Mth 254 Clculus Exm 1 Review Your first exm is Fridy, April 26. I will provide one pge of notes. You my bring in one 3- inch by 5-inch note crd with notes on both sides. You should hve been working on the odd-numbered ssigned problems s you worked through ech section. Test questions re often VERY much like the odd nd even ssigned problems. You might find it useful to mke sure you cn do ll of them. Additionlly I WILL sk you to explin concepts nd you will likely find some true/flse questions. 12.1 Three-Dimensionl Coordinte System Be ble to grph points in the three-dimensionl coordinte system. Be ble to find the distnce between two points in the three-dimensionl coordinte system. Be ble to find the eqution of sphere in spce nd stte its center nd rdius. You my need to complete the squre like problem 15. Be ble to describe region of three-spce given n eqution or inequlity like problems 23 to 34. Be ble to crete n inequlity tht would crete region tht is described such s in problems 35 to 38. 12.2 Vectors Be ble to grph vector in three-spce. Know the component form of the three stndrd bsis vectors. Know properties of vectors nd be sure you re using proper nottion (rrows on vectors). Be ble to find the position vector if you know two points. Be ble to find the mgnitude of vector nd the direction it mkes with the positive x-xis such s in problem 33. Be ble to find the component form of vector such s in problem 31. Be ble to dd vectors both grphiclly nd using components such s in problems 15, 17, 35. Be ble to find unit vector in prticulr direction such s in problems 23 nd 25. 12.3 The Dot Product Be ble to clculte the dot product of two vectors given in ny form. Look crefully t problem 1 nd mke sure you hve good hndle on meningful expressions involving the dot product. If you understnd wht is involved with dot product then you cn nlyze problem 1 nd figure out which ones hve mening nd which ones re nonsense. Be ble to find the ngle between two vectors. Be ble to find the direction cosines nd direction ngles of vector in 3-D. Cn the direction cosines ever be negtive? How re those direction cosines defined? Be ble to find the sclr nd vector projection of one vector onto nother. Know properties of the dot product such s the vlue of the dot product when two vectors re orthogonl. Be ble to solve pplied (work) problems like problem 49 nd 51.
12.4 The Cross Product Be ble to clculte the cross product of two vectors given in ny form, such in problems 1 nd 3. You my use direct formul or the 3by3 determinnt method to get the cross product. Be ble to find unit vector orthogonl to two given vectors. Be ble to use the Right Hnd Rule to determine the direction of the cross product vector. Study problems 9, 11, 16b. Be ble to use vectors nd the cross product to find the re of prllelogrm or tringle given its vertices. Be ble to find the volume of prllelepiped given its vertices. Know when opertions on vectors give vectors or sclrs. Study problem 13. Know the properties of the cross product: see Theorem 11. Know the vlue of the cross product if two vectors re prllel. Be ble to find the torque (mgnitude nd direction) such s in problems 39 nd 40. 12.5 Equtions of Lines nd Plnes Be ble to find the vector, prmetric, nd symmetric equtions of lines in spce. Be ble to explin in WORDS how we use vectors to crete line in spce. Be ble to determine whether two lines in spce re prllel, skew or intersecting. Be ble to find the eqution of plne in spce. Be ble to explin in WORDS how we use vectors to crete the eqution of plne in spce. Be ble to use intercepts to grph plne in spce. Be ble to find where line intersects plne. Be ble to determine whether two plnes in spce re prllel, perpendiculr or neither. Be ble to find the line of intersection of two plnes nd the ngle between two plnes. 12.6 Cylinders nd Qudric Surfces Be ble to describe nd sketch the grph of cylinder in 3-spce. You will recognize cylinder becuse one of the vribles will be missing, so it cn tke on ny vlue. Sketch one trce nd then drg tht curve long the missing vrible s xis. This type should be EASY. Be ble to find nd grph the trces of qudric surfce. Some of you did not sketch trces. I will specificlly test this skill. Be ble to sketch trces in ech coordinte plne. Let z = 0 to get trce in the xy-plne, etc. Be ble to mtch the grph of qudric surfce with its eqution. If I give you more complicted eqution you should be ble to reson out wht it would look like nd mtch it to its grph.
14.1 Functions of Severl Vribles Be ble to find vlues of function of severl vribles from tble nd interpret their mening. Be ble to find nd sketch the domin of function of severl vribles. Some of you were tripped up by this question in the homework. If we re deling with surfces in 3 dimensions, then the domin is just in the xy-plne. So if I sk you to sketch the domin then you re sketching 2 dimensionl region. Go study exmple 1 crefully. Be ble to evlute function of severl vribles. Be ble to describe the shpe of function of severl vribles given contour mp (which we cll level curves). Go study Figure 11 in Section 14.1 to see how to use the level curves to mentlly recrete the three dimensionl surfce. Study Exmple 10 nd exmple 11. You should be ble to drw severl level curves for function. Alwys lbel ech level curve. 14.2 Limits nd Continuity Be ble to find where function of severl vribles is continuous. Be ble to find the limit of function of severl vribles, or explin why it doesn't exist. You my need to check the limit long severl pths so you cn sy the limit does not exist. 14.3 Prtil Derivtives Be ble to find the first, second nd mixed prtil derivtives of function of severl vribles. Be ble to use implicit differentition to find the prtil derivtives of function of severl vribles. Be ble to EXPLAIN in words wht the prtil derivtive with respect to x or y represents. Study Exmple 2 (especilly Figures 2 nd 3) in this section to give you insight into how to explin this concept. Note how one vrible is fixed. Note the direction of the tngent lines in the two figures. You cn tell the sign of the prtil derivtive in ech figure. Be sure you cn work the odd nd even ssigned problems for the Chpter 12 nd Chpter 14 sections we covered. If you wnt more then go to the chpter end mteril for Chpter 12. Chpter 12 Concept Check 1 through 19 (not 17) Chpter 12 True-Flse Quiz gives you some good questions to mull over. Chpter 12 Review Exercises but not 22, 27, 37, or 38. If you wnt more then go to the chpter end mteril for Chpter 14. Chpter 14 Concept Check 1 through 6 Chpter 14 Review Exercises 1, 3, 5, 8, 9, 10, 13, 19 On the next pge you will find SOME of the formuls you will need for the test. These re the criticl formuls I wnted you to hve hndy while you were working on homework. You will wnt to flip through the sections nd dd ny other formuls you need to your note crd.
Vectors = 2 b + c = b + c b + c = b + c b = b cosθ Projection b comp b = b proj b = Direction Cosines =,, 1 cosα = 1 2 3 cos β = cosγ = 2 3 2 2 2 cos α + cos β + cos γ = 1 = cos α,cos β,cosγ Some Cross Product Formuls i j k 2 3 1 2 1 2 b = 1 2 3 = i j + k b2 b3 b1 b2 b1 b2 b b b 1 2 3 b + c = b + c b c = b c b = b sinθ Lines v =, b, c direction vector, ro = xo, yo, zo, nd r = x, y, z Vector Form: r = r + tv so x, y, z = x, y, z + t, b, c o o o o Prmetric Form: x = xo + t, y = yo + bt, z = zo + ct x xo y yo z zo Symmetric Equtions: = =, for, b, c nonzero b c Plnes n =, b, c norml vector, r = x, y, z Vector Form: n ( r r o ) = 0 or n r = n ro o o o o Sclr Form: ( x x ) + b( y y ) + c( z z ) = 0 o o o Liner Form: x + by + cz + d = 0, nd r = x, y, z