MTH 234 Final Exam Study Guide Disclaimer This Final Exam Study Guide is meant to help you start studying It is not necessarily a complete list of everything you need to know The MTH 234 nal exam mainly consists of standard response questions where students must justify their work In addition to these the Final Exam may consist of ll in the blank true false or multiple choice questions Most instructors agree that a good way to study for the nal is to do lots of problems to help familiarize yourself with all of the concepts covered Sections containing similar concepts have been grouped in blue boxes Most MTH 234 nal exam writers agree that the items below contain crucial material for showcasing MTH 234 knowledge and are therefore very important Expect at least one problem from each group on the nal exam Important Items from Each Section 12 3 The Dot Product Recall that a b a b cos and use this to solve for angles between vectors Remember the projection of b onto a formula projab a b a 2 a 12 4 The Cross Product Know how to calculate a b using the determinate 12 5 Equations of Lines and Planes Remember how to parametrize straight lines r r0 tv Recall the equation of a plane a x x0 b y y0 c z z0 0 Be able to calculate the angle between two planes Determine the distance between combinations of points lines and planes Study this section well There are many types of problems here Good Final Exam Review Problems 12 3 17 12 3 43 12 4 4 12 4 27 12 4 44 12 5 5 12 5 9 12 5 33 12 5 38 12 5 45 12 5 64 12 5 69 12 5 71 12 5 73 12 5 78 1 MTH 234 Final Exam Study Guide Important Items from Each Section 13 2 Derivatives and Integrals of Vector Functions Recall that r cid 48 t is the tangent vector to r t Be able to derive and integrate vector valued functions 13 3 Arc Length and Curvature Know the arc length formula L r cid 48 t dt Good Final Exam Review Problems 13 2 26 13 3 1 13 4 4 13 4 18a cid 90 b a 13 3 2 13 3 15 Important Items from Each Section 14 4 Tangent Planes and Linear Approximations Recall the formula for the tangent plane Be able to calculate the linearization L x y and use it to estimate the value of a function 14 6 Directional Derivatives and the Gradient Vector Be able to calculate the directional derivative using the dot product Know how to maximize and minimize the directional derivative Determine equations of tangent planes to level surfaces Good Final Exam Review Problems 14 4 4 14 4 21 14 6 7 14 6 12 14 6 25 14 6 44 Important Items from Each Section 14 5 The Chain Rule Recall the Chain Rule dz dt z x dx dt z y dy dt Know the Implicit Function Theorem which gives us Good Final Exam Review Problems 14 5 2 14 5 23 14 5 30 dy dx Fx Fy 2 MTH 234 Final Exam Study Guide Important Items from Each Section 14 7 Maximum and Minimum Values Know how to nd critical points Be able to classify critical points using the Second Derivatives Test Recall how to nd absolute Maximums and Minimums on closed bounded regions Good Final Exam Review Problems 14 7 9 14 7 14 14 7 30 14 7 32 14 7 41 14 7 43 Important Items from Each Section 15 6 Surface Area Know the surface area formula A s cid 115 cid 90 cid 90 D 1 cid 17 2 cid 16 z x cid 16 z y cid 17 2 dA 15 7 Triple Integrals Know how to sketch regions of integration Be able to determine how when to switch the order of integration Double Integrals over General Regions see 15 3 is an easier version of this 15 8 Triple Integrals in Cylindrical Coordinates Recall how to switch from rectangular to cylindrical coordinates E f x y z dV cid 82 cid 82 cid 82 E f r cos r sin z r dz dr d Double Integrals in Polar Coordinates see 15 4 is an easier version of this 15 9 Triple Integrals in Spherical Coordinates Recall how to switch from rectangular to spherical coordinates E f x y z dV cid 82 cid 82 cid 82 E f sin cos sin sin cos 2 sin d d d Good Final Exam Review Problems x2 y2 r2 x r cos y r sin y x tan cid 82 cid 82 cid 82 x2 y2 z2 2 x sin cos y sin sin z cos cid 82 cid 82 cid 82 15 3 51 15 4 13 15 6 4 15 6 12 15 7 13 15 7 20 15 8 18 15 8 21 3 15 8 24 15 9 24 15 9 28 15 9 39 MTH 234 Final Exam Study Guide Important Items from Each Section 16 2 Line Integrals Know how to evaluate line integrals over scalar functions using cid 90 C cid 90 b a f x y ds f x t y t cid 114 cid 16 dx dt cid 17 2 cid 17 2 cid 16 dy dt dt Know how to evaluate line integrals over vector elds using cid 82 C F Tds cid 82 b a F r t r cid 48 t dt Know when to apply Green s Theorem cid 82 C P dx Q dy cid 82 cid 82 D cid 16 Q x P y cid 17 dA 16 4 Green s Theorem 16 5 Curl and Divergence Know the normal form of Green s Theorem cid 82 C F n ds cid 82 C P dy Q dx cid 82 cid 82 D cid 16 P x Q y cid 17 dA Be able to evaluate line integrals with respect to the normal component of F Good Final Exam Review Problems 16 2 2 16 2 7 16 2 10 16 4 4 16 4 8 16 4 18 Important Items from Each Section 16 3 The Fundamental Theorem for Line Integrals Recall the component test for conservative vector elds P y Q x If F is conservative know how to nd a function f such that f F Be able to use the fundamental theorem of line integrals cid 82 C f dr f r b f r a 16 5 Curl and Divergence Recall the component test for conservative vector elds on R3 F 0 Good Final Exam Review Problems 16 3 6 16 3 15 16 3 20 16 5 13 16 5 16 16 5 18 4 MTH 234 Final Exam Study Guide Important Items from Each Section 16 5 Curl and Divergence Recall the formulas for curl F F and div F F 16 6 Parametric Surfaces and Their Areas Know how to parametrize and sketch a variety of surfaces …