Math 250B Study tips for the Final Exam
Math 250B Study tips for the Final Exam
Math 250B Study tips for the Final Exam
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<strong>Math</strong> <strong>250B</strong><strong>Study</strong> <strong>tips</strong> <strong>for</strong> <strong>the</strong> <strong>Final</strong> <strong>Exam</strong>1. 13.3 Make sure that you know how to take partial derivatives of any function.2. 13.4 You should know that if z = f(x, y), <strong>the</strong>n <strong>for</strong> “small” ∆x and ∆y we have∆z ≈ ∂f ∂f∆x +∂x ∂y ∆y.For Error Analysis questions, ∆z = absolute error, and ∆zz= relative error, which is often expressed asa percentage.For example, if z = 5.0 ± 0.1, <strong>the</strong>n ∆z = 0.1 and ∆zz = 0.15.0 = 2%.3. 13.5 Drawing a tree diagram can be helpful when we use <strong>the</strong> chain rule <strong>for</strong> functions of several variables.Instead of memorizing <strong>for</strong>mulas <strong>for</strong> implicit differentiation, it is much easier to draw a tree diagram.4. 13.6 The <strong>for</strong>mula <strong>for</strong> directional derivativeD u f(x, y) = ∇f(x, y) · uis only valid if u is a unit vector. Remember that if v is a nonzero vector, <strong>the</strong>n u =v‖v‖is a unit vectorthat points in <strong>the</strong> same direction as v.5. 13.6 Make sure you know <strong>the</strong> following properties of <strong>the</strong> gradient of f(x, y).(a) For all (x, y), <strong>the</strong> vector ∇f(x, y) points in <strong>the</strong> direction of maximum increase of f.(b) The maximum value of D u f(x, y) is ‖∇f(x, y)‖.(c) ∇f(x 0 , y 0 ) is a vector normal to <strong>the</strong> level curve f(x, y) = c at point (x 0 , y 0 ).6. 13.6 For <strong>the</strong> Heat-Seeking Particle example (p. 937), observe that <strong>the</strong> path of <strong>the</strong> particle is alwaystangent to ∇T . There<strong>for</strong>e, <strong>the</strong> slope of <strong>the</strong> tangent line of <strong>the</strong> path ( dydx) is always equal to <strong>the</strong> “slope”of <strong>the</strong> vector ∇T . We <strong>the</strong>n havedydx = T y.T x7. 13.7 You should know that if F (x, y, z) = c is a surface in space, <strong>the</strong>n at point (x 0 , y 0 , z 0 ) on <strong>the</strong> surface,<strong>the</strong> vectorN = ∇F (x 0 , y 0 , z 0 )is a vector normal to <strong>the</strong> surface. Once you know vector N and point (x 0 , y 0 , z 0 ), you can ei<strong>the</strong>r findequations <strong>for</strong> a tangent plane or a normal line.8. 13.7 To get a vector normal to a surface described as z = f(x, y). ConsiderF (x, y, z) = f(x, y) − zand think of <strong>the</strong> surface as F (x, y, z) = 0 so that ∇F gives a normal vector.9. 13.7 If (x 0 , y 0 , z 0 ) is a point on <strong>the</strong> curve of intersection of <strong>the</strong> two surfaces<strong>the</strong>n vectorF (x, y, z) = c and G(x, y, z) = d∇F (x 0 , y 0 , z 0 ) × ∇G(x 0 , y 0 , z 0 )is a vector tangent to <strong>the</strong> curve at point (x 0 , y 0 , z 0 ).
10. 13.8 You do not have to memorize it but make sure that you know how to use <strong>the</strong> Second PartialDerivative Test.11. 13.8 For problems asking <strong>for</strong> absolute extremum of a function over a closed, bounded region, rememberthat <strong>the</strong> absolute extremum can occur ei<strong>the</strong>r(a) at a critical point inside <strong>the</strong> region;(b) or at a boundary point of <strong>the</strong> region.12. 13.10 You should know how to use <strong>the</strong> method of Lagrange Multiplier <strong>for</strong> problems involving one or twoconstraints.13. 14.1 Make sure that you know <strong>the</strong> method of substitution and integration by parts since some integralsmight require it.14. 14.2 Make sure that you understand <strong>the</strong> process of reversing <strong>the</strong> order of integration <strong>for</strong> double integrals.See example 4, p. 996.15. 14.3 When we evaluate integrals in polar coordinates, we often encounter trigonometric integrals involvingsin θ and cos θ. Make sure that you know how to evaluate integrals such as∫∫sin 3 θ cos θ dθ or cos 2 θ dθ.16. 14.4 - 14.5 These sections include some important applications of Double Integrals. The <strong>for</strong>mulas <strong>for</strong>center of mass, moments of inertia, and surface area are on <strong>the</strong> <strong>for</strong>mula sheet.17. 14.6 The more difficult part with triple integrals is usually not <strong>the</strong> computation but setting-up <strong>the</strong> tripleintegral. I suggest to study example 4 (p. 1028) and to redo questions from practice test 2.18. 14.7 This section is very important. Converting a triple integral to cylinder coordinates is very similar toconverting a double integral to polar coordinates, simply note that <strong>the</strong> extra variable z remains unchanged.For spherical coordinates, I suggest to study example 4 (p. 1039) and look at <strong>the</strong> suggested problems fromthis section.19. 14.8 When you per<strong>for</strong>m a change of variables to evaluate a double integral, do not <strong>for</strong>get <strong>the</strong> Jacobian.20. 15.1 Make sure that you know how to test if a vector field is conservative and how to find a potentialfunction.21. 15.2 You should know how to evaluate line integrals ∫ C f(x, y, z) ds and ∫ F · dr <strong>for</strong> a piecewise smoothCcurve C of parametrization r(t), a ≤ t ≤ b.22. 15.3 For conservative fields, you should know <strong>the</strong> Fundamental Theorem of Line Integrals and <strong>the</strong>Independence of Path property.23. 15.4 You should know how to use Green’s <strong>the</strong>orem.24. 15.5 For a parametric surface given by r(u, v), <strong>the</strong> vector r u × r v is a vector normal to <strong>the</strong> surface and<strong>the</strong> element of surface area is dS = ||r u × r v || dA25. 15.6 You should know how to evaluate Surface Integrals and Flux Integrals.26. 15.7 - 15.8 You should know how to use both <strong>the</strong> Divergence and Stoke’s <strong>the</strong>orem . Since we were nottested on <strong>the</strong>se sections, I suggest to do <strong>the</strong> suggested homework problems from <strong>the</strong> book. You shouldalso look at <strong>Exam</strong>ple 1 from 15.7 (p. 1122) and <strong>Exam</strong>ple 1 from 15.8 (p. 1129).