Outcomes List for Math 200-200935 Multivariable Calculus (9 th edition of text) Spring 2009-2010 The purpose of the Outcomes List is to give you a concrete summary of the material you should know, and the skills you should acquire, by the end of this course. As an overall summary, you should be able to, after completing this course: Do basic calculations with vectors and related geometric shapes (lines and planes), using dot products, cross products, and vector calculations Do calculus with space curves, including finding velocity and equations of tangent lines Work with plots of multivariable functions, including the computation of level curves and level surfaces Use partial derivatives, including chain rule formulas Compute tangent planes to surfaces, find critical points, and check for max/min Work with cylindrical and spherical coordinates Work with parametric surfaces Do basic multivariable integrals Do change of variables in multivariable integrals This outcomes list will be updated with specific review problems and topics for each exam of the quarter. The following information is for reviewing for the new material since Exam 3. The final exam is COMPREHENSIVE. It will cover the old material from the Outcomes Lists for Exam 1, Exam 2, and Exam 3. In addition it will cover this new material from Chapters 14.1, 14.2, 14.3, 11.8, 14.4, 14.5, 14.6, and 14.7. Approximately 60-70% of the final exam will come from this new material. 14.1 Compute double integral calculations over rectangular regions using partial integration. Inspect an integral, and decide if the order of integration is easier one way (y first, x second) or the other (x first, y second). Interpret a double integral as the volume under a surface.
In addition to reviewing assigned problems from 14.1, look at (all references to section 14.1): Example 3 (both ways!); Regular problem 33 14.2 Sketch the region of integration for a double integral over a non-rectangular region based on a description in terms of equations and inequalities. After inspecting the region, set up the integral. If possible, set it up in both possible orders (x first, or y first). Decide, on the basis of inspection of the region of integration, which order is most efficient. Do a given integral both ways, in order to check your answer. Reverse the order of integration to simplify the evaluation of a double integral. In addition to reviewing assigned problems from 14.2, look at (all references to section 14.2): Example 4 (both ways!), Regular problem 55 14.3 Convert rectangular double integrals to polar double integrals, including converting the limits of integration, the function to be integrated, and converting the differential da to rdrd. In addition to reviewing assigned problems from 14.3, look at (all references to section 14.3): Examples 1, 4; Regular problem 29 11.8 Convert between rectangular, cylindrical, and spherical coordinates. Describe simple surfaces in terms of cylindrical and spherical coordinates. Be able to use Table 11.8.1, and understand Table 11.8.2. In addition to reviewing assigned problems from 11.8, look at (all references to section 11.8): Example 3; Regular problems 24, 30, 31 14.4 Compute the surface area of a surface of the form z f( x, y). Describe a surface parametrically. Eliminate the parameters to identify a surface. Compute the tangent plane of a parametric surface. Compute the surface area of a parametric surface. In addition to reviewing assigned problems from 14.4, look at (all references to section 14.4): Examples 10, 12; Regular problems 9, 26, 30
14.5 Evaluate triple integrals over rectangular boxes. Evaluate over more general regions, using Theorem 14.5.2 (projecting the solid onto the xy-plane), as well as by projecting the solid onto the xz- and yz-planes. In addition to reviewing assigned problems from 14.5, look at (all references to section 14.5): Examples 3, 5; Regular problem 16 14.6 Perform simple triple integrals in spherical and cylindrical coordinates. Convert integrals from rectangular coordinates to these other coordinate systems, remembering that dv r dr d dz 2 sin( ) d d d. Use Table 14.6.1. In addition to reviewing assigned problems from 14.6, look at (all references to section 14.6): Examples 3, 4; Quick Check problems 2, 3 14.7 Use the change of variable formulas (14.7.2 and 14.7.4) for converting an integral from rectangular coordinates to another coordinate system by changing the integrand, the region of integration, and including the Jacobian factor. In addition to reviewing assigned problems from 14.7, look at (all references to section 14.7): Examples 3, 4; Regular problem 22 The preceding information is for reviewing for the new material since Exam 3.
The following information is for reviewing for the material of Exam 3: Exam 3 will cover sections 13.4, 13.5, 13.6, 13.7, and 13.8 13.4 Compute the local linear approximation to a function of two or more variables. Use total differentials to estimate maximum possible (and percentage) errors. In addition to reviewing assigned problems from 13.4, look at (all references to section 13.4): Examples 3, 4; Quick Check problem 4; Regular problems 42, 54 13.5 Compute partial derivatives with the various versions of the multi-dimensional chain rule. Compare your answer with the direct method of computing the partial derivatives. In addition to reviewing assigned problems from 13.5, look at (all references to section 13.5): Quick Check problems 1, 4; Regular problem 50 13.6 Compute the gradient, and use it to compute a directional derivative of any given function in any given direction. Use the fact that the gradient of a function f(x,y) is perpendicular (normal) to the level curves f(x,y) = c. In addition to reviewing assigned problems from 13.6, look at (all references to section 13.6): Example 5; Quick Check problems 3, 4; Regular problems 57, 72 13.7 Use the local linear approximation to f(x,y) to compute the equation of a tangent plane. Given an implicitly defined level surface F(x,y,z) = c, be able to compute the equation of the tangent plane at a point on the surface. Compute the parametric equations for the normal line. Use gradients to find tangent lines to the intersection curve of two surfaces. Find (acute) angles between tangent planes and other planes. In addition to reviewing assigned problems from 13.7, look at (all references to section 13.7): Example 3; Quick Check problems 1, 2; Regular problems 2, 26 13.8 Use the partial derivatives to find critical points (possible locations of maxima or minima). Use the Second Partials Test for functions of two variables to determine whether a critical point is a relative maximum, relative minimum, or a
saddle point. Solve word problems involving maxima and minima. Compute absolute maxima and minima on closed regions. In addition to reviewing assigned problems from 13.8, look at (all references to section 13.8): Example 5; Quick Check problem 2; Regular problems 12, 43 The preceding information is for reviewing for the material of Exam 3
The following information is for reviewing for the material of Exam 2: Exam 2 will cover sections 12.1, 12.2, 13.1, 11.7, and 13.3 12.1 Work with and recognize simple parameterized curves (i.e. graphs of vectorvalued functions), including lines and helices. Find the domain of vector-valued functions. In addition to reviewing assigned problems from 12.1, look at (all references to section 12.1): Example 6; Quick Check problems 1, 3: Regular problem 42 12.2 Differentiate vector-valued functions for curves in the plane and in space. Compute the equation for a tangent line to a curve; describe the tangent line as a vector equation and as a set of parametric equations. Determine (acute) angles between tangent lines. Use differentiation formulas involving cross-products and dot products. Evaluate indefinite and definite integrals of vector-valued functions. Solve vector initial-value problems. In addition to reviewing assigned problems from 12.2, look at (all references to section 12.2): Examples 3, 4, 9; Quick Check problem 4; Regular problems 24, 50 13.1 Describe and sketch the domain of a function of two or more variables. Evaluate functions of two or more variables. For functions of two variables, compute level curves. In addition to reviewing assigned problems from 13.1, look at (all references to section 13.1): Quick Check problem 3; Regular problems 22, 39, 41, 46 11.7 Compute traces of quadric surfaces, and recognize the resulting conic sections in the given plane. Given an equation for a quadric in standard form, be able to recognize the type of the quadric (and in particular, its graph). In addition to reviewing assigned problems from 11.7, look at (all references to section 11.7): Tables 11.7.1, 11.7.2; Quick Check problems 1, 4; Regular problems 27, 29, 34, 36
13.3 Compute first-order and second-order partial derivatives. Do implicit partial differentiation. Do various word problems involving rates of change, which use partial derivatives. In addition to reviewing assigned problems from 13.3, look at (all references to section 13.3): Quick Check problems 2, 3; Regular problems 53, 62 The preceding information is for reviewing for the material of Exam 2
The following information is for reviewing for the material of Exam 1: Exam 1 will cover sections 11.1, 11.2, 11.3, 11.4, 11.5, and 11.6 11.1 Work with rectangular coordinates and work with equations of spheres, and "cylindrical surfaces", which are generalizations of regular cylinders. In addition to reviewing assigned problems from 11.1, look at (all references to section 11.1): Quick Check problem 4; Regular problems 18, 33, 55 11.2 Add vectors and do scalar multiplications. Compute lengths of vectors. Use vectors for basic geometry and force problems. In addition to reviewing assigned problems from 11.2, look at (all references to section 11.2): Example 8; Quick Check problem 4; Regular problems 27, 40 11.3 Compute a dot product, and use it to find the length of a vector. Use dot products to compute the angle between two vectors; in particular, how to check if two vectors are orthogonal. Compute the direction cosines of a vector. Decompose vectors into orthogonal components. Compute the orthogonal projection of one vector onto another. In addition to reviewing assigned problems from 11.3, look at (all references to section 11.3): Example 7; Quick Check problems 2, 4; Regular problems 4, 22 11.4 Compute a cross product, particularly using the determinant formula. Use a cross product to find a vector perpendicular to two given vectors. Use a cross product to find areas of parallelograms, and use a cross product together with a dot product to compute volumes of parallelepipeds. In addition to reviewing assigned problems from 11.4, look at (all references to section 11.4): Regular problems 12, 30, 34 11.5 Find the parametric equations of a line that satisfies certain conditions by finding a point on the line and a vector parallel to the line. Check if two lines in space are parallel, skew, or intersect. Determine where a line intersects a surface.
In addition to reviewing assigned problems from 11.5, look at (all references to section 11.5): Example 3; Quick Check problems 2, 4; Regular problems 17, 27, 47, 55 11.6 Find the equation of a plane that satisfies certain conditions by finding a point on the plane and a vector normal to the plane. Find the parametric equations of the line of intersection of two (non-parallel) planes. Find the (acute) angle of intersection between two planes. In addition to reviewing assigned problems from 11.6, look at (all references to section 11.6): Examples 6, 7; Regular problems 26, 29 The preceding information is for reviewing for the material of Exam 1.