Math 251 Midterm II Information Spring 2018

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1 Math 251 Midterm II Information Spring 2018 WHEN: Thursday, April 12 (in class). You will have the entire period (125 minutes) to work on the exam. RULES: No books or notes. You may bring a non-graphing calculator to use during the exam. EXTRA OFFICE HOUR: Thursday, April 12: I will be at the Fullerton College cafeteria from 1:00 3:45 PM. COVERAGE: The midterm will cover the material discussed from Sections Please note that this ONLY refers to topics that we have covered during the lecture. (There are topics in the book from these sections that are unrelated to the content we have covered, but those topics will not be on the exam.) STUDYING: Here is an overview of the topics we have covered. You should be comfortable with all of the following phrases below: Chapter 13.1: Function of several variables: dependent vs. independent variables; domain; range; types of functions: polynomial, rational, etc.; graphing of functions of multiple variables; sections/traces of a function; level curves (i.e., contours) of a function; contour diagram/map; level surface. Chapter 13.2: Limit of a function of two (or more) variables; methods for showing a limit does or does not exist (e.g., continuity, different paths, domain considerations, algebraic manipulations: rationalization, cancellation, etc.); continuous functions. Chapter 13.3: Partial derivatives: limit definitions, slope interpretation, method for practical computation of; higher-order partial derivatives; notations for partial derivatives; equality of mixed partial derivatives (Clairaut s Theorem). Chapter 13.4: Total differential; using differentials to approximate change in a function; real-world applications of differentials. Chapter 13.5: Chain Rule for functions of several variables; tree diagram; implicit partial differentiation. Chapter 13.6: Directional derivative; gradient vector: points in direction of maximum increase of a function, and points orthogonal to the level curves/surfaces of a function. Chapter 13.7: Tangent plane equation for a level surface; general tangent plane

2 equation for any surface; normal line to a surface using gradient vector. Chapter 13.8: Local or relative maximums and minimums (extrema) of a function; critical points; saddle points; Second Partial Derivative Test using D = f xx f yy f 2 xy; absolute extrema: finding candidates, testing the boundary of a region, etc. Chapter 13.9: Applied extrema problems: minimizing or maximizing physical quantities such as distance, area, volume, revenue, etc. Chapter 13.10: Lagrange multipliers; constrained optimization; Lagrange s Theorem. It is not enough to simply know what all of the words and phrases above mean. You need to be able to solve problems that involve the concepts that occur in the list above. To help you, I ve made a list of some of the most important things that you should be able to do confidently, without notes, books, etc. by the time you reach the midterm. THINGS TO BE ABLE TO DO: Evaluate a function of several variables at a point. Determine the domain of a function of several variables. Determine the range of a function of several variables. Determine whether a given multivariable function is a polynomial, rational, power, radical, logarithmic, exponential, trigonometric, or some other type of function. More specifically, should be able to identify such surfaces as spheres, ellipsoids, planes, cones, hyperboloids (one or two sheets), paraboloids, hyperbolic paraboloids, etc. Draw sections/traces of a graph of a function of two variables. Identify (i.e., classify) sections/traces of a graph. Draw level curves of a graph of a function of two variables. That is, sketch a contour diagram of a function of two variables. Draw level surfaces of a graph of a function of three variables (provided they are familiar surfaces, such as those we discussed in Section 11.6). Identify (i.e., classify) level curves or surfaces of a function.

3 Determine whether or not limits of a function exist. For limits that do exist, be able to do calculations and show work to obtain the limit value. Common strategies include direct substitution, rationalizing a radical expression, factoring and cancelling, converting to polar or cylindrical or spherical coordinates, applying L Hopital s Rule, and so on. For limits that do not exist, be able to show why: different paths yielding different limit values, edge-of-domain considerations, etc. Understand the physical meaning of partial derivatives f x and f y. Be able to compute all orders of partial derivatives of a function f. Utilize the fact that mixed partial derivatives can be computed in any order to simplify your work. Be able to explain in words what a partial derivative means in the context of a real-world problem (e.g. problem 111 on page 898). Be able to compute the total differential of a function. Be able to use the total differential to approximate values of a function or the change in a function of multiple variables. Be able to create a function and apply differentials to estimate values of an expression at points that could not be evaluated easily without a calculator. Understand how to derive the Chain Rule formulas for a variety of situations by using tree diagrams. Be able to use the Chain Rule to compute derivatives and partial derivatives. Be able to perform implicit partial differentiation. Understand how to use the Chain Rule to derive partial differentiation formulas see, for instance, Problem 3 on Group Work #5 or problem 51 on page 914 of the book. Be able to compute the directional derivative of a function in a specified direction. Understand the physical meaning of the numerical value of a directional derivative.

4 Be able to compute the gradient vector for any function. Understand the physical meaning of the vector value of the gradient at a given point. Understand the relationship between the directional derivative and the gradient vector. Be clear about what the magnitude of the gradient vector tells you. Understand how the gradient vector relates to the level curves or level surfaces of a function of two or three variables, respectively. Be able to use the relationship between the level surfaces of a function and the gradient vector to write the equation of the tangent plane to a surface at a given point. Be able to write the equation of a tangent plane to any function z = f(x, y) at a point (x 0, y 0 ) in the domain of f. Be able to write the equations of a normal line to a level surface at a point (x 0, y 0, z 0 ). Understand the relationship between differentials and the tangent plane to a surface. Be able to find critical points of a multivariable function. Be able to use the D-test to classify critical points as to whether they are relative maximums, minimums, saddle points, etc. Understand that a relative extremum can only occur at a critical point of a function. Understand the difference between a relative extremum and an absolute extremum. Can you give an example of a relative extremum that is not absolute? How about an absolute extremum that is not relative? Be clear on how the region on which extrema are sought for a given function affects the answers you find. Be able to test the boundary of a region in order to seek candidates for absolute extrema on the region in question.

5 Know how to make a list of candidates for absolute extrema on a region and then successfully test them. Be able to use the theory of extrema in order to solve real-world optimization problems (e.g. minimizing distance, maximizing revenue, minimizing surface area, etc.) Be able to apply the method of Lagrange multipliers to solve constrained optimization problems. ADVICE: I suggest reviewing the group work, beginning with Group Work #4. If you have time, it might be best to try re-working those exercises from scratch and looking up the answers afterwards. You will not have time to re-do all of the homework, but you might try some of them again, especially the ones that will assist you with the THINGS TO BE ABLE TO DO listed above. TRY TO RE- DO PROBLEMS FROM SCRATCH, RATHER THAN JUST REVIEWING YOUR ALREADY-COMPLETED SOLUTIONS. Also, quiz each other in study groups. Finally, you can me questions as much as you want, and I will be happy to review or pop-quiz a topic with you if you feel shaky. Basically, I m here to help and I want everybody to do well, so please don t be shy :=)!! GOOD PRACTICE PROBLEMS FROM THE BOOK: (Detailed solutions will be posted in a separate document) Page 960: #3 6, 9 14, 18 24, 28, 29, 32, 33, 37(a), 39(a), 41, 43, 45, 47, 50, 51, 55 58, 61, 64, 67, 69, 77, 80.

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