MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm. Bring a calculator and something to write with. Also, you will be allowed to bring in one 8.5 11 sheet of paper (either typed or handwritten) containing any formulas, definitions, or theorems that you want. All trigonometric identities will be provided for you on the exam. Section 12.1 We defined an ordered 3-tuple, the 3-dimensional coordinate system R 3, and spheres. You should be able to find (and use) distances between points in R 3. Section 12.2 Definitions included vectors, equivalent vectors, vector addition, scalars, scalar multiplication, differences of vectors, the position vector of a point P, magnitude of a vector, V n (the set of all n-dimensional vectors), the standard basis vectors i, j, and k, and unit vectors. Be able to compute the above mentioned operations on vectors, understand the properties of these operations, and express a vector in the form a, b, c when just its endpoints are given. You should also have a good understanding of the geometric significance of the above topics. Section 12.3 In this section, we defined the dot product of two vectors, orthogonal vectors, direction angles, direction cosines, scalar projection, and vector projection. We proved some properties of dot products, interpreted them geometrically, and proved the property a b = a b cos θ, 0 θ π. Section 12.4 We defined the cross product of two vectors in V 3, the determinant of order 2, the determinant of order 3, the scalar triple product, and torque. Be familiar with the geometric interpretations of the cross product and triple scalar product and be able to use the property a b = a b sin θ, 0 θ π. Section 12.5 Given a point on a line and a vector describing its direction, you should be able to find the direction numbers of the line, the vector equation of the line, the parametric equations of the line, and the symmetric equations of the line. You should also be able to give a similar description of a line segment from a point P 0 to a point P 1. If you are given a point on a plane and vector normal to the plane, you should be able to find the vector equation of the plane, the scalar equation of the plane, and the linear equation of the plane. Of course, in all of these case, the necessary components may not be explicitly given. Other definitions in this section included skew lines and parallel planes. Section 12.6 Be familiar with the definitions of cylinders and quadric surfaces. The main quadric surfaces that we saw include ellipsoids, elliptic paraboloids, hyperbolic 1
paraboloids, cones, and hyperboloids of one and two sheets. graphs of these solids by graphing their cross-sections. Be able to sketch the Section 12.7 Be able to convert between cylindrical, spherical, and rectangular coordinates and describe the shapes of surfaces given by equations in any one of the three coordinate systems. Section 13.1 Definitions included vector-valued functions, limits and continuity of vector-valued functions, and space curves. Section 13.2 Our focus in this section was on the differentiation and integration of vector-valued functions. We defined the derivative, tangent vector, unit tangent vector, smooth, and piecewise smooth. The differentiation rules were stated (and some were proved) and we applied the Fundamental Theorem of Calculus to this setting. Section 13.3 Be able to find the arc length of a curve defined by a vector-valued function over a closed interval. Section 13.4 We worked velocity and acceleration problems using vector-valued functions. We also stated Newton s Second Law of Motion in this setting. Section 14.1 Be familiar with all of the terminology surrounding functions of more than one variable. Be able to find (and graph) domains, graph surfaces in R 3, and graph level curves/surfaces. Section 14.2 Be familiar with the ε-δ definitions of limits of functions of 2, 3, and n variables as well at continuity of such functions. Be able to determine (and show) whether or not a given limit exists. Section 14.3 We defined partial derivatives of a functions of more than one variable. Higher partial derivatives were considered and we stated Clairaut s Theorem. You should be able to check whether or not a given function is harmonic (ie., a solution to Laplace s equation). Section 14.4 Tangent planes to a surface z = f(x, y) and the linearization of f at a point (a, b) were defined. Be able to determine if a function of two variables is differentiable at a point. Differentiate between the increment of z, z and the differential dz. Section 14.5 Three different versions of the Chain Rule were discussed. The final version encompassed the previous two. We also stated the Implicit Function Theorem for functions of two and three variables. Section 14.6 We defined directional derivatives for functions of 2, 3, and n variables along with the gradient vectors in each of these cases. The main theorem of this section stated that the maximum value of the directional derivative D u f(x), where f is a function of two or three variables, is f(x) and it occurs when u has the same direction as 2
f(x). We also saw that if F is a function of three variables, then the tangent plane to the level surface F (x, y, z) = k at a point P = (x 0, y 0, z 0 ) is the plane through P with normal vector F (x 0, y 0, z 0 ) (provided F (x 0, y 0, z 0 ) 0). You should also be able to find the symmetric equations of a normal line to a level surface F (x, y, z) = k at a point P. Section 14.7 For functions of two variables, we defined local minimum/maximum points, absolute minimum/maximum points, critical points, saddle points, closed sets, and bounded sets. The 2 nd Derivative Test helped us in finding local extreme points and the Extreme Value Theorem guaranteed the existence of absolute extreme points for a continuous function defined on a closed, bounded set. We also described an algorithmic process for finding absolute extreme points. Section 15.1 We defined the double integral and used it to find volume. Be familiar with the midpoint rule for double integrals and the average value of a function of two variables. Section 15.2 Fubini s Theorem gave us a way of evaluating double integrals via iterated integrals. Note the special case where an integrand is given by a product of a function of x with a function of y. Section 15.3 Previous definitions and applications of double integrals involved rectangular regions in R 2. In this section, we extended the definition to arbitrary regions. Several important properties of double integrals were given at the end of this section. Section 15.4 We used polar coordinates to evaluate double integrals. Section 15.5 One application that we looked at involved finding the center of mass of a flat plate with continuous density function. Section 15.6 Be able to find the surface area of a surface z = f(x, y). Section 15.7 We defined triple integrals and used Fubini s Theorem to evaluate them. Be able to find volumes of solids using triple integrals. Section 15.8 Section 15.9 integrals. Evaluate triple integrals using cylindrical and spherical coordinates. We discussed making a change of variables in both double and triple Section 16.1 We were introduced to vector fields of R 2 and R 3. One particular example that you should be familiar with is the gradient vector field. Recall that if a vector field F is the gradient vector field of a scalar function f, then F is called a conservative vector field and f is called a potential function of F. Section 16.2 Definitions included line integrals of a function along a smooth curve (and hence, along a piecewise-smooth curve), line integrals with respect to x or y, line 3
integrals in space, and line integrals of vector fields. Section 16.3 We saw the Fundamental Theorem for Line Integrals and several important theorems regarding conservative vector fields. Section 16.4 The analogue of the Fundamental Theorem of Calculus for double integrals is Green s Theorem. Be familiar with the ways in which we can use Green s Theorem either to evaluate a line integral or to evaluate a double integral. Also, be able to find an area using a line integral. Section 16.5 In this section, we defined curl and divergence. Use curl to determine if a vector field is conservative. Section 16.6 We defined parametric surfaces and found a general form of a double integral (more general than the previous form) for finding surface area. Section 16.7 of vector fields. Section 16.8 Section 16.9 We defined the surface integrals, oriented surfaces, and surface integrals Be familiar with Stoke s Theorem. Be familiar with the Divergence Theorem. Other In addition to the topics on the front of this page, you should be able to work the examples covered in class as well as the assigned homework problems listed below. ASSIGNED HOMEWORK PROBLEMS Section 12.1 # 2, 3, 5, 9, 11, 13, 15, 19, 21, 22, 29, 32, 35 Section 12.2 # 1, 4, 6, 7, 9, 13, 17, 19, 21, 23, 25, 27, 28, 29, 31, 35 Section 12.3 # 2, 4, 5, 7, 9, 11, 13, 14, 17, 20, 21, 23, 26, 35, 38, 41, 57, 58 Section 12.4 # 1, 3, 5, 7, 10, 15, 18-22, 25, 29, 35, 41 Section 12.5 # 2, 4, 5, 7, 10, 15, 17, 19, 24, 26, 32, 35, 39, 53, 55 Section 12.6 # 1-8, 9, 11, 13, 14, 33, 35 Section 12.7 # 9, 15, 17, 21, 22, 23, 25, 26, 29, 37, 39, 41, 43, 45, 59, 60 Section 13.1 # 1, 3, 5, 7, 9, 10, 11, 12, 15, 17, 19 Section 13.2 # 3, 4, 9, 12, 13, 17, 19, 21, 23, 25, 32, 33, 37 4
Section 13.3 # 1-5 Section 13.4 # 3, 5, 7, 9, 13, 15, 16, 19, 20, 22, 23, 27 Section 14.1 # 5, 6, 8, 11, 13, 16, 19, 23, 25, 27, 30, 37, 43, 45, 59, 63 Section 14.2 # 5-16, 18, 37 Section 14.3 # 14, 15, 16, 17, 19, 21, 24, 27, 31, 33, 35, 37, 39, 41, 43, 45, 47, 55, 60, 68, 82, 87 Section 14.4 # 2, 3, 4, 11, 12, 13, 17, 23, 24, 29 Section 14.5 # 2, 3, 6, 7, 10, 12, 13, 21, 23, 25, 27, 29, 31, 33, 43, 44, 48, 53 Section 14.6 # 4, 5, 8, 9, 11, 12, 15, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 37, 39, 40 Section 14.7 # 1, 3, 5, 6, 7, 13, 14, 15, 27, 29, 32, 33, 38, 39 Section 15.1 # 3, 4, 7, 11, 13 Section 15.2 # 3, 5-8, 13-15, 20, 27, 33 Section 15.3 # 7-18, 20, 22, 26, 37, 38, 50, 51 Section 15.4 # 9-14, 17, 18, 21, 22, 29 Section 15.5 # 3, 6, 7, 10, 11, 13 Section 15.6 # 2, 3, 5, 6, 11 Section 15.7 # 7, 10, 11, 13, 15, 17, 18, 31 Section 15.8 # 5-8, 10, 12, 17, 18, 20, 22, 33, 35, 39 (a) Section 15.9 # 2-5, 7-9, 11-13, 19-21 Section 16.1 # 1-7, 10-18, 25, 29-32 Section 16.2 # 2-5, 6, 9, 11, 14, 16, 17, 19, 21 Section 16.3 # 3-5, 7, 12, 16, 26, 29, 31 Section 16.4 # 1-3, 7, 10, 13, 19 Section 16.5 # 1, 3, 5, 9, 10, 13, 15, 19 Section 16.6 # 1, 3, 31, 33, 35, 40 Section 16.7 # 5-7, 20, 21, 23 5
Section 16.8 # 3, 8, 13 Section 16.9 # 4, 7, 9 6