Math 250B Study tips for the Final Exam

Math 250BStudy tips for the Final Exam1. 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), then for “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, then ∆z = 0.1 and ∆zz = 0.15.0 = 2%.3. 13.5 Drawing a tree diagram can be helpful when we use the chain rule for functions of several variables.Instead of memorizing formulas for implicit differentiation, it is much easier to draw a tree diagram.4. 13.6 The formula for 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, then u =v‖v‖is a unit vectorthat points in the same direction as v.5. 13.6 Make sure you know the following properties of the gradient of f(x, y).(a) For all (x, y), the vector ∇f(x, y) points in the 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 the level curve f(x, y) = c at point (x 0 , y 0 ).6. 13.6 For the Heat-Seeking Particle example (p. 937), observe that the path of the particle is alwaystangent to ∇T . Therefore, the slope of the tangent line of the path ( dydx) is always equal to the “slope”of the vector ∇T . We then havedydx = T y.T x7. 13.7 You should know that if F (x, y, z) = c is a surface in space, then at point (x 0 , y 0 , z 0 ) on the surface,the vectorN = ∇F (x 0 , y 0 , z 0 )is a vector normal to the surface. Once you know vector N and point (x 0 , y 0 , z 0 ), you can either findequations for 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 the 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 the curve of intersection of the two surfacesthen 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 the 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 the Second PartialDerivative Test.11. 13.8 For problems asking for absolute extremum of a function over a closed, bounded region, rememberthat the absolute extremum can occur either(a) at a critical point inside the region;(b) or at a boundary point of the region.12. 13.10 You should know how to use the method of Lagrange Multiplier for problems involving one or twoconstraints.13. 14.1 Make sure that you know the method of substitution and integration by parts since some integralsmight require it.14. 14.2 Make sure that you understand the process of reversing the order of integration for 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 formulas forcenter of mass, moments of inertia, and surface area are on the formula sheet.17. 14.6 The more difficult part with triple integrals is usually not the computation but setting-up the 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 the extra variable z remains unchanged.For spherical coordinates, I suggest to study example 4 (p. 1039) and look at the suggested problems fromthis section.19. 14.8 When you perform a change of variables to evaluate a double integral, do not forget the 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 for a piecewise smoothCcurve C of parametrization r(t), a ≤ t ≤ b.22. 15.3 For conservative fields, you should know the Fundamental Theorem of Line Integrals and theIndependence of Path property.23. 15.4 You should know how to use Green’s theorem.24. 15.5 For a parametric surface given by r(u, v), the vector r u × r v is a vector normal to the surface andthe 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 the Divergence and Stoke’s theorem . Since we were nottested on these sections, I suggest to do the suggested homework problems from the book. You shouldalso look at Example 1 from 15.7 (p. 1122) and Example 1 from 15.8 (p. 1129).