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# 7 Review Sheet for MATH 231 at PSU

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 26 views.

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Date Created: 02/06/15
Math 231 Fall 2008 Review sheet for the nal Exam Our nal exam will cover Chapters 137 147 and 15 This review sheet deals only with Chapter 15 since the midterm review sheet covered the main topics from Chapters 13 and 15 Some important skills 151 152 153 Functions of several variables Finding the domain and the range of a function of two or more variables Sketch the domain of a function of two variables Finding the level curves of a function of two variables Limits and continuity Finding the limit of a function at a point7 provided that it exists or show that the function has no limit at the given point Finding the set of points where a given function is continuous Partial derivatives Finding the rst partial derivatives of a function of two or more variables Finding all second partial derivatives of a function of two or three variables Apply Clairaut7s Theorem7 if it is applicable Finding the indicated higher partial derivatives Tangent planes and linear approximations Finding an equation of the tangent plane to a surface at a given point Finding an equation of the normal line to a surface at a given point Finding the linearization L of a function f at a given point P0 zmyo 20 Use the linearization function at P0 to estimate the value of the function at some point P Finding the total differential of a function 155 The Chain Rule 0 Chain Rule for function of two variables Case 1 0 Chain Rule for function of two variables Case 2 0 Applying the Chain Rule for a function of two or three variables 156 Directional Derivatives and the Gradient Vector o How to nd the directional derivative at a point if the direction is given by a nonunit vector 0 What is the maximum rate of change of a function of two or more variables at a given point 0 Finding the directions in which the directional derivative has a given value 0 Finding the tangent plane and the normal line to a given level surface Fz7 y z k at a given point 157 Maximum and Minimum Values 0 How to nd the critical points of a function of two or three variables 0 How to nd the extreme values local minima and maxima and the saddle points of a function of two or three variables 0 Apply the rst derivative Test for nding critcal points 0 Classify a critical point using the Second derivative test 0 How to nd the absolute minimum and maximum values of ay on a given closed and bounded set 158 Lagrange Multipliers 0 Applying the Method of Lagrange Multipliers for functions of two or three variables subject to one constraint 0 Finding the absolute minimum or maximum values of a function f of two or three variables on the boundary of a given region D in 3 space using the Method of Lagrange Multipliers Other important facts 1 The directional derivative Duf of a function f of three variables at a point P Ly7 z in the direction of a given unit vector u can be written as DufVfu or DuflVflcost97 where 6 is the angle between Vf and u The gradient vector Vf at P gives the direction of most rapidly increase of f at that point The gradient vector Vf at P is normal to the level surface of f passing through P

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