 0 P23 72 75 P31 4 74 P40 3 2 Consider the points A4 723 B8 776 and C6 746 7 7 Find the area of the parallelogram spanned by the vectors AB and AC AA C793 VV Find an equation of the plane containing the points A B 0 Find an equation of the sphere centered at the origin and tangent to the plane containing the points A B C A O V A D V Find the area of the triangle with vertices A1 0 0 B0 1 0 and C00 1 and com pare it to the area of its projection on the zy plane A C7 V Find the distance from the point P14 48 to the plane 6x 7 2y 32 2 A O V Find an equation of the sphere centered at the point P1448 and touching the plane 6x 7 2y 32 2 A D V Find an equation of the plane containing the line L I39t 23ti1 72tj 71 tk and the point P131 Find a parametric equation of the line of intersections of the planes z 7 y 22 2 and 3x y 7 z 4 b Consider the points A2 771 B654 C6 23 and D7 744 a Find the area of the parallelogram de ned by E and AC 7 7 7 b Find the volume of the parallelipiped de ned by AB AC and AD c Find the distance from the point D to the plane passing through A B and C Find the vector component of v along b and the vector component of v orthogonal to b where v lt3 72 76 and b lt1 72 2 Find the area of the triangle with vertices P1 10 Q101 and R0110 13 Find parametric equations for the line a through 3 718 and parallel to V 235 b through 31 71 and 32 76 14 Where does the line parallel to zyz 1 2t3t5 7 7t and through the point 02 71 intersect the coordinate planes 15 Where does the line zyz 2tt 7 1 73t intersect the hyperboloid 27i1 19 9 16 Determine whether the lines L1 and L2 are parallel skew or intersecting If they intersect nd the point of intersection L1x1t y27t 23t L2z27t y12t z4t 17 Show that the lines L1 and L2 are not paralleland do not intersect one another L1z27t y3t z4t L2x573t yt z23t 18 Find an equation of the plane 75 1 2 with normal vector n 3 752 123 with normal vector n 159 712 10 73 0 72 74 4 1 6 2 1 73 5 71 4 2 72 4 e passing through 71 73 2 and containing the line zyz 71 7 2t4t 2 t passing through AAAA c passing through and 2z7y321 19 Determine whether the planes are parallel perpendicular or neither If neither nd the angle between them a snug7321 73z6y720 b 2z2y724 73z76y3z10 20 a Determine whether the line L and the plane 73 intersect or are parallel Lx774t y36t 295t 734xy2217 b Do the same for these Lx32t y675t z23t 733x2y7421 2 H 2 D 23 2 a 2 U 2 2 2 00 2 CD 3 O 31 7 Derive an equation in s y and z for the plane that contains the point P0z0y020 and is perpendicular to an arrowvector represented by the cartesian vector n lta bcgt Show the distance between the parallel planes cw by 02 d1 and ax by 02 d2 is ldl dzl D Given rectangular coordinates convert them to cylindrical coordinates and spherical coordi nates a 07270 17 717 Given cylindrical coordinates convert them to rectangular coordinates and spherical coordi nates a xi 73 0 27 07 2 Given spherical coordinates convert them to cylindrical coordinates and rectangular coordi nates a 0 3 b 57 07 0 Convert the given equation in cylindrical or spherical coordinates to an equivalent equation in cartesian coordinates and identify the surface represented by it a p 2 sec b b r2 cos 20 2 Match the equations Match each rectangular equation from the rst column with an equiva lent cylindrical equation in the second column Then match each cylindrical equation in the second column with an equivalent spherical equation in the third column 1 z 1x2y2 a rcos02sin052 10 15 2x2y5210 b 27 Let I39t cost i sintj tk and to E Find the vector Iquott0 Find a parametric equation of the line tangent to the graph of I39t 62ti 7 2 cos3tj at the point where t 1 1 Evaluate 62 i e tj tk dt 0 Solve the vector initial value problem for I39t by integrating antidifferentiating and using the initial conditions to nd the constants of integration I39 t 9sin ti 9cos tj 4k I390 3i 4j r 0 2i 7 7j ii cos0sin 2sin0sin 5cos 10 3 3 3 D 03 F Find an arc length parametrization of the curve that has the same orientation as the given parametrization and has It 1 as the reference point rt2t2it3 193 Find the arc length of I39t tZi cost tsin tj sint 7 tcos tk for 0 S t S 7139 Find an arc length parametrization of the curve that has the same orientation as the given parametrization and has It 0 as the reference point I39t QCOSStiSin3tj 0 S t S 1 2 3 Some Answers and Solutions Answer 2501bft lt07071gt a Answer b Answer 4 Solution Let 2 ltu1u2u3gt Then 12 X12 U1 uz Us u2u3U3u2Zyi U1U3iuSu1ju1u27u2u1 k U1 uz U3 D 5 Hint Consider 171 gtlt 132 6 Hint See problem 28 from section 124 in the book 7 Solution a We have Elt4753gt a a i J k a a A A gtlt 04 5 3lt79762gt ABgtltA l81364 1211 AC lt27273gt 2 i2 3 6 6 Answer AreaAB O T B gtlt OT 11 b Since AB and A0 are two vectors In the plane n B gtlt AC Taking A as a reference point we get 79z7476y22273 0 Answer The equation of the plane is 9x 6y 7 22 18 c Radius of this sphere is equal to the distance from this plane to the origin Since comp lt4723gt o lt79762gt 7187 ilt 97 672gti 11 the radius is r distOA lcompn 1811 The equation of the sphere is Answer 8 Solution a Since the area of the triangle AABC is halfthe area of the parallelogram spanned 6 by AB and AC we get E07E1lt010gt7lt100gtlt7110gt Elti101gt 6 i k 6 BXAO i110 lt111gt 1 BxA01121212 71 0 1 1 3 AreaAABC AreafBT The area of its shadow or projection on the gy plane is 1 1 1 AreaAABO 5AreaABA0 k th component of the vector AB gtlt A Answer The area is AreaAABO xg2 its shadow it is xZ times larger than the area of b From the equation of the plane H we can see that its normal vector n lt6 723 Also the point P00 71 0 is clearly a point on this plane Then 1458 6723 84710 24 98 distPH Compn POP 1439 1lt677273gt1 62 72 32 149 Answer The distance from the point P14 48 to the plane 6x 72y32 2 is c From part b we know that the distance from P to the plane H is 14 and so the radius of the sphere touching H and centered at P is 14 Answer The equation of the sphere centered at the point P1448 and touching the plane 672y322 is x7142y7422782 142 D 9 Solution a The point on L corresponding to t 0 is P02171 and the direction vector of L is E V lt3721 it is the vector of coefficients of t Then P0P lt7122 and V are two vectors in the plane and we can get the normal vector of the plane by taking their cross product i J E n P0P gtlt v 71 2 3 2 lt6 7 74 HMW and so the equation of the plane is 6z727y7174210 or 6z7y74223 Answer The equation ofthe pane containing the line L and the point P is 6 7y 7 42 23 b Taking the cross product of the normal vectors of the planes n1 lt1712gt and n2 lt3171gt we get the direction vector of the line of their intersection i j k Vl 11gtltl 12 1 71 2 lt7174gt 3 1 71 To find a reference point on this line we need to find a solution to the following system of equations adding the equations we obtain 4x z 6 7y222 3y724 sowecantakez026yz227210andweget z 7 t rltt lt0106gt lt7174gtt or in components y 10 7t 2 6 4t Answer The line of intersection of two given planes is rltt lt0106gt lt71 74gtt El 10 Solution a First we calculate the vectors 1 j k 6 XA l 412 3 llt24727127836748gtl 4 9 2 lt734712gtlx9161441913 Answer The area of the parallelogramm is 13 square units b The vector E lt533gt so 5 3 3 6 VolumeltABA A 4 12 3 lA A XA lilt533gtlt734712gti 4 9 2 l71512736ll739l39 Answer The volume of the parallelipiped is 20 21 22 23 Hint c Since Volumeparallelipiped Areabase gtlt height VolumeABACAD i 39 i 3 CM H he39ght AreaABAC E Alternatively 7ABXACAD 7 lti34712gt lt533gt l dD7H lCOHIPABxAC ADl l AB X AC 73747712 Answer The distance from the point D to the ABC plane is Answer Component of V along b is Component of V orthogonal to b is lt15 7 7 V AreallzTQ gtlt ail a Answer l 3y1t27175t z32ty713t285tl b Answer 773973 13 Solution The line in question is given by z 2t y 2 3t z 717 7t The line will intersect the y plane when 2 0 Solving 0 717 7t we get that the line intersects the y plane when t71 7 which corresponds to the point 7 710 the line intersects the z plane when y 0 which is found to be 7 0 intersects the yz plane when x 0 which is found to be 02 71 a Answer b Answer Following the same process we find that Finally the line D 24 25 26 27 28 29 30 31 32 33 34 a Answer b Answer a Answer b Answer Answer 3793 v Wigv as 2072 7 2 0s 07 07 7 07 9 005R 7 0705 1bi are equivalent as are 2aii

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