MTH 234 Final Exam Study GuideDisclaimer: 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 final exam mainly consists of standard response questions where students must justify their work. In additionto these, the Final Exam may consist of: fill in the blank, true/false, or multiple choice questions.Most instructors agree that a good way to study for the final is to do lots of problems to help familiarize yourself with all ofthe concepts covered.Sections containing similar concepts have been grouped in blue boxes. Most MTH 234 final exam writers agree that the itemsbelow contain crucial material for showcasing MTH 234 knowledge and are therefore very important. Expect at least oneproblem from each group on the final exam.Important Items from Each Section:12.3 - The Dot Product• Recall that a · b = |a| |b| cos θ and use this to solve for angles between vectors.• Remember the projection of b onto a formula: projab =a·b|a|2a.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• 12.5.781MTH 234 Final Exam Study GuideImportant Items from Each Section:13.2 - Derivatives and Integrals of Vector Functions• Recall that r0(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 =Zba|r0(t)| dtGood Final Exam Review Problems:• 13.2.26• 13.3.1• 13.3.2• 13.3.15• 13.4.4• 13.4.18aImportant 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• 14.6.44Important Items from Each Section:14.5 - The Chain Rule• Recall the Chain Rule:dzdt=∂z∂xdxdt+∂z∂ydydt• Know the Implicit Function Theorem which gives us:dydx= −FxFy.Good Final Exam Review Problems:• 14.5.2 • 14.5.23 • 14.5.302MTH 234 Final Exam Study GuideImportant Items from Each Section:14.7 - Maximum and Minimum Values• Know how to find critical points.• Be able to classify critical points using the Second Derivatives Test.• Recall how to find 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• 14.7.43Important Items from Each Section:15.6 - Surface Area• Know the surface area formula: A(s) =ZZDs1 +∂z∂x2+∂z∂y2dA15.7 - 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:• x2+ y2= r2• x = r cos θ• y = r sin θ• y/x = tan θ•RRREf(x, y, z) dV =RRREf(r cos θ, r sin θ, z) r dz dr dθ• 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:• x2+ y2+ z2= ρ2• x = ρ sin φ cos θ• y = ρ sin φ sin θ• z = ρ cos φ•RRREf(x, y, z) dV =RRREf(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ2sin φ dρ dφ dθGood Final Exam Review Problems:• 15.3.51• 15.4.13• 15.6.4• 15.6.12• 15.7.13• 15.7.20• 15.8.18• 15.8.21• 15.8.24• 15.9.24• 15.9.28• 15.9.393MTH 234 Final Exam Study GuideImportant Items from Each Section:16.2 - Line Integrals• Know how to evaluate line integrals over scalar functions using:ZCf(x, y)ds =Zbaf(x(t), y(t))rdxdt2+dydt2dt.• Know how to evaluate line integrals over vector fields using:RCF · Tds =RbaF(r(t)) · r0(t) dt.16.4 - Green’s Theorem• Know when to apply Green’s Theorem:RCP dx + Q dy =RRD∂Q∂x−∂P∂ydA.16.5 - Curl and Divergence• Know the normal form of Green’s Theorem:RCF · n ds =RCP dy − Q dx =RRD∂P∂x+∂Q∂y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• 16.4.18Important Items from Each Section:16.3 - The Fundamental Theorem for Line Integrals• Recall the component test for conservative vector fields:∂P∂y=∂Q∂x.• If F is conservative know how to find a function f such that ∇f = F.• Be able to use the fundamental theorem of line integrals:RC∇f · dr = f(r(b)) − f(r(a)).16.5 - Curl and Divergence• Recall the component test for conservative vector fields on R3: ∇ × F = 0.Good Final Exam Review Problems:• 16.3.6• 16.3.15• 16.3.20• 16.5.13• 16.5.16• 16.5.184MTH 234 Final Exam Study GuideImportant 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.• Remember the formula for surface area: A(S) =RRD|ru× rv| dA• Recall that if your surface is of the form z = f(x, y), then you can useA(S) =RRDr1 +∂z∂x2+∂z∂y2dA for surface area.16.7 - Surface Integrals• Be able to calculate various surface integrals using the equations:•RRSf(x, y, z)dS =RRDf(r(u, v)) |ru× rv| dA•RRSf(x, y, z)dS =RRDf(x, y, g(x, y))r1 +∂z∂x2+∂z∂y2dAwhen appropriate.• Know how to calculate the flux integral:RRF · dS =RRSF · n dS• Recall the formulas:•RRSF · dS