Math32a R Kozhan Final Summary Material for the Final includes everything that we covered in the course Namely Chapter 12 Chapter 13 except Section 13 4 Sections 10 1 10 2 Basics of the conic sections Section 10 5 without foci and directrices Chapter 14 Bring your ID card to the exam No calculators no books no notes no cheatsheets no cell phones no computers will be allowed on the exam Below is the summary list of the concepts methods theorems statements and formulas that students should know and understand 1 Everything from the summary for midterm1 http www math ucla edu kozhan 32a 1 11f 32a midterm1 summary pdf 2 Everything from the summary for midterm2 http www math ucla edu kozhan 32a 1 11f 32a midterm2 summary pdf 3 Know the definition and interpretations of the directional derivatives for function of two vari2 f a b if u is unit Be able to compute ables the definition is D u f a b lim f a hu1 b hu h directional derivatives 4 Know and understand that if u i then D u f a b is f a b and if u j then D u f a b is x f a b y 5 Know that if f is differentiable and u hu1 u2 i is any unit vector then D u f a b fx a b u1 fy a b u2 6 Know what the gradient of a function of two variables is Know how many components it has Know that the formula from 5 can also be written as D u f a b f a b u 1 7 Know that for a fixed point a b the directional derivative D u f a b is maximal if u has the same direction as f a b and minimal if u has the opposite direction to f a b Know that this maximal value of D u f a b is equal to f a b and the minimal is f a b 8 Know the analogues of 3 7 for the functions of three and more variables 9 Be able to write the equation of the tangent line to a curve given by the equation F x y 0 at a given point x0 y0 Compare and distinguish this from the situation when the curve is given by y f x 10 Be able to write the equation of the tangent plane to a surface given by the equation F x y z 0 at a given point x0 y0 z0 Compare and distinguish this from the situation when the surface is given by z f x y 11 Be able to write the equation of the normal line to a surface given by the equation F x y z 0 at a given point x0 y0 z0 Be able to write the equation of the normal line when the surface is given in the form z f x y 12 Suppose that a b lies on the level curve F x y k of some function F x y Know that F a b is perpendicular to this level curve at the point a b in the sense that it s perpendicular to the tangent line at a b of F x y k 13 Suppose that a b c lies on the level surface F x y z k of some function F x y z Know that F a b c is perpendicular to this level surface in the sense that it s perpendicular to the tangent plane at a b c of F x y z k N B It s NOT true that f a b is perpendicular to the surface z f x y Why not 14 Know the definition of a local maximum minimum of a function of two variables f x y Know the definition of a local maximum minimum of a function of one variable f x Know the definition of a local maximum minimum of a function of three variables f x y z 15 Know the definitions of the absolute or global maximum minimum of a function 16 Know the first derivative test for functions of several variables 17 Know the second derivative test for functions of two variables 18 Know the definition of a critical point of a function Be able to find and classify them local max local min saddle point 19 Know the definition of a boundary point of a set D in R2 Know the definition of a boundary point of a set D in R Know the definition of a boundary point of a set D in R3 20 Know the definition of a closed set 21 Know the definition of a bounded set in R R2 R3 22 Know that any function f of several variables that is continuous on a closed bounded set attains its absolute maximum and absolute minimum These values are achieved either in the interior of D in which case it s a critical point of f or on the boundary of D Be able to use this fact to find absolute maximum and absolute minimum of a function on a given closed bounded set D 23 Know and be able to apply the Lagrange multipliers method for finding a maximum minimum under possibly several constraints 2